The Republic
And, for all these reasons, arithmetic is a kind of knowledge in which
the best natures should be trained, and which must not be given up.
I agree.
Let this then be made one of our subjects of education. And next, shall
we enquire whether the kindred science also concerns us?
You mean geometry?
Exactly so.
Clearly, he said, we are concerned with that part of geometry which
relates to war; for in pitching a camp, or taking up a position, or
closing or extending the lines of an army, or any other military
manoeuvre, whether in actual battle or on a march, it will make all the
difference whether a general is or is not a geometrician.
Yes, I said, but for that purpose a very little of either geometry or
calculation will be enough; the question relates rather to the greater
and more advanced part of geometry—whether that tends in any degree to
make more easy the vision of the idea of good; and thither, as I was
saying, all things tend which compel the soul to turn her gaze towards
that place, where is the full perfection of being, which she ought, by
all means, to behold.
True, he said.
Then if geometry compels us to view being, it concerns us; if becoming
only, it does not concern us?
Yes, that is what we assert.
Yet anybody who has the least acquaintance with geometry will not deny
that such a conception of the science is in flat contradiction to the
ordinary language of geometricians.
How so?
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 science.
Certainly, he said.
Then must not a further admission be made?
What admission?
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 has
not.
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 own
behalf.
Then take a step backward, for we have gone wrong in the order of the
sciences.
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
these subjects.
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
fourth.
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 sight.
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 real use.
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 already
named.
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: I might carry on the metaphor
and speak after their manner of the blows which the plectrum gives, and