Book VII, 523–530

523a “It seems to be one of the studies we seek, naturally leading toward understanding; yet no one uses it rightly, though it has a power to draw the soul entirely toward being.”

“What do you mean?” he said.

“I will try,” I said, “to explain what I think. Join me in looking at the things I distinguish for myself as leading where we have said, or not leading there; agree or disagree, so that we may see more clearly whether I have guessed rightly.”

“Show me,” he said.

“I will,” I said. “If you can see it, some things in perception 523b do not summon the understanding to examine them, because perception judges them adequately. Others insist that it examine them, because perception gives no sound account of them.”

“You must mean,” he said, “things seen from a distance and things painted with deceptive shading.”

“That is not quite what I mean,” I said.

“Then what do you mean?” he said.

“Those which do not summon it,” I said, “are the things that do not issue in opposite perceptions at once 523c. Those that do, I set down as summoning it, whenever perception shows one thing no more clearly than its opposite, whether it strikes from near or far. You will understand what I mean more clearly this way. Let us say these are three fingers: the little finger, the next, and the middle finger.”

“Certainly,” he said.

“Understand that I am speaking of them seen close at hand. Now consider this about them.”

“What?”

“Each appears equally a finger 523d, and in that respect it makes no difference whether it is seen in the middle or at the end, whether white or black, thick or thin, or anything of the sort. For in all these cases the soul of most people is not compelled to ask the understanding what a finger is. Sight has never indicated to it that a finger is at the same time the opposite of a finger.”

“No, it has not,” he said.

“So naturally such a perception would neither summon 523e nor awaken understanding.”

“Naturally.”

“But what of their size and smallness? Does sight judge these adequately, and does it make no difference to it whether a finger is placed in the middle or at the end? And similarly, does touch judge thickness and thinness, softness and hardness? Do the other senses not also disclose such things inadequately? Or does

524a each of them act like this: first, the sense appointed to perceive hardness must also perceive softness, and it reports to the soul that it perceives the same thing as both hard and soft?”

“That is so,” he said.

“Then in such cases must the soul not be at a loss as to what that sense means by hard, if it also calls the same thing soft? And as for the sense that perceives light and heavy, what does light or heavy mean if it reports that the heavy is light and the light is heavy?” 524b

“These reports are strange to the soul indeed,” he said, “and need examination.”

“Then it is natural,” I said, “that in such cases the soul first tries to summon calculation and understanding to examine whether each thing reported to it is one or two.”

“Of course.”

“And if they appear to be two, does each appear both distinct and one?”

“Yes.”

“So if each is one, and both are two, it will conceive the two as separate 524c. It could not conceive two that were inseparable as two, but only as one.”

“Right.”

“Now we say sight saw both large and small, but did not see them separate: it saw a kind of mixture. Is that so?”

“Yes.”

“To make this clear, understanding was compelled in turn to see large and small not as mixed but as distinct—just the opposite of sight.”

“True.”

“Is it not from here that the question first occurs to us: what, then, are the large and the small?”

“Absolutely.”

“And so we named one intelligible, the other visible.” 524d

“Quite right,” he said.

“That is what I was trying to say just now: some things summon thought and some do not. I define as summoning it those things that enter perception along with their opposites, and as not awakening understanding those that do not.”

“Now I understand,” he said, “and that seems right to me.”

“What then? To which kind do you think number and the one belong?”

“I do not see,” he said.

“Reason from what we have said,” I replied. “If the one itself is adequately seen by itself, or apprehended by some other sense 524e, it would not draw us toward being, as we said of the finger. But if some opposite is always seen along with it, so that it seems no more one than its opposite, there would be need of something to decide the matter. The soul would then be compelled to puzzle over it, to search, to stir thought within itself, and to ask what the one itself is. In this way the study of

525a the one would be among the studies that lead the soul and turn it toward the vision of being.”

“Yet sight of the one most certainly has this character,” he said. “We see the very same thing both as one and as an unlimited multitude.”

“And if this is true of the one,” I said, “is the same true of every number?”

“Of course.”

“Now calculation and arithmetic deal entirely with number.”

“Certainly.” 525b

“And these plainly lead toward truth.”

“Wonderfully so.”

“Then they would seem to be among the studies we seek. The warrior needs to learn them for arranging troops; the philosopher, because he must emerge from becoming and grasp being, or he will never become capable of calculation.”

“That is so,” he said.

“And our guardian happens to be both warrior and philosopher.”

“Of course.”

“It would be fitting, then, Glaucon, to establish this study by law and persuade those who will take part in the highest affairs of our city 525c to pursue calculation and engage with it, not as amateurs, but until they reach contemplation of the nature of numbers through understanding itself. They must practice it not for buying and selling, like merchants and shopkeepers, but for war and for the soul’s own easier turning from becoming toward truth and being.”

“Beautifully said,” he replied.

“And now,” I said, “having spoken of the study of calculation 525d, I realize how refined it is, and how useful in many ways for our purpose, provided a person practices it for knowledge and not for trade.”

“How so?” he said.

“In the very way we just discussed: it powerfully leads the soul upward and compels it to discuss numbers themselves, never accepting a discussion in which someone offers it numbers embodied in things visible or tangible. You know how those expert in these matters react if anyone tries in argument to divide the one itself 525e: they laugh and refuse to accept it. If you cut it into fractions, they multiply it, taking care that the one should never appear not as one, but as many parts.”

“What you say is entirely true,” he said.

526a “What do you think, Glaucon, they would answer if someone asked them: ‘My remarkable friends, what numbers are you talking about, in which the one has the qualities you require—each unit equal to every other in every way, without the slightest difference, and containing no part within itself?’ What do you think they would say?”

“That they are talking about numbers accessible only to thought, impossible to handle in any other way.”

“Then you see, my friend,” I said, “that this study really does seem necessary 526b for us: it plainly compels the soul to use understanding itself in pursuit of truth itself.”

“Indeed,” he said, “it does that most powerfully.”

“And have you noticed this? Those naturally gifted at calculation are, as a rule, quick at every study; and even the slow, if educated and exercised in this one, all make progress in becoming quicker than they were, even if they gain no other benefit.”

“That is so,” he said. 526c

“And in my opinion you would not easily find many studies, if any, that demand more labor in learning and practicing than this one.”

“You would not.”

“For all these reasons we must not neglect it, but must educate our best natures in it.”

“I agree,” he said.

“Let this be our first study,” I said. “Now let us see whether the second, which follows it, is of any use to us.”

“What is it? Do you mean geometry?” he said.

“Precisely,” I said. 526d

“Insofar as it bears on warfare,” he said, “its usefulness is clear. For laying out camps, seizing positions, drawing up and extending armies, and all the other formations of troops in battles and on the march, a man trained in geometry would be quite different from the same man without it.”

“But for such purposes,” I said, “a small portion of geometry and calculation would suffice. We must examine whether its greater, more advanced part tends toward that other goal 526e: making it easier to behold the Form of the Good. Everything tends there, we say, that compels the soul to turn toward the region where the happiest part of being is, which it must see by every means.”

“You speak rightly,” he said.

“So if geometry compels the soul to contemplate being, it is relevant; if becoming, it is not.”

“That is what we say.”

527a “Now no one with even a little experience of geometry will dispute this with us,” I said: “the nature of this science is the exact opposite of the language its practitioners use when they discuss it.”

“How so?” he said.

“They speak in a fashion both laughable and unavoidable. They talk as if they were doing things, and making all their arguments for the sake of action: ‘squaring,’ ‘extending,’ ‘adding,’ and so on. Yet the entire study 527b is pursued for the sake of knowledge.”

“Exactly,” he said.

“Then we must agree on this further point.”

“What point?”

“That its knowledge concerns what always is, not something that comes to be at one time and perishes at another.”

“Easy to agree to,” he said. “Geometric knowledge concerns what always is.”

“It would therefore draw the soul toward truth, my noble friend, and foster a philosopher’s mind by turning upward what we now wrongly direct downward.”

“As far as anything can,” he said. 527c

“Then as strongly as you can,” I said, “you must require that the people of your beautiful city by no means neglect geometry. Its incidental benefits are not small either.”

“What are they?” he asked.

“Those you mentioned relating to war,” I said. “And for the better grasp of every study, we know there is an immense difference between someone acquainted with geometry and someone who is not.”

“By Zeus, an immense difference,” he said.

“Shall we set this down as the second study for the young?”

“Let us,” he said. 527d

“What next? Shall we make astronomy third? Or do you disagree?”

“I certainly agree,” he said. “A keener awareness of seasons, months, and years is useful not only for farming and navigation but no less for generalship.”

“You are charming,” I said, “for you seem afraid of the crowd, lest they think you are prescribing useless studies. But it is a considerable thing, and difficult to believe, that each of these studies purifies and rekindles an instrument of the soul 527e that other pursuits destroy and blind—an instrument worth saving more than ten thousand eyes, since by it alone truth is seen. Those who share this belief will find your words wonderfully sound; those who have no awareness of it will naturally think you are saying nothing, since they can see no other benefit in these studies worth mentioning. So consider

528a at once to which people you are speaking. Or perhaps you address neither group, but conduct this discussion chiefly for your own sake, without begrudging another any benefit they might gain from it.”

“That is what I choose,” he said: “to speak, ask, and answer chiefly for my own sake.”

“Then go back a step with me,” I said, “for we did not choose correctly what comes next after geometry.”

“What did we choose?” he asked.

“After the plane,” I said, “we chose a solid in motion 528b before considering the solid by itself. The right sequence after the second dimension is the third, concerned with the dimension of cubes and whatever has depth.”

“Yes,” he said, “but that subject does not seem to have been discovered yet, Socrates.”

“There are two reasons,” I said. “No city honors it, so it is studied feebly, being difficult; and researchers need a director, without whom they could not make discoveries. Such a person is difficult to find in the first place, and even if found, as things stand 528c those who pursue this study, full of pride, would not listen to him. But if the whole city directed the study alongside them and honored it, they would listen, and through continuous, vigorous investigation its nature would be revealed. Even now, despised and stunted by the many, and pursued by researchers unable to explain its use, it grows despite all these obstacles through its own appeal. It would be no wonder if its nature came to light.” 528d

“Indeed,” he said, “its appeal is extraordinary. But explain more clearly what you said just now. You counted the study of the plane as geometry.”

“Yes,” I said.

“Then at first,” he said, “you put astronomy after it, but later you went back.”

“In my haste to go through everything quickly,” I said, “I have slowed myself down. The study of the third dimension follows next, but because it is in a ridiculous state of neglect, I passed it over and named astronomy after geometry 528e, astronomy being the motion of bodies with depth.”

“You are right,” he said.

“So let us set down astronomy as the fourth study,” I said, “assuming that the study now passed over comes into being, if the city takes it up.”

“That is reasonable,” he said. “And since you reproached me just now, Socrates, for praising astronomy crudely, I now praise it on the grounds on which you approach it 529a. It is obvious to everyone, I think, that astronomy compels the soul to look upward and leads it from things here to things there.”

“Perhaps obvious to everyone but me,” I said. “That is not how it seems to me.”

“How, then?” he said.

“As now practiced by those who claim to lead people toward philosophy, it makes the soul look very much downward.”

“What do you mean?” he asked.

“You have a generous conception,” I said, “of what the study of things above consists in. Apparently 529b you would think that if someone craned his head back to study painted figures on a ceiling, he was contemplating them with his understanding, not with his eyes. Perhaps your view is right and mine is naive. But I cannot think that any study makes the soul look upward except one concerned with what is and what is invisible. If someone tries to learn something perceptible, whether gaping upward or squinting downward, I say he will never learn it—for knowledge 529c has no such object—and his soul looks downward, not upward, even if he learns while floating on his back on land or at sea.”

“I stand corrected,” he said. “You were right to rebuke me. But how did you mean astronomy should be learned differently from its present practice, if people are to learn it profitably for the end we have described?”

“Like this,” I said. “The patterns embroidered in the heavens, since they are embroidered in the visible realm, must be considered the finest and most exact of visible things 529d, yet far inferior to the true things: the movements in which true speed and true slowness, in true number and in all true shapes, move in relation to one another and carry what is within them. These are grasped by reason and thought, not sight. Or do you think otherwise?”

“Not at all,” he said.

“Then the embroidered heavens must be used as models for the study of those other things, just as we might use 529e exceptionally well-drawn and carefully worked diagrams by Daedalus or some other craftsman or painter. A person experienced in geometry, seeing such works, would consider them beautifully executed, but would find it ridiculous to examine them in earnest in the hope of obtaining from them the truth

530a about equal or double proportions, or any other ratio.”

“How could that not be ridiculous?” he said.

“Do you not think,” I said, “that a genuine astronomer would feel the same on looking at the movements of the stars? He would believe that the maker of the heavens arranged them and everything in them as beautifully as such works could be arranged. But as for the proportions of night to day, and of these to the month, of the month to the year, and of the other stars 530b to these and to one another, would he not find it absurd to suppose that, because they are bodily and visible, they always remain exactly the same without the slightest variation, and to seek by every means to grasp their truth?”

“That is how it seems to me now,” he said, “as I hear you.”

“Then we will pursue astronomy, like geometry, by means of problems,” I said, “and leave aside the things in the heavens, if by truly engaging in astronomy we are to turn the soul’s naturally intelligent part from useless 530c to useful.”

“The task you prescribe is many times greater than the astronomy now practiced,” he said.

“I think,” I said, “we shall prescribe the other studies in the same way, if we are to be of any use as lawgivers. But can you remind me of another suitable study?”

“Not just now,” he said.

“Motion, though, presents not one but several kinds 530d, as I think. A wise person might be able to name them all; two, at least, are evident even to us.”

“What are they?”

“Besides astronomy, its counterpart.”

“What is that?”

“It seems,” I said, “that as eyes are made for astronomy, so ears are made for harmonious motion, and these two sciences are sisters, as the Pythagoreans say, and as we, Glaucon, agree. Or what do we say?”

“We agree,” he said. 530e

“Then since the task is vast,” I said, “we will ask them what they say about these subjects, and whether there is anything else besides. Through all this we shall keep watch over our own concern.”

“What is that?”

“That those we educate must never attempt to learn any part of these studies in an incomplete way, failing to reach the end toward which everything must tend, as we were saying just now about astronomy. Or do you not know that”

523a “It seems to be one of the studies we are looking for, one that naturally leads to understanding. Yet nobody uses it properly, though it can certainly draw the soul toward what is real.”

“What do you mean?” he said.

“I'll try to explain what I think,” I said. “As I sort out for myself which things lead where we're talking about and which do not, look at them with me. Agree or disagree, so we can see more clearly whether my guess is right.”

“Show me,” he said.

“Here is what I mean,” I said. “Some things we sense 523b do not call understanding to examine them, because the senses judge them well enough. Others insist that understanding look into them, because the senses produce no reliable result.”

“Clearly you mean distant objects,” he said, “and things painted to create an illusion.”

“That's not quite what I mean,” I said.

“Then what do you mean?” he asked.

“Things that don't call for thought,” I said, “are those whose perception doesn't also produce 523c an opposite perception. Things that do produce one, I count as calling for thought, because the sense reports a thing and its opposite with equal force, whether it encounters the thing nearby or far away. Here is a clearer example. We say these are three fingers: the little finger, the next one, and the middle finger.”

“Yes,” he said.

“Keep in mind that I'm talking about seeing them close up. Now consider this about them.”

“What?”

“Each looks equally like a finger. 523d It makes no difference in that respect whether it's in the middle or at the end, white or black, thick or thin, or anything else of that kind. In all these cases, most people's souls aren't forced to ask their understanding what a finger is. Sight has never reported to the soul that a finger is, at the same time, the opposite of a finger.”

“No, it hasn't,” he said.

“So it makes sense,” I said, “that this sort of thing would not 523e call understanding into action or wake it up.”

“It does.”

“But what about their size? Does sight see whether they are large or small well enough? Does it make no difference to sight whether one is in the middle or at the end? Does touch judge their thickness and thinness, softness and hardness, equally well? Don't the other senses give inadequate reports about such things? Or

524a does each sense work this way? The sense responsible for hardness must also register softness. It tells the soul that it has sensed the same thing as both hard and soft.”

“Yes,” he said.

“So in cases like these,” I said, “the soul must be puzzled by what that sense means by ‘hard,’ if it calls the same thing soft as well. And it must ask what the senses mean by ‘light’ and ‘heavy’ if they report what is heavy as light and what is light as heavy.” 524b

“Yes,” he said, “those reports are strange to the soul and need investigating.”

“So it makes sense,” I said, “that in these cases the soul first calls on calculation and understanding. It tries to find out whether each of the things reported to it is one or two.”

“Of course.”

“And if they appear to be two, each appears separate and one in itself?”

“Yes.”

“If each is one and the two together make two, the soul will think of them as separate. 524c If they were not separate, it would think of them not as two but as one.”

“Right.”

“But we say sight saw both large and small, not separately but mixed up together. Isn't that so?”

“Yes.”

“To make this clear, understanding had to see large and small separately, not mixed together. It did the opposite of sight.”

“True.”

“And isn't this where we first come to ask, ‘What are large and small, anyway?’”

“Exactly.”

“And this is how we came to call one thing intelligible and the other visible.” 524d

“Quite right,” he said.

“This is what I was trying to say just now. Some things call thought into action and others don't. I defined the first as things whose sensations include their own opposites at the same time, and the second as things that don't wake understanding up.”

“Now I understand,” he said, “and that seems right to me.”

“So which kind do you think number and the one belong to?”

“I don't see,” he said.

“Then reason it out from what we've said,” I replied. “If the one can be seen clearly in itself, or grasped by some other sense, it will not draw us toward what is real, as we said about the finger. 524e But if it is always seen along with something that contradicts it, so that it appears no more one than its opposite, we will need something to judge between them. The soul will then have to puzzle over it and investigate. It will stir its own thinking and ask what the one itself is. In that way, the study of the one will be among the studies

525a that lead and turn the soul toward the sight of what is.”

“But that is especially true of the sight of the one,” he said. “We see the same thing at once as one and as unlimited in number.”

“And if that's true of the one,” I said, “isn't it also true of every number?”

“Of course.”

“And calculation and arithmetic are entirely concerned with numbers.”

“Yes.” 525b

“So they clearly lead us toward truth.”

“They certainly do.”

“Then they seem to be among the studies we are looking for. A soldier needs to learn them in order to arrange troops. A philosopher needs them in order to reach what is real, rising out of the world of change. Otherwise that person will never be able to calculate.”

“That's right,” he said.

“And our guardian is both a soldier and a philosopher.”

“Of course.”

“So, Glaucon, it would be right to make this study a requirement by law and persuade the people who will hold the most important positions 525c in the city to take up calculation. They must study it not as amateurs, but until they reach an understanding of the nature of numbers through thought itself. They should not practice it for buying and selling, like merchants or shopkeepers. They should do it for war, and to make it easier for the soul itself to turn from the world of change to truth and what is real.”

“Well said,” he replied.

“Now that we have mentioned the study of calculation,” I said, “I see how valuable it is. It is useful in many ways for what we want, if a person practices it for knowledge and not for trade.” 525d

“How?” he asked.

“In the way we just described. It powerfully leads the soul upward. It forces the soul to discuss numbers themselves, and never accepts anyone offering it visible or tangible objects that have numbers and discussing those instead. You know how experts in this subject behave. If someone tries to divide 525e the one itself in an argument, they laugh and reject the attempt. If you break it into fractions, they multiply those fractions, careful not to let the one turn out to be many parts instead of one.”

“That's exactly right,” he said.

526a “So, Glaucon, suppose someone asked them, ‘My good friends, what sort of numbers are you talking about? In these numbers, the one is just as you say: each unit is entirely equal to every other, differs not at all, and contains no parts.’ What do you think they would answer?”

“I think they would say they mean numbers that can only be thought about, not handled in any other way.”

“Do you see, my friend,” I said, “that this study really does seem necessary 526b for us? It clearly forces the soul to use understanding itself to reach truth itself.”

“It certainly does,” he said.

“And have you noticed this? People who have a natural gift for calculation tend to learn almost every subject quickly. Those who are slow learners improve if they're taught and trained in it. Even if they gain nothing else, every one of them becomes quicker than they were before.”

“That's true,” he said. 526c

“And I think you would have trouble finding many subjects, if any, that take more work to learn and practice.”

“You would,” he said.

“For all these reasons we must not leave this study out. Our most gifted people must be taught it.”

“I agree,” he said.

“Let that be our first study,” I said. “Now let's look at the one that comes next and see whether it suits our purpose.”

“Which one? Do you mean geometry?” he asked.

“Exactly,” I said. 526d

“It obviously suits military purposes,” he said. “In choosing places to camp or occupy, bringing troops together or spreading them out, and arranging an army in all the other ways needed during battles and marches, a general's knowledge of geometry will make a difference.”

“But for that,” I said, “a small part of geometry and calculation would do. We must look at the greater, more advanced part and see whether it serves 526e our aim: helping people see the Form of the Good more easily. We say that everything serves this aim if it forces the soul to turn toward that place where the happiest of all realities is, which it must see by every possible means.”

“That's right,” he said.

“So if geometry forces us to study what is real, it suits our purpose. If it studies the world of change, it doesn't.”

“We agree on that.”

527a “And no one with even a little experience of geometry,” I said, “will dispute this: the nature of that subject is the exact opposite of what the people working at it say about it.”

“How so?” he asked.

“They speak in a funny way, though they can hardly help it. They talk as if they are doing things for practical purposes. They say they're ‘squaring,’ ‘extending,’ and ‘adding,’ and use all sorts of terms like that. But the whole subject 527b is pursued for the sake of knowledge.”

“Exactly,” he said.

“Then we also have to agree on this, don't we?”

“On what?”

“That it gives knowledge of what always is, not of something that comes into being at one time and passes away at another.”

“That's easy to agree to,” he said. “Geometry is knowledge of what always is.”

“Then, my good friend, it would draw the soul toward truth. It would develop a philosopher's mind so that it turned upward, rather than staying turned downward as ours now do when they shouldn't.”

“As much as anything could,” he said. 527c

“Then we must do everything we can,” I said, “to make sure that the people of your beautiful city never neglect geometry. Even its secondary benefits are important.”

“Which ones?” he asked.

“The military uses you mentioned,” I said. “And we know it makes a huge difference to how well someone grasps every other subject, whether that person has studied geometry or not.”

“A huge difference, by Zeus,” he said.

“Shall we make this the second study for the young?”

“Let's,” he said. 527d

“What about astronomy? Shall we make it the third?”

“I think so,” he said. “A better grasp of seasons, months, and years matters not only for farming and sailing, but just as much for commanding an army.”

“You amuse me,” I said. “You seem afraid that most people will think you're prescribing useless studies. But it is no small matter—and it is hard to believe—that these studies clean and rekindle an instrument in each person's soul. Other pursuits are destroying it and making it blind, yet saving it is worth more than saving ten thousand eyes. Only with that instrument do we see truth. 527e People who agree with us on this will think you speak remarkably well. People who have never understood it will naturally think you're talking nonsense, because they see no other benefit from these studies worth mentioning. So decide right now which people you're talking to. Or are you talking chiefly for your own sake, rather than for either group? Still, you wouldn't begrudge anyone else the benefit they might get from what you say.”

528a “That's what I choose,” he said. “I speak, ask, and answer mainly for my own sake.”

“Then go back a step,” I said. “Just now we chose the wrong subject to follow geometry.”

“How did we?” he asked.

“After studying surfaces,” I said, “we went straight to solids in motion before studying solids themselves. But the right order after the second dimension is to take the third. That concerns the growth of cubes and things that have depth.” 528b

“Yes,” he said, “but that subject does not seem to have been developed yet, Socrates.”

“There are two reasons,” I said. “No city values it, so people do little research into it, difficult as it is. And researchers need someone to direct them; without such a person they won't make discoveries. First, it's hard to find a director. Even if one were found, researchers in this field as things stand are too proud to listen. 528c But if a whole city helped direct them and valued the subject, they would listen. If research went on steadily and energetically, its nature would become clear. Even now, although most people disrespect it and hinder its growth, and even though its researchers cannot explain its use, its appeal drives it to develop in spite of all this. It would not be surprising if it came to light.”

“It certainly has a special appeal,” he said. “But explain what you were saying more clearly. You called the study of surfaces geometry.” 528d

“Yes,” I said.

“Then you put astronomy after it at first, but later you went back.”

“In my rush to get through everything quickly,” I said, “I'm actually slowing myself down. The study of the third dimension should have come next. But because its research is in such a sorry state, I skipped it and put astronomy—the study of solids in motion—after geometry.” 528e

“That's right,” he said.

“Then let's make astronomy the fourth study,” I said, “assuming that the city takes up the study we are now leaving out.”

“That makes sense,” he said. “Socrates, you just criticized me for praising astronomy in a crude way. Now I'll praise it on your terms. Surely anyone can see that it forces the soul to look upward and leads it away from things down here and toward things up there.”

529a “Maybe everyone can see that but me,” I said. “I don't think it does.”

“Why not?” he asked.

“As practiced today by people who claim it leads to philosophy, it makes the soul look very much downward.”

“What do you mean?” he asked.

“I think you're being rather generous in your own idea of what studying the things above means,” I said. “You might think that if someone leaned his head back to study decorations on a ceiling, 529b he was observing them with his understanding, not his eyes. Maybe you're right and I'm being foolish. But I cannot believe any subject makes the soul look upward except one concerned with what is real and invisible. If someone tries to learn about a visible thing by staring up with his mouth open or looking down with his eyes half shut, I say he will never learn it. There is no knowledge of such things. His soul will look down, not up, 529c even if he studies while floating on his back on land or at sea.”

“I deserve that,” he said. “You were right to criticize me. But how did you mean astronomy should be studied differently from the way people study it now, if it is to help them reach the goal we are talking about?”

“Like this,” I said. “We should recognize that the patterns in the sky, because they are set in the visible world, are the most beautiful and exact visible things of their kind. But they fall far short 529d of the true things: the motions by which true speed and true slowness move relative to each other according to true number and every true figure, and move the things they contain. These can be grasped by reasoning and thought, not by sight. Don't you agree?”

“Completely,” he said.

“Then we must use the patterns of the sky as models to help us study those true things. It would be like finding drawings carefully made and executed by Daedalus or another craftsman or painter. 529e A person who knew geometry would think such drawings were very beautifully made, but would find it ridiculous to examine them seriously in hopes of finding in them the exact truth about equal or double lengths or any other proportion.”

530a “Of course it would be ridiculous,” he said.

“Don't you think a real astronomer will feel the same way when he looks at the motions of the stars?” I asked. “He will believe that the maker of the sky has arranged the sky and everything in it as beautifully as visible things can be arranged. But think of the proportions between night and day, between them and the month, between the month and the year, and between the other stars 530b and both these measures and one another. Won't he think it strange if anyone believes these relationships always stay exactly the same and never vary in any way, even though these things have bodies and can be seen? Won't he think it strange to try in every way to find the exact truth about them?”

“Now that I hear you say it, I do,” he replied.

“Then we'll study astronomy by working on problems, just as we do in geometry,” I said. “We'll leave the things in the sky alone, if we want our study of astronomy to make the part of the soul that is naturally capable of understanding useful 530c instead of useless.”

“You're demanding far more work than current astronomy involves,” he said.

“I think we'll make similar demands of the other studies,” I said, “if we're to be any use as lawmakers. Can you suggest another suitable subject?”

“Not just now,” he said.

“Motion has more than one form,” I said. “A wise person might be able to name them all, but two are obvious even to us.” 530d

“What are they?”

“Alongside the one we've mentioned, there is its counterpart.”

“What's that?”

“It seems,” I said, “that just as eyes are fitted for astronomy, ears are fitted for harmonious motion. These two fields are like sisters, as the Pythagoreans say, and we agree with them, Glaucon. Don't we?”

“Yes,” he said. 530e

“Since there's a great deal to do, we'll ask them what they say about these subjects, and whether there's anything else to add. But through it all we'll keep our own requirement in view.”

“What requirement?”

“That those we educate must never try to learn any part of these subjects without carrying it through to its end. Each study must always reach the goal toward which all studies should lead, as we were just saying about astronomy. Or don't you know that