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into two kinds？
Pro。 What are the two kinds？
Soc。 In the first place； arithmetic is of two kinds； one of which is
popular； and the other philosophical。
Pro。 How would you distinguish them？
Soc。 There is a wide difference between them； Protarchus； some
arithmeticians reckon unequal units； as for example； two armies； two
oxen； two very large things or two very small things。 The party who
are opposed to them insist that every unit in ten thousand must be the
same as every other unit。
Pro。 Undoubtedly there is； as you say； a great difference among
the votaries of the science； and there may be reasonably supposed to
be two sorts of arithmetic。
Soc。 And when we compare the art of mensuration which is used in
building with philosophical geometry； or the art of computation
which is used in trading with exact calculation； shall we say of
either of the pairs that it is one or two？
Pro。 On the analogy of what has preceded； I should be of opinion
that they were severally two。
Soc。 Right； but do you understand why I have discussed the subject？
Pro。 I think so； but I should like to be told by you。
Soc。 The argument has all along been seeking a parallel to pleasure；
and true to that original design； has gone on to ask whether one
sort of knowledge is purer than another； as one pleasure is purer than
another。
Pro。 Clearly； that was the intention。
Soc。 And has not the argument in what has preceded； already shown
that the arts have different provinces； and vary in their degrees of
certainty？
Pro。 Very true。
Soc。 And just now did not the argument first designate a
particular art by a common term； thus making us believe in the unity
of that art； and then again； as if speaking of two different things；
proceed to enquire whether the art as pursed by philosophers； or as
pursued by non philosophers； has more of certainty and purity？
Pro。 That is the very question which the argument is asking。
Soc。 And how； Protarchus； shall we answer the enquiry？
Pro。 O Socrates； we have reached a point at which the difference
of clearness in different kinds of knowledge is enormous。
Soc。 Then the answer will be the easier。
Pro。 Certainly； and let us say in reply； that those arts into
which arithmetic and mensuration enter； far surpass all others； and
that of these the arts or sciences which are animated by the pure
philosophic impulse are infinitely superior in accuracy and truth。
Soc。 Then this is your judgment； and this is the answer which；
upon your authority； we will give to all masters of the art of
misinterpretation？
Pro。 What answer？
Soc。 That there are two arts of arithmetic； and two of
mensuration； and also several other arts which in like manner have
this double nature； and yet only one name。
Pro。 Let us boldly return this answer to the masters of whom you
speak； Socrates； and hope for good luck。
Soc。 We have explained what we term the most exact arts or sciences。
Pro。 Very good。
Soc。 And yet； Protarchus； dialectic will refuse to acknowledge us；
if we do not award to her the first place。
Pro。 And pray； what is dialectic？
Soc。 Clearly the science which has to do with all that knowledge
of which we are now speaking； for I am sure that all men who have a
grain of intelligence will admit that the knowledge which has to do
with being and reality； and sameness and unchangeableness； is by far
the truest of all。 But how would you decide this question； Protarchus？
Pro。 I have often heard Gorgias maintain； Socrates； that the art
of persuasion far surpassed every other； this； as he says； is by far
the best of them all； for to it all things submit； not by
compulsion； but of their own free will。 Now； I should not like to
quarrel either with you or with him。
Soc。 You mean to say that you would like to desert； if you were
not ashamed？
Pro。 As you please。
Soc。 May I not have led you into a misapprehension？
Pro。 How？
Soc。 Dear Protarchus； I never asked which was the greatest or best
or usefullest of arts or sciences； but which had clearness and
accuracy； and the greatest amount of truth； however humble and
little useful an art。 And as for Gorgias； if you do not deny that
his art has the advantage in usefulness to mankind； he will not
quarrel with you for saying that the study of which I am speaking is
superior in this particular of essential truth； as in the comparison
of white colours； a little whiteness； if that little be only pure； was
said to be superior in truth to a great mass which is impure。 And
now let us give our best attention and consider well； not the
comparative use or reputation of the sciences； but the power or
faculty； if there be such； which the soul has of loving the truth； and
of doing all things for the sake of it； let us search into the pure
element of mind and intelligence； and then we shall be able to say
whether the science of which I have been speaking is most likely to
possess the faculty； or whether there be some other which has higher
claims。
Pro。 Well； I have been considering； and I can hardly think that
any other science or art has a firmer grasp of the truth than this。
Soc。 Do you say so because you observe that the arts in general
and those engaged in them make use of opinion； and are resolutely
engaged in the investigation of matters of opinion？ Even he who
supposes himself to be occupied with nature is really occupied with
the things of this world； how created； how acting or acted upon。 Is
not this the sort of enquiry in which his life is spent？
Pro。 True。
Soc。 He is labouring； not after eternal being； but about things
which are becoming； or which will or have become。
Pro。 Very true。
Soc。 And can we say that any of these things which neither are nor
have been nor will be unchangeable； when judged by the strict rule
of truth； ever become certain？
Pro。 Impossible。
Soc。 How can anything fixed be concerned with that which has no
fixedness？
Pro。 How indeed？
Soc。 Then mind and science when employed about such changing
things do not attain the highest truth？
Pro。 I should imagine not。
Soc。 And now let us bid farewell； a long farewell； to you or me or
Philebus or Gorgias； and urge on behalf of the argument a single
point。
Pro。 What point？
Soc。 Let us say that the stable and pure and true and unalloyed
has to do with the things which are eternal and unchangeable and
unmixed； or if not； at any rate what is most akin to them has； and
that all other things are to be placed in a second or inferior class。
Pro。 Very true。
Soc。 And of the names expressing cognition； ought not the fairest to
be given to the fairest things？
Pro。 That is natural。