Plato, Republic 527a–531e: Book VII: Geometry, solids, astronomy and harmonics

“This at least,” said I, “will not be disputed by those who have even a slight acquaintance with geometry, that this science is in direct contradiction with the language employed in it by its adepts.” “How so?” he said. “Their language is most ludicrous, though they cannot help it, for they speak as if they were doing something and as if all their words were directed towards action. For all their talk is of squaring and applying and adding and the like, whereas in fact the real object of the entire study is pure knowledge.” “That is absolutely true,” he said. “And must we not agree on a further point?” “What?” “That it is the knowledge of that which always is, and not of a something which at some time comes into being and passes away.” “That is readily admitted,” he said, “for geometry is the knowledge of the eternally existent.” “Then, my good friend, it would tend to draw the soul to truth, and would be productive of a philosophic attitude of mind, directing upward the faculties that now wrongly are turned earthward.” “Nothing is surer,” he said. “Then nothing is surer,” said I, “than that we must require that the men of your Fair City shall never neglect geometry, for even the by-products of such study are not slight.” “What are they?” said he. “What you mentioned,” said I, “its uses in war, and also we are aware that for the better reception of all studies there will be an immeasurable difference between the student who has been imbued with geometry and the one who has not.” “Immense indeed, by Zeus,” he said. “Shall we, then, lay this down as a second branch of study for our lads?” “Let us do so,” he said. “Shall we set down astronomy as a third, or do you dissent?” “I certainly agree,” he said; “for quickness of perception about the seasons and the courses of the months and the years is serviceable, not only to agriculture and navigation, but still more to the military art.” “I am amused,” said I, “at your apparent fear lest the multitude may suppose you to be recommending useless studies. It is indeed no trifling task, but very difficult to realize that there is in every soul an organ or instrument of knowledge that is purified and kindled afresh by such studies when it has been destroyed and blinded by our ordinary pursuits, a faculty whose preservation outweighs ten thousand eyes; for by it only is reality beheld. Those who share this faith will think your words superlatively true. But those who have and have had no inkling of it will naturally think them all moonshine. For they can see no other benefit from such pursuits worth mentioning.

Decide, then, on the spot, to which party you address yourself. Or are you speaking to neither, but chiefly carrying on the discussion for your own sake, without however judging any other who may be able to profit by it?” “This is the alternative I choose,” he said, “that it is for my own sake chiefly that I speak and ask questions and reply.” “Fall back a little, then,” said I; “for we just now did not rightly select the study that comes next after geometry.” “What was our mistake?” he said. “After plane surfaces,” said I, “we went on to solids in revolution before studying them in themselves. The right way is next in order after the second dimension to take the third. This, I suppose, is the dimension of cubes and of everything that has depth.” “Why, yes, it is,” he said; “but this subject, Socrates, does not appear to have been investigated yet.” “There are two causes of that,” said I: “first, inasmuch as no city holds them in honor, these inquiries are languidly pursued owing to their difficulty. And secondly, the investigators need a director, who is indispensable for success and who, to begin with, is not easy to find, and then, if he could be found, as things are now, seekers in this field would be too arrogant to submit to his guidance. But if the state as a whole should join in superintending these studies and honor them, these specialists would accept advice, and continuous and strenuous investigation would bring out the truth. Since even now, lightly esteemed as they are by the multitude and hampered by the ignorance of their students as to the true reasons for pursuing them, they nevertheless in the face of all these obstacles force their way by their inherent charm and it would not surprise us if the truth about them were made apparent.” “It is true,” he said, “that they do possess an extraordinary attractiveness and charm. But explain more clearly what you were just speaking of. The investigation of plane surfaces, I presume, you took to be geometry?” “Yes,” said I. “And then,” he said, “at first you took astronomy next and then you drew back.” “Yes,” I said, “for in my haste to be done I was making less speed. For, while the next thing in order is the study of the third dimension or solids, I passed it over because of our absurd neglect to investigate it, and mentioned next after geometry astronomy, which deals with the movements of solids.” “That is right,” he said. “Then, as our fourth study,” said I, “let us set down astronomy, assuming that this science, the discussion of which has been passed over, is available, provided, that is, that the state pursues it.”

“That is likely,” said he; “and instead of the vulgar utilitarian commendation of astronomy, for which you just now rebuked me, Socrates, I now will praise it on your principles. For it is obvious to everybody, I think, that this study certainly compels the soul to look upward and leads it away from things here to those higher things.” “It may be obvious to everybody except me,” said I, “for I do not think so.” “What do you think?” he said. “As it is now handled by those who are trying to lead us up to philosophy, I think that it turns the soul’s gaze very much downward.” “What do you mean?” he said. “You seem to me in your thought to put a most liberal interpretation on the “study of higher things,”” I said, “for apparently if anyone with back-thrown head should learn something by staring at decorations on a ceiling, you would regard him as contemplating them with the higher reason and not with the eyes. Perhaps you are right and I am a simpleton. For I, for my part, am unable to suppose that any other study turns the soul’s gaze upward than that which deals with being and the invisible. But if anyone tries to learn about the things of sense, whether gaping up or blinking down, I would never say that he really learns—for nothing of the kind admits of true knowledge—nor would I say that his soul looks up, but down, even though he study floating on his back on sea or land.” “A fair retort,” he said; “your rebuke is deserved. But how, then, did you mean that astronomy ought to be taught contrary to the present fashion if it is to be learned in a way to conduce to our purpose?” “Thus,” said I, “these sparks that paint the sky, since they are decorations on a visible surface, we must regard, to be sure, as the fairest and most exact of material things but we must recognize that they fall far short of the truth, the movements, namely, of real speed and real slowness in true number and in all true figures both in relation to one another and as vehicles of the things they carry and contain. These can be apprehended only by reason and thought, but not by sight; or do you think otherwise?” “By no means,” he said. “Then,” said I, “we must use the blazonry of the heavens as patterns to aid in the study of those realities, just as one would do who chanced upon diagrams drawn with special care and elaboration by Daedalus or some other craftsman or painter.

For anyone acquainted with geometry who saw such designs would admit the beauty of the workmanship, but would think it absurd to examine them seriously in the expectation of finding in them the absolute truth with regard to equals or doubles or any other ratio.” “How could it be otherwise than absurd?” he said. “Do you not think,” said I, “that one who was an astronomer in very truth would feel in the same way when he turned his eyes upon the movements of the stars? He will be willing to concede that the artisan of heaven fashioned it and all that it contains in the best possible manner for such a fabric; but when it comes to the proportions of day and night, and of their relation to the month, and that of the month to the year, and of the other stars to these and one another, do you not suppose that he will regard as a very strange fellow the man who believes that these things go on for ever without change or the least deviation—though they possess bodies and are visible objects—and that his unremitting quest the realities of these things?” “I at least do think so,” he said, “now that I hear it from you.” “It is by means of problems, then,” said I, “as in the study of geometry, that we will pursue astronomy too, and we will let be the things in the heavens, if we are to have a part in the true science of astronomy and so convert to right use from uselessness that natural indwelling intelligence of the soul.” “You enjoin a task,” he said, “that will multiply the labor of our present study of astronomy many times.” “And I fancy,” I said, “that our other injunctions will be of the same kind if we are of any use as lawgivers. “However, what suitable studies have you to suggest?” “Nothing,” he said, “thus off-hand.” “Yet, surely,” said I, “motion in general provides not one but many forms or species, according to my opinion. To enumerate them all will perhaps be the task of a wise man, but even to us two of them are apparent.” “What are they?” “In addition to astronomy, its counterpart, I replied.” “What is that?” “We may venture to suppose,” I said, “that as the eyes are framed for astronomy so the ears are framed, for the movements of harmony; and these are in some sort kindred sciences, as the Pythagoreans affirm and we admit, do we not, Glaucon?” “We do,” he said. “Then,” said I, since the task is, so great, shall we not inquire of them what their opinion is and whether they have anything to add? And we in all this will be on the watch for what concerns us.” “What is that?” “To prevent our fosterlings from attempting to learn anything that does not conduce to the end we have in view, and does not always come out at what we said ought to be the goal of everything, as we were just now saying about astronomy.

Or do you not know that they repeat the same procedure in the case of harmonies? They transfer it to hearing and measure audible concords and sounds against one another, expending much useless labor just as the astronomers do.” “Yes, by heaven,” he said, “and most absurdly too. They talk of something they call minims and, laying their ears alongside, as if trying to catch a voice from next door, some affirm that they can hear a note between and that this is the least interval and the unit of measurement, while others insist that the strings now render identical sounds, both preferring their ears to their minds.” “You,” said I, “are speaking of the worthies who vex and torture the strings and rack them on the pegs; but—not to draw out the comparison with strokes of the plectrum and the musician’s complaints of too responsive and too reluctant strings—I drop the figure, and tell you that I do not mean these people, but those others whom we just now said we would interrogate about harmony. Their method exactly corresponds to that of the astronomer; for the numbers they seek are those found in these heard concords, but they do not ascend to generalized problems and the consideration which numbers are inherently concordant and which not and why in each case.” “A superhuman task,” he said. “Say, rather, useful, said I, for the investigation of the beautiful and the good, but if otherwise pursued, useless.” “That is likely,” he said. “And what is more,” I said, I take it that if the investigation of all these studies goes far enough to bring out their community and kinship with one another, and to infer their affinities, then to busy ourselves with them contributes to our desired end, and the labor taken is not lost; but otherwise it is vain.” “I too so surmise,” said he; “but it is a huge task of which you speak, Socrates.” “Are you talking about the prelude,” I said, “or what? Or do we not know that all this is but the preamble of the law itself, the prelude of the strain that we have to apprehend? For you surely do not suppose that experts in these matters are reasoners and dialecticians? “ “No, by Zeus,” he said, “except a very few whom I have met.” “But have you ever supposed,” I said, “that men who could not render and exact an account of opinions in discussion would ever know anything of the things we say must be known?” ““No” is surely the answer to that too.”