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the ordinary language of geometricians.
They have in view practice only, and are always speaking? in a
narrow and ridiculous manner, of squaring and extending and applying
and the like --they confuse the necessities of geometry with those
of daily life; whereas knowledge is the real object of the whole
Certainly, he said.
Then must not a further admission be made?
That the knowledge at which geometry aims is knowledge of the
eternal, and not of aught perishing and transient.
That, he replied, may be readily allowed, and is true.
Then, my noble friend, geometry will draw the soul towards truth,
and create the spirit of philosophy, and raise up that which is now
unhappily allowed to fall down.
Nothing will be more likely to have such an effect.
Then nothing should be more sternly laid down than that the
inhabitants of your fair city should by all means learn geometry.
Moreover the science has indirect effects, which are not small.
Of what kind? he said.
There are the military advantages of which you spoke, I said; and in
all departments of knowledge, as experience proves, any one who has
studied geometry is infinitely quicker of apprehension than one who
Yes indeed, he said, there is an infinite difference between them.
Then shall we propose this as a second branch of knowledge which our
youth will study?
Let us do so, he replied.
And suppose we make astronomy the third --what do you say?
I am strongly inclined to it, he said; the observation of the
seasons and of months and years is as essential to the general as it
is to the farmer or sailor.
I am amused, I said, at your fear of the world, which makes you
guard against the appearance of insisting upon useless studies; and
I quite admit the difficulty of believing that in every man there is
an eye of the soul which, when by other pursuits lost and dimmed, is
by these purified and re-illumined; and is more precious far than
ten thousand bodily eyes, for by it alone is truth seen. Now there are
two classes of persons: one class of those who will agree with you and
will take your words as a revelation; another class to whom they
will be utterly unmeaning, and who will naturally deem them to be idle
tales, for they see no sort of profit which is to be obtained from
them. And therefore you had better decide at once with which of the
two you are proposing to argue. You will very likely say with neither,
and that your chief aim in carrying on the argument is your own
improvement; at the same time you do not grudge to others any
benefit which they may receive.
I think that I should prefer to carry on the argument mainly on my
Then take a step backward, for we have gone wrong in the order of
What was the mistake? he said.
After plane geometry, I said, we proceeded at once to solids in
revolution, instead of taking solids in themselves; whereas after
the second dimension the third, which is concerned with cubes and
dimensions of depth, ought to have followed.
That is true, Socrates; but so little seems to be known as yet about
Why, yes, I said, and for two reasons: --in the first place, no
government patronises them; this leads to a want of energy in the
pursuit of them, and they are difficult; in the second place, students
cannot learn them unless they have a director. But then a director can
hardly be found, and even if he could, as matters now stand, the
students, who are very conceited, would not attend to him. That,
however, would be otherwise if the whole State became the director
of these studies and gave honour to them; then disciples would want to
come, and there would be continuous and earnest search, and
discoveries would be made; since even now, disregarded as they are
by the world, and maimed of their fair proportions, and although
none of their votaries can tell the use of them, still these studies
force their way by their natural charm, and very likely, if they had
the help of the State, they would some day emerge into light.
Yes, he said, there is a remarkable charm in them. But I do not
clearly understand the change in the order. First you began with a
geometry of plane surfaces?
Yes, I said.
And you placed astronomy next, and then you made a step backward?
Yes, and I have delayed you by my hurry; the ludicrous state of
solid geometry, which, in natural order, should have followed, made me
pass over this branch and go on to astronomy, or motion of solids.
True, he said.
Then assuming that the science now omitted would come into existence
if encouraged by the State, let us go on to astronomy, which will be
The right order, he replied. And now, Socrates, as you rebuked the
vulgar manner in which I praised astronomy before, my praise shall
be given in your own spirit. For every one, as I think, must see
that astronomy compels the soul to look upwards and leads us from this
world to another.
Every one but myself, I said; to every one else this may be clear,
but not to me.
And what then would you say?
I should rather say that those who elevate astronomy into philosophy
appear to me to make us look downwards and not upwards.
What do you mean? he asked.
You, I replied, have in your mind a truly sublime conception of
our knowledge of the things above. And I dare say that if a person
were to throw his head back and study the fretted ceiling, you would
still think that his mind was the percipient, and not his eyes. And
you are very likely right, and I may be a simpleton: but, in my
opinion, that knowledge only which is of being and of the unseen can
make the soul look upwards, and whether a man gapes at the heavens
or blinks on the ground, seeking to learn some particular of sense,
I would deny that he can learn, for nothing of that sort is matter
of science; his soul is looking downwards, not upwards, whether his
way to knowledge is by water or by land, whether he floats, or only
lies on his back.
I acknowledge, he said, the justice of your rebuke. Still, I
should like to ascertain how astronomy can be learned in any manner
more conducive to that knowledge of which we are speaking?
I will tell you, I said: The starry heaven which we behold is
wrought upon a visible ground, and therefore, although the fairest and
most perfect of visible things, must necessarily be deemed inferior
far to the true motions of absolute swiftness and absolute slowness,
which are relative to each other, and carry with them that which is
contained in them, in the true number and in every true figure. Now,
these are to be apprehended by reason and intelligence, but not by
True, he replied.
The spangled heavens should be used as a pattern and with a view
to that higher knowledge; their beauty is like the beauty of figures
or pictures excellently wrought by the hand of Daedalus, or some other
great artist, which we may chance to behold; any geometrician who
saw them would appreciate the exquisiteness of their workmanship,
but he would never dream of thinking that in them he could find the
true equal or the true double, or the truth of any other proportion.
No, he replied, such an idea would be ridiculous.
And will not a true astronomer have the same feeling when he looks
at the movements of the stars? Will he not think that heaven and the
things in heaven are framed by the Creator of them in the most perfect
manner? But he will never imagine that the proportions of night and
day, or of both to the month, or of the month to the year, or of the
stars to these and to one another, and any other things that are
material and visible can also be eternal and subject to no deviation
--that would be absurd; and it is equally absurd to take so much pains
in investigating their exact truth.
I quite agree, though I never thought of this before.
Then, I said, in astronomy, as in geometry, we should employ
problems, and let the heavens alone if we would approach the subject
in the right way and so make the natural gift of reason to be of any
That, he said, is a work infinitely beyond our present astronomers.
Yes, I said; and there are many other things which must also have
a similar extension given to them, if our legislation is to be of
any value. But can you tell me of any other suitable study?
No, he said, not without thinking.
Motion, I said, has many forms, and not one only; two of them are
obvious enough even to wits no better than ours; and there are others,
as I imagine, which may be left to wiser persons.
But where are the two?
There is a second, I said, which is the counterpart of the one
And what may that be?
The second, I said, would seem relatively to the ears to be what the
first is to the eyes; for I conceive that as the eyes are designed
to look up at the stars, so are the ears to hear harmonious motions;
and these are sister sciences --as the Pythagoreans say, and we,
Glaucon, agree with them?
Yes, he replied.
But this, I said, is a laborious study, and therefore we had
better go and learn of them; and they will tell us whether there are
any other applications of these sciences. At the same time, we must
not lose sight of our own higher object.
What is that?
There is a perfection which all knowledge ought to reach, and
which our pupils ought also to attain, and not to fall short of, as
I was saying that they did in astronomy. For in the science of
harmony, as you probably know, the same thing happens. The teachers of
harmony compare the sounds and consonances which are heard only, and
their labour, like that of the astronomers, is in vain.
Yes, by heaven! he said; and 'tis as good as a play to hear them
talking about their condensed notes, as they call them; they put their
ears close alongside of the strings like persons catching a sound from
their neighbour's wall --one set of them declaring that they
distinguish an intermediate note and have found the least interval
which should be the unit of measurement; the others insisting that the
two sounds have passed into the same --either party setting their ears
before their understanding.
You mean, I said, those gentlemen who tease and torture the
strings and rack them on the pegs of the instrument: might carry on
the metaphor and speak after their manner of the blows which the
plectrum gives, and make accusations against the strings, both of
backwardness and forwardness to sound; but this would be tedious,
and therefore I will only say that these are not the men, and that I
am referring to the Pythagoreans, of whom I was just now proposing
to enquire about harmony. For they too are in error, like the
astronomers; they investigate the numbers of the harmonies which are
heard, but they never attain to problems-that is to say, they never
reach the natural harmonies of number, or reflect why some numbers are
harmonious and others not.
That, he said, is a thing of more than mortal knowledge.
A thing, I replied, which I would rather call useful; that is, if
sought after with a view to the beautiful and good; but if pursued
in any other spirit, useless. Very true, he said.
Now, when all these studies reach the point of inter-communion and
connection with one another, and come to be considered in their mutual
affinities, then, I think, but not till then, will the pursuit of them
have a value for our objects; otherwise there is no profit in them.
I suspect so; but you are speaking, Socrates, of a vast work.