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the same object； at one time a greater； at another a smaller number of
signs。 Thus； one person may cogitate in his conception of gold； in
addition to its properties of weight； colour； malleability； that of
resisting rust； while another person may be ignorant of this
quality。 We employ certain signs only so long as we require them for
the sake of distinction； new observations abstract some and add new
ones； so that an empirical conception never remains within permanent
limits。 It is； in fact； useless to define a conception of this kind。
If； for example； we are speaking of water and its properties； we do
not stop at what we actually think by the word water； but proceed to
observation and experiment； and the word； with the few signs
attached to it； is more properly a designation than a conception of
the thing。 A definition in this case would evidently be nothing more
than a determination of the word。 In the second place； no a priori
conception； such as those of substance； cause； right； fitness； and
so on； can be defined。 For I can never be sure； that the clear
representation of a given conception （which is given in a confused
state） has been fully developed； until I know that the
representation is adequate with its object。 But； inasmuch as the
conception； as it is presented to the mind； may contain a number of
obscure representations； which we do not observe in our analysis；
although we employ them in our application of the conception； I can
never be sure that my analysis is plete； while examples may make
this probable； although they can never demonstrate the fact。 instead
of the word definition； I should rather employ the term exposition…
a more modest expression； which the critic may accept without
surrendering his doubts as to the pleteness of the analysis of
any such conception。 As； therefore； neither empirical nor a priori
conceptions are capable of definition； we have to see whether the only
other kind of conceptions… arbitrary conceptions… can be subjected
to this mental operation。 Such a conception can always be defined； for
I must know thoroughly what I wished to cogitate in it； as it was I
who created it； and it was not given to my mind either by the nature
of my understanding or by experience。 At the same time； I cannot say
that； by such a definition； I have defined a real object。 If the
conception is based upon empirical conditions； if； for example； I have
a conception of a clock for a ship； this arbitrary conception does not
assure me of the existence or even of the possibility of the object。
My definition of such a conception would with more propriety be termed
a declaration of a project than a definition of an object。 There
are no other conceptions which can bear definition； except those which
contain an arbitrary synthesis； which can be constructed a priori。
Consequently； the science of mathematics alone possesses
definitions。 For the object here thought is presented a priori in
intuition； and thus it can never contain more or less than the
conception； because the conception of the object has been given by the
definition… and primarily； that is； without deriving the definition
from any other source。 Philosophical definitions are； therefore；
merely expositions of given conceptions； while mathematical
definitions are constructions of conceptions originally formed by
the mind itself； the former are produced by analysis； the pleteness
of which is never demonstratively certain； the latter by a
synthesis。 In a mathematical definition the conception is formed； in a
philosophical definition it is only explained。 From this it follows：
*The definition must describe the conception pletely that is；
omit none of the marks or signs of which it posed； within its own
limits； that is； it must be precise； and enumerate no more signs
than belong to the conception； and on primary grounds； that is to say；
the limitations of the bounds of the conception must not be deduced
from other conceptions； as in this case a proof would be necessary；
and the so…called definition would be incapable of taking its place at
the bead of all the judgements we have to form regarding an object。
（a） That we must not imitate； in philosophy； the mathematical
usage of mencing with definitions… except by way of hypothesis or
experiment。 For； as all so…called philosophical definitions are merely
analyses of given conceptions； these conceptions； although only in a
confused form； must precede the analysis； and the inplete
exposition must precede the plete； so that we may be able to draw
certain inferences from the characteristics which an inplete
analysis has enabled us to discover； before we attain to the
plete exposition or definition of the conception。 In one word； a
full and clear definition ought； in philosophy； rather to form the
conclusion than the mencement of our labours。* In mathematics； on
the contrary； we cannot have a conception prior to the definition；
it is the definition which gives us the conception； and it must for
this reason form the mencement of every chain of mathematical
reasoning。
*Philosophy abounds in faulty definitions； especially such as
contain some of the elements requisite to form a plete
definition。 If a conception could not be employed in reasoning
before it had been defined； it would fare ill with all philosophical
thought。 But； as inpletely defined conceptions may always be
employed without detriment to truth； so far as our analysis of the
elements contained in them proceeds； imperfect definitions； that is；
propositions which are properly not definitions； but merely
approximations thereto； may be used with great advantage。 In
mathematics； definition belongs ad esse； in philosophy ad melius esse。
It is a difficult task to construct a proper definition。 Jurists are
still without a plete definition of the idea of right。
（b） Mathematical definitions cannot be erroneous。 For the conception
is given only in and through the definition； and thus it contains only
what has been cogitated in the definition。 But although a definition
cannot be incorrect； as regards its content； an error may sometimes；
although seldom； creep into the form。 This error consists in a want of
precision。 Thus the mon definition of a circle… that it is a curved
line； every point in which is equally distant from another point
called the centre… is faulty； from the fact that the determination
indicated by the word curved is superfluous。 For there ought to be a
particular theorem； which may be easily proved from the definition； to
the effect that every line； which has all its points at equal
distances from another point； must be a curved line… that is； that not
even the smallest part of it can be straight。 Analytical
definitions； on the other hand； may be erroneous in many respects；
either by the introduction of signs which do not actually exist in the
conception； or by wanting in that pleteness which forms the
essential of a definition。 In the latter case； the definition is
necessarily defective； because we can never be fully certain of the
pleteness of our analysis。 For these reasons； the method of
definition employed in mathematics cannot be imi