Unit VI — Parmenides, Philebus & Laws VII: Straightness, Limit & Incommensurability¶
Optical Straightness & Equidistant Roundness (Parm. 137d–138a, 145a–b), Limit & Pure Geometric Beauty by Compass & Rule (Phil. 16c–26d, 51b–52c), and Dimensional Incommensurability (Laws VII 817e–820d)
«καὶ γὰρ εὐθύ γε, οἶμαι, ἐστὶν οὗ ἂν τὸ μέσον ἀμφοῖν τοῖν ἐσχάτοιν ἐπίπροσθεν ᾖ.»
“For straight, I take it, is that of which the middle stands in front of both extremes.” — Parmenides 137e3–4
VI.1 Round Equidistance & Optical Straightness: First Hypothesis (Parm. 137d–138a)¶
English Translation by Benjamin Jowett (3rd ed., 1892) · Project Gutenberg #1687 (Jowett 3rd ed., 1892)
But if it has no parts, it will have neither beginning, middle, nor end; for these would of course be parts of it.
Right.
But then, again, a beginning and an end are the limits of everything?
Certainly.
Then the one, having neither beginning nor end, is unlimited?
Yes, unlimited.
And therefore formless; for it cannot partake either of round or straight.
But why?
Why, because the round is that of which all the extreme points are equidistant from the centre?
Yes.
And the straight is that of which the centre intercepts the view of the extremes?
True.
Then the one would have parts and would be many, if it partook either of a straight or of a circular form?
Assuredly.
But having no parts, it will be neither straight nor round?
Right.
VI.2 Beginning, Middle, Extremes & Figure: Second Hypothesis (Parm. 145a–b)¶
English Translation by Benjamin Jowett (3rd ed., 1892) · Project Gutenberg #1687 (Jowett 3rd ed., 1892)
Further, inasmuch as the parts are parts of a whole, the one, as a whole, will be limited; for are not the parts contained by the whole?
Certainly.
And that which contains, is a limit?
Of course.
Then the one if it has being is one and many, whole and parts, having limits and yet unlimited in number?
Clearly.
And because having limits, also having extremes?
Certainly.
And if a whole, having beginning and middle and end. For can anything be a whole without these three? And if any one of them is wanting to anything, will that any longer be a whole?
No.
Then the one, as appears, will have beginning, middle, and end.
It will.
But, again, the middle will be equidistant from the extremes; or it would not be in the middle?
Yes.
Then the one will partake of figure, either rectilinear or round, or a union of the two?
True.
VI.3 Limit (Peras), the Unlimited (Apeiron) & Mathematical Measure (Phil. 16c–26d)¶
English Translation by Benjamin Jowett (3rd ed., 1892) · Project Gutenberg #1744 (Jowett 3rd ed., 1892)
SOCRATES: The reverse of unimportant, my boys, as Philebus calls you, and there neither is nor ever will be a better than my own favourite way, which has nevertheless already often deserted me and left me helpless in the hour of need.
PROTARCHUS: Tell us what that is.
SOCRATES: One which may be easily pointed out, but is by no means easy of application; it is the parent of all the discoveries in the arts.
PROTARCHUS: Tell us what it is.
SOCRATES: A gift of heaven, which, as I conceive, the gods tossed among men by the hands of a new Prometheus, and therewith a blaze of light; and the ancients, who were our betters and nearer the gods than we are, handed down the tradition, that whatever things are said to be are composed of one and many, and have the finite and infinite implanted in them: seeing, then, that such is the order of the world, we too ought in every enquiry to begin by laying down one idea of that which is the subject of enquiry; this unity we shall find in everything. Having found it, we may next proceed to look for two, if there be two, or, if not, then for three or some other number, subdividing each of these units, until at last the unity with which we began is seen not only to be one and many and infinite, but also a definite number; the infinite must not be suffered to approach the many until the entire number of the species intermediate between unity and infinity has been discovered,–then, and not till then, we may rest from division, and without further troubling ourselves about the endless individuals may allow them to drop into infinity. This, as I was saying, is the way of considering and learning and teaching one another, which the gods have handed down to us. But the wise men of our time are either too quick or too slow in conceiving plurality in unity. Having no method, they make their one and many anyhow, and from unity pass at once to infinity; the intermediate steps never occur to them. And this, I repeat, is what makes the difference between the mere art of disputation and true dialectic.
PROTARCHUS: I think that I partly understand you Socrates, but I should like to have a clearer notion of what you are saying.
SOCRATES: I may illustrate my meaning by the letters of the alphabet, Protarchus, which you were made to learn as a child.
PROTARCHUS: How do they afford an illustration?
SOCRATES: The sound which passes through the lips whether of an individual or of all men is one and yet infinite.
PROTARCHUS: Very true.
SOCRATES: And yet not by knowing either that sound is one or that sound is infinite are we perfect in the art of speech, but the knowledge of the number and nature of sounds is what makes a man a grammarian.
PROTARCHUS: Very true.
SOCRATES: And the knowledge which makes a man a musician is of the same kind.
PROTARCHUS: How so?
SOCRATES: Sound is one in music as well as in grammar?
PROTARCHUS: Certainly.
SOCRATES: And there is a higher note and a lower note, and a note of equal pitch:–may we affirm so much?
PROTARCHUS: Yes.
SOCRATES: But you would not be a real musician if this was all that you knew; though if you did not know this you would know almost nothing of music.
PROTARCHUS: Nothing.
SOCRATES: But when you have learned what sounds are high and what low, and the number and nature of the intervals and their limits or proportions, and the systems compounded out of them, which our fathers discovered, and have handed down to us who are their descendants under the name of harmonies; and the affections corresponding to them in the movements of the human body, which when measured by numbers ought, as they say, to be called rhythms and measures; and they tell us that the same principle should be applied to every one and many;–when, I say, you have learned all this, then, my dear friend, you are perfect; and you may be said to understand any other subject, when you have a similar grasp of it. But the infinity of kinds and the infinity of individuals which there is in each of them, when not classified, creates in every one of us a state of infinite ignorance; and he who never looks for number in anything, will not himself be looked for in the number of famous men.
PROTARCHUS: I think that what Socrates is now saying is excellent, Philebus.
PHILEBUS: I think so too, but how do his words bear upon us and upon the argument?
SOCRATES: Philebus is right in asking that question of us, Protarchus.
PROTARCHUS: Indeed he is, and you must answer him.
SOCRATES: I will; but you must let me make one little remark first about these matters; I was saying, that he who begins with any individual unity, should proceed from that, not to infinity, but to a definite number, and now I say conversely, that he who has to begin with infinity should not jump to unity, but he should look about for some number representing a certain quantity, and thus out of all end in one. And now let us return for an illustration of our principle to the case of letters.
PROTARCHUS: What do you mean?
SOCRATES: Some god or divine man, who in the Egyptian legend is said to have been Theuth, observing that the human voice was infinite, first distinguished in this infinity a certain number of vowels, and then other letters which had sound, but were not pure vowels (i.e., the semivowels); these too exist in a definite number; and lastly, he distinguished a third class of letters which we now call mutes, without voice and without sound, and divided these, and likewise the two other classes of vowels and semivowels, into the individual sounds, and told the number of them, and gave to each and all of them the name of letters; and observing that none of us could learn any one of them and not learn them all, and in consideration of this common bond which in a manner united them, he assigned to them all a single art, and this he called the art of grammar or letters.
PHILEBUS: The illustration, Protarchus, has assisted me in understanding the original statement, but I still feel the defect of which I just now complained.
SOCRATES: Are you going to ask, Philebus, what this has to do with the argument?
PHILEBUS: Yes, that is a question which Protarchus and I have been long asking.
SOCRATES: Assuredly you have already arrived at the answer to the question which, as you say, you have been so long asking?
PHILEBUS: How so?
SOCRATES: Did we not begin by enquiring into the comparative eligibility of pleasure and wisdom?
PHILEBUS: Certainly.
SOCRATES: And we maintain that they are each of them one?
PHILEBUS: True.
SOCRATES: And the precise question to which the previous discussion desires an answer is, how they are one and also many (i.e., how they have one genus and many species), and are not at once infinite, and what number of species is to be assigned to either of them before they pass into infinity (i.e. into the infinite number of individuals).
PROTARCHUS: That is a very serious question, Philebus, to which Socrates has ingeniously brought us round, and please to consider which of us shall answer him; there may be something ridiculous in my being unable to answer, and therefore imposing the task upon you, when I have undertaken the whole charge of the argument, but if neither of us were able to answer, the result methinks would be still more ridiculous. Let us consider, then, what we are to do:–Socrates, if I understood him rightly, is asking whether there are not kinds of pleasure, and what is the number and nature of them, and the same of wisdom.
SOCRATES: Most true, O son of Callias; and the previous argument showed that if we are not able to tell the kinds of everything that has unity, likeness, sameness, or their opposites, none of us will be of the smallest use in any enquiry.
PROTARCHUS: That seems to be very near the truth, Socrates. Happy would the wise man be if he knew all things, and the next best thing for him is that he should know himself. Why do I say so at this moment? I will tell you. You, Socrates, have granted us this opportunity of conversing with you, and are ready to assist us in determining what is the best of human goods. For when Philebus said that pleasure and delight and enjoyment and the like were the chief good, you answered–No, not those, but another class of goods; and we are constantly reminding ourselves of what you said, and very properly, in order that we may not forget to examine and compare the two. And these goods, which in your opinion are to be designated as superior to pleasure, and are the true objects of pursuit, are mind and knowledge and understanding and art, and the like. There was a dispute about which were the best, and we playfully threatened that you should not be allowed to go home until the question was settled; and you agreed, and placed yourself at our disposal. And now, as children say, what has been fairly given cannot be taken back; cease then to fight against us in this way.
SOCRATES: In what way?
PHILEBUS: Do not perplex us, and keep asking questions of us to which we have not as yet any sufficient answer to give; let us not imagine that a general puzzling of us all is to be the end of our discussion, but if we are unable to answer, do you answer, as you have promised. Consider, then, whether you will divide pleasure and knowledge according to their kinds; or you may let the matter drop, if you are able and willing to find some other mode of clearing up our controversy.
SOCRATES: If you say that, I have nothing to apprehend, for the words ‘if you are willing’ dispel all my fear; and, moreover, a god seems to have recalled something to my mind.
PHILEBUS: What is that?
SOCRATES: I remember to have heard long ago certain discussions about pleasure and wisdom, whether awake or in a dream I cannot tell; they were to the effect that neither the one nor the other of them was the good, but some third thing, which was different from them, and better than either. If this be clearly established, then pleasure will lose the victory, for the good will cease to be identified with her:–Am I not right?
PROTARCHUS: Yes.
SOCRATES: And there will cease to be any need of distinguishing the kinds of pleasures, as I am inclined to think, but this will appear more clearly as we proceed.
PROTARCHUS: Capital, Socrates; pray go on as you propose.
SOCRATES: But, let us first agree on some little points.
PROTARCHUS: What are they?
SOCRATES: Is the good perfect or imperfect?
PROTARCHUS: The most perfect, Socrates, of all things.
SOCRATES: And is the good sufficient?
PROTARCHUS: Yes, certainly, and in a degree surpassing all other things.
SOCRATES: And no one can deny that all percipient beings desire and hunt after good, and are eager to catch and have the good about them, and care not for the attainment of anything which is not accompanied by good.
PROTARCHUS: That is undeniable.
SOCRATES: Now let us part off the life of pleasure from the life of wisdom, and pass them in review.
PROTARCHUS: How do you mean?
SOCRATES: Let there be no wisdom in the life of pleasure, nor any pleasure in the life of wisdom, for if either of them is the chief good, it cannot be supposed to want anything, but if either is shown to want anything, then it cannot really be the chief good.
PROTARCHUS: Impossible.
SOCRATES: And will you help us to test these two lives?
PROTARCHUS: Certainly.
SOCRATES: Then answer.
PROTARCHUS: Ask.
SOCRATES: Would you choose, Protarchus, to live all your life long in the enjoyment of the greatest pleasures?
PROTARCHUS: Certainly I should.
SOCRATES: Would you consider that there was still anything wanting to you if you had perfect pleasure?
PROTARCHUS: Certainly not.
SOCRATES: Reflect; would you not want wisdom and intelligence and forethought, and similar qualities? would you not at any rate want sight?
PROTARCHUS: Why should I? Having pleasure I should have all things.
SOCRATES: Living thus, you would always throughout your life enjoy the greatest pleasures?
PROTARCHUS: I should.
SOCRATES: But if you had neither mind, nor memory, nor knowledge, nor true opinion, you would in the first place be utterly ignorant of whether you were pleased or not, because you would be entirely devoid of intelligence.
PROTARCHUS: Certainly.
SOCRATES: And similarly, if you had no memory you would not recollect that you had ever been pleased, nor would the slightest recollection of the pleasure which you feel at any moment remain with you; and if you had no true opinion you would not think that you were pleased when you were; and if you had no power of calculation you would not be able to calculate on future pleasure, and your life would be the life, not of a man, but of an oyster or ‘pulmo marinus.’ Could this be otherwise?
PROTARCHUS: No.
SOCRATES: But is such a life eligible?
PROTARCHUS: I cannot answer you, Socrates; the argument has taken away from me the power of speech.
SOCRATES: We must keep up our spirits;–let us now take the life of mind and examine it in turn.
PROTARCHUS: And what is this life of mind?
SOCRATES: I want to know whether any one of us would consent to live, having wisdom and mind and knowledge and memory of all things, but having no sense of pleasure or pain, and wholly unaffected by these and the like feelings?
PROTARCHUS: Neither life, Socrates, appears eligible to me, nor is likely, as I should imagine, to be chosen by any one else.
SOCRATES: What would you say, Protarchus, to both of these in one, or to one that was made out of the union of the two?
PROTARCHUS: Out of the union, that is, of pleasure with mind and wisdom?
SOCRATES: Yes, that is the life which I mean.
PROTARCHUS: There can be no difference of opinion; not some but all would surely choose this third rather than either of the other two, and in addition to them.
SOCRATES: But do you see the consequence?
PROTARCHUS: To be sure I do. The consequence is, that two out of the three lives which have been proposed are neither sufficient nor eligible for man or for animal.
SOCRATES: Then now there can be no doubt that neither of them has the good, for the one which had would certainly have been sufficient and perfect and eligible for every living creature or thing that was able to live such a life; and if any of us had chosen any other, he would have chosen contrary to the nature of the truly eligible, and not of his own free will, but either through ignorance or from some unhappy necessity.
PROTARCHUS: Certainly that seems to be true.
SOCRATES: And now have I not sufficiently shown that Philebus’ goddess is not to be regarded as identical with the good?
PHILEBUS: Neither is your ‘mind’ the good, Socrates, for that will be open to the same objections.
SOCRATES: Perhaps, Philebus, you may be right in saying so of my ‘mind’; but of the true, which is also the divine mind, far otherwise. However, I will not at present claim the first place for mind as against the mixed life; but we must come to some understanding about the second place. For you might affirm pleasure and I mind to be the cause of the mixed life; and in that case although neither of them would be the good, one of them might be imagined to be the cause of the good. And I might proceed further to argue in opposition to Philebus, that the element which makes this mixed life eligible and good, is more akin and more similar to mind than to pleasure. And if this is true, pleasure cannot be truly said to share either in the first or second place, and does not, if I may trust my own mind, attain even to the third.
PROTARCHUS: Truly, Socrates, pleasure appears to me to have had a fall; in fighting for the palm, she has been smitten by the argument, and is laid low. I must say that mind would have fallen too, and may therefore be thought to show discretion in not putting forward a similar claim. And if pleasure were deprived not only of the first but of the second place, she would be terribly damaged in the eyes of her admirers, for not even to them would she still appear as fair as before.
SOCRATES: Well, but had we not better leave her now, and not pain her by applying the crucial test, and finally detecting her?
PROTARCHUS: Nonsense, Socrates.
SOCRATES: Why? because I said that we had better not pain pleasure, which is an impossibility?
PROTARCHUS: Yes, and more than that, because you do not seem to be aware that none of us will let you go home until you have finished the argument.
SOCRATES: Heavens! Protarchus, that will be a tedious business, and just at present not at all an easy one. For in going to war in the cause of mind, who is aspiring to the second prize, I ought to have weapons of another make from those which I used before; some, however, of the old ones may do again. And must I then finish the argument?
PROTARCHUS: Of course you must.
SOCRATES: Let us be very careful in laying the foundation.
PROTARCHUS: What do you mean?
SOCRATES: Let us divide all existing things into two, or rather, if you do not object, into three classes.
PROTARCHUS: Upon what principle would you make the division?
SOCRATES: Let us take some of our newly-found notions.
PROTARCHUS: Which of them?
SOCRATES: Were we not saying that God revealed a finite element of existence, and also an infinite?
PROTARCHUS: Certainly.
SOCRATES: Let us assume these two principles, and also a third, which is compounded out of them; but I fear that I am ridiculously clumsy at these processes of division and enumeration.
PROTARCHUS: What do you mean, my good friend?
SOCRATES: I say that a fourth class is still wanted.
PROTARCHUS: What will that be?
SOCRATES: Find the cause of the third or compound, and add this as a fourth class to the three others.
PROTARCHUS: And would you like to have a fifth class or cause of resolution as well as a cause of composition?
SOCRATES: Not, I think, at present; but if I want a fifth at some future time you shall allow me to have it.
PROTARCHUS: Certainly.
SOCRATES: Let us begin with the first three; and as we find two out of the three greatly divided and dispersed, let us endeavour to reunite them, and see how in each of them there is a one and many.
PROTARCHUS: If you would explain to me a little more about them, perhaps I might be able to follow you.
SOCRATES: Well, the two classes are the same which I mentioned before, one the finite, and the other the infinite; I will first show that the infinite is in a certain sense many, and the finite may be hereafter discussed.
PROTARCHUS: I agree.
SOCRATES: And now consider well; for the question to which I invite your attention is difficult and controverted. When you speak of hotter and colder, can you conceive any limit in those qualities? Does not the more and less, which dwells in their very nature, prevent their having any end? for if they had an end, the more and less would themselves have an end.
PROTARCHUS: That is most true.
SOCRATES: Ever, as we say, into the hotter and the colder there enters a more and a less.
PROTARCHUS: Yes.
SOCRATES: Then, says the argument, there is never any end of them, and being endless they must also be infinite.
PROTARCHUS: Yes, Socrates, that is exceedingly true.
SOCRATES: Yes, my dear Protarchus, and your answer reminds me that such an expression as ‘exceedingly,’ which you have just uttered, and also the term ‘gently,’ have the same significance as more or less; for whenever they occur they do not allow of the existence of quantity–they are always introducing degrees into actions, instituting a comparison of a more or a less excessive or a more or a less gentle, and at each creation of more or less, quantity disappears. For, as I was just now saying, if quantity and measure did not disappear, but were allowed to intrude in the sphere of more and less and the other comparatives, these last would be driven out of their own domain. When definite quantity is once admitted, there can be no longer a ‘hotter’ or a ‘colder’ (for these are always progressing, and are never in one stay); but definite quantity is at rest, and has ceased to progress. Which proves that comparatives, such as the hotter and the colder, are to be ranked in the class of the infinite.
PROTARCHUS: Your remark certainly has the look of truth, Socrates; but these subjects, as you were saying, are difficult to follow at first. I think however, that if I could hear the argument repeated by you once or twice, there would be a substantial agreement between us.
SOCRATES: Yes, and I will try to meet your wish; but, as I would rather not waste time in the enumeration of endless particulars, let me know whether I may not assume as a note of the infinite–
PROTARCHUS: What?
SOCRATES: I want to know whether such things as appear to us to admit of more or less, or are denoted by the words ‘exceedingly,’ ‘gently,’ ‘extremely,’ and the like, may not be referred to the class of the infinite, which is their unity, for, as was asserted in the previous argument, all things that were divided and dispersed should be brought together, and have the mark or seal of some one nature, if possible, set upon them–do you remember?
PROTARCHUS: Yes.
SOCRATES: And all things which do not admit of more or less, but admit their opposites, that is to say, first of all, equality, and the equal, or again, the double, or any other ratio of number and measure–all these may, I think, be rightly reckoned by us in the class of the limited or finite; what do you say?
PROTARCHUS: Excellent, Socrates.
SOCRATES: And now what nature shall we ascribe to the third or compound kind?
PROTARCHUS: You, I think, will have to tell me that.
SOCRATES: Rather God will tell you, if there be any God who will listen to my prayers.
PROTARCHUS: Offer up a prayer, then, and think.
SOCRATES: I am thinking, Protarchus, and I believe that some God has befriended us.
PROTARCHUS: What do you mean, and what proof have you to offer of what you are saying?
SOCRATES: I will tell you, and do you listen to my words.
PROTARCHUS: Proceed.
SOCRATES: Were we not speaking just now of hotter and colder?
PROTARCHUS: True.
SOCRATES: Add to them drier, wetter, more, less, swifter, slower, greater, smaller, and all that in the preceding argument we placed under the unity of more and less.
PROTARCHUS: In the class of the infinite, you mean?
SOCRATES: Yes; and now mingle this with the other.
PROTARCHUS: What is the other.
SOCRATES: The class of the finite which we ought to have brought together as we did the infinite; but, perhaps, it will come to the same thing if we do so now;–when the two are combined, a third will appear.
PROTARCHUS: What do you mean by the class of the finite?
SOCRATES: The class of the equal and the double, and any class which puts an end to difference and opposition, and by introducing number creates harmony and proportion among the different elements.
PROTARCHUS: I understand; you seem to me to mean that the various opposites, when you mingle with them the class of the finite, takes certain forms.
SOCRATES: Yes, that is my meaning.
PROTARCHUS: Proceed.
SOCRATES: Does not the right participation in the finite give health–in disease, for instance?
PROTARCHUS: Certainly.
SOCRATES: And whereas the high and low, the swift and the slow are infinite or unlimited, does not the addition of the principles aforesaid introduce a limit, and perfect the whole frame of music?
PROTARCHUS: Yes, certainly.
SOCRATES: Or, again, when cold and heat prevail, does not the introduction of them take away excess and indefiniteness, and infuse moderation and harmony?
PROTARCHUS: Certainly.
SOCRATES: And from a like admixture of the finite and infinite come the seasons, and all the delights of life?
PROTARCHUS: Most true.
SOCRATES: I omit ten thousand other things, such as beauty and health and strength, and the many beauties and high perfections of the soul: O my beautiful Philebus, the goddess, methinks, seeing the universal wantonness and wickedness of all things, and that there was in them no limit to pleasures and self-indulgence, devised the limit of law and order, whereby, as you say, Philebus, she torments, or as I maintain, delivers the soul.–What think you, Protarchus?
PROTARCHUS: Her ways are much to my mind, Socrates.
SOCRATES: You will observe that I have spoken of three classes?
PROTARCHUS: Yes, I think that I understand you: you mean to say that the infinite is one class, and that the finite is a second class of existences; but what you would make the third I am not so certain.
SOCRATES: That is because the amazing variety of the third class is too much for you, my dear friend; but there was not this difficulty with the infinite, which also comprehended many classes, for all of them were sealed with the note of more and less, and therefore appeared one.
PROTARCHUS: True.
SOCRATES: And the finite or limit had not many divisions, and we readily acknowledged it to be by nature one?
PROTARCHUS: Yes.
SOCRATES: Yes, indeed; and when I speak of the third class, understand me to mean any offspring of these, being a birth into true being, effected by the measure which the limit introduces.
PROTARCHUS: I understand.
SOCRATES: Still there was, as we said, a fourth class to be investigated, and you must assist in the investigation; for does not everything which comes into being, of necessity come into being through a cause?
PROTARCHUS: Yes, certainly; for how can there be anything which has no cause?
SOCRATES: And is not the agent the same as the cause in all except name; the agent and the cause may be rightly called one?
PROTARCHUS: Very true.
SOCRATES: And the same may be said of the patient, or effect; we shall find that they too differ, as I was saying, only in name–shall we not?
PROTARCHUS: We shall.
SOCRATES: The agent or cause always naturally leads, and the patient or effect naturally follows it?
PROTARCHUS: Certainly.
SOCRATES: Then the cause and what is subordinate to it in generation are not the same, but different?
PROTARCHUS: True.
SOCRATES: Did not the things which were generated, and the things out of which they were generated, furnish all the three classes?
PROTARCHUS: Yes.
SOCRATES: And the creator or cause of them has been satisfactorily proven to be distinct from them,–and may therefore be called a fourth principle?
PROTARCHUS: So let us call it.
SOCRATES: Quite right; but now, having distinguished the four, I think that we had better refresh our memories by recapitulating each of them in order.
PROTARCHUS: By all means.
SOCRATES: Then the first I will call the infinite or unlimited, and the second the finite or limited; then follows the third, an essence compound and generated; and I do not think that I shall be far wrong in speaking of the cause of mixture and generation as the fourth.
PROTARCHUS: Certainly not.
VI.4 Pure Geometric Beauty Constructed by Compass, Rule & Square (Phil. 51b–52c)¶
English Translation by Benjamin Jowett (3rd ed., 1892) · Project Gutenberg #1744 (Jowett 3rd ed., 1892)
SOCRATES: True pleasures are those which are given by beauty of colour and form, and most of those which arise from smells; those of sound, again, and in general those of which the want is painless and unconscious, and of which the fruition is palpable to sense and pleasant and unalloyed with pain.
PROTARCHUS: Once more, Socrates, I must ask what you mean.
SOCRATES: My meaning is certainly not obvious, and I will endeavour to be plainer. I do not mean by beauty of form such beauty as that of animals or pictures, which the many would suppose to be my meaning; but, says the argument, understand me to mean straight lines and circles, and the plane or solid figures which are formed out of them by turning-lathes and rulers and measurers of angles; for these I affirm to be not only relatively beautiful, like other things, but they are eternally and absolutely beautiful, and they have peculiar pleasures, quite unlike the pleasures of scratching. And there are colours which are of the same character, and have similar pleasures; now do you understand my meaning?
PROTARCHUS: I am trying to understand, Socrates, and I hope that you will try to make your meaning clearer.
SOCRATES: When sounds are smooth and clear, and have a single pure tone, then I mean to say that they are not relatively but absolutely beautiful, and have natural pleasures associated with them.
PROTARCHUS: Yes, there are such pleasures.
SOCRATES: The pleasures of smell are of a less ethereal sort, but they have no necessary admixture of pain; and all pleasures, however and wherever experienced, which are unattended by pains, I assign to an analogous class. Here then are two kinds of pleasures.
PROTARCHUS: I understand.
SOCRATES: To these may be added the pleasures of knowledge, if no hunger of knowledge and no pain caused by such hunger precede them.
PROTARCHUS: And this is the case.
SOCRATES: Well, but if a man who is full of knowledge loses his knowledge, are there not pains of forgetting?
PROTARCHUS: Not necessarily, but there may be times of reflection, when he feels grief at the loss of his knowledge.
SOCRATES: Yes, my friend, but at present we are enumerating only the natural perceptions, and have nothing to do with reflection.
PROTARCHUS: In that case you are right in saying that the loss of knowledge is not attended with pain.
SOCRATES: These pleasures of knowledge, then, are unmixed with pain; and they are not the pleasures of the many but of a very few.
PROTARCHUS: Quite true.
SOCRATES: And now, having fairly separated the pure pleasures and those which may be rightly termed impure, let us further add to our description of them, that the pleasures which are in excess have no measure, but that those which are not in excess have measure; the great, the excessive, whether more or less frequent, we shall be right in referring to the class of the infinite, and of the more and less, which pours through body and soul alike; and the others we shall refer to the class which has measure.
PROTARCHUS: Quite right, Socrates.
VI.5 Commensurable & Incommensurable Lengths, Breadths, and Depths (Laws VII 817e–820d)¶
English Translation by Benjamin Jowett (3rd ed., 1892) · Project Gutenberg #1750 (Jowett 3rd ed., 1892)
ATHENIAN: There still remain three studies suitable for freemen. Arithmetic is one of them; the measurement of length, surface, and depth is the second; and the third has to do with the revolutions of the stars in relation to one another. Not every one has need to toil through all these things in a strictly scientific manner, but only a few, and who they are to be we will hereafter indicate at the end, which will be the proper place; not to know what is necessary for mankind in general, and what is the truth, is disgraceful to every one: and yet to enter into these matters minutely is neither easy, nor at all possible for every one; but there is something in them which is necessary and cannot be set aside, and probably he who made the proverb about God originally had this in view when he said, that ‘not even God himself can fight against necessity;’ he meant, if I am not mistaken, divine necessity; for as to the human necessities of which the many speak, when they talk in this manner, nothing can be more ridiculous than such an application of the words.
CLEINIAS: And what necessities of knowledge are there, Stranger, which are divine and not human?
ATHENIAN: I conceive them to be those of which he who has no use nor any knowledge at all cannot be a God, or demi-god, or hero to mankind, or able to take any serious thought or charge of them. And very unlike a divine man would he be, who is unable to count one, two, three, or to distinguish odd and even numbers, or is unable to count at all, or reckon night and day, and who is totally unacquainted with the revolution of the sun and moon, and the other stars. There would be great folly in supposing that all these are not necessary parts of knowledge to him who intends to know anything about the highest kinds of knowledge; but which these are, and how many there are of them, and when they are to be learned, and what is to be learned together and what apart, and the whole correlation of them, must be rightly apprehended first; and these leading the way we may proceed to the other parts of knowledge. For so necessity grounded in nature constrains us, against which we say that no God contends, or ever will contend.
CLEINIAS: I think, Stranger, that what you have now said is very true and agreeable to nature.
ATHENIAN: Yes, Cleinias, that is so. But it is difficult for the legislator to begin with these studies; at a more convenient time we will make regulations for them.
CLEINIAS: You seem, Stranger, to be afraid of our habitual ignorance of the subject: there is no reason why that should prevent you from speaking out.
ATHENIAN: I certainly am afraid of the difficulties to which you allude, but I am still more afraid of those who apply themselves to this sort of knowledge, and apply themselves badly. For entire ignorance is not so terrible or extreme an evil, and is far from being the greatest of all; too much cleverness and too much learning, accompanied with an ill bringing up, are far more fatal.
CLEINIAS: True.
ATHENIAN: All freemen I conceive, should learn as much of these branches of knowledge as every child in Egypt is taught when he learns the alphabet. In that country arithmetical games have been invented for the use of mere children, which they learn as a pleasure and amusement. They have to distribute apples and garlands, using the same number sometimes for a larger and sometimes for a lesser number of persons; and they arrange pugilists and wrestlers as they pair together by lot or remain over, and show how their turns come in natural order. Another mode of amusing them is to distribute vessels, sometimes of gold, brass, silver, and the like, intermixed with one another, sometimes of one metal only; as I was saying they adapt to their amusement the numbers in common use, and in this way make more intelligible to their pupils the arrangements and movements of armies and expeditions, and in the management of a household they make people more useful to themselves, and more wide awake; and again in measurements of things which have length, and breadth, and depth, they free us from that natural ignorance of all these things which is so ludicrous and disgraceful.
CLEINIAS: What kind of ignorance do you mean?
ATHENIAN: O my dear Cleinias, I, like yourself, have late in life heard with amazement of our ignorance in these matters; to me we appear to be more like pigs than men, and I am quite ashamed, not only of myself, but of all Hellenes.
CLEINIAS: About what? Say, Stranger, what you mean.
ATHENIAN: I will; or rather I will show you my meaning by a question, and do you please to answer me: You know, I suppose, what length is?
CLEINIAS: Certainly.
ATHENIAN: And what breadth is?
CLEINIAS: To be sure.
ATHENIAN: And you know that these are two distinct things, and that there is a third thing called depth?
CLEINIAS: Of course.
ATHENIAN: And do not all these seem to you to be commensurable with themselves?
CLEINIAS: Yes.
ATHENIAN: That is to say, length is naturally commensurable with length, and breadth with breadth, and depth in like manner with depth?
CLEINIAS: Undoubtedly.
ATHENIAN: But if some things are commensurable and others wholly incommensurable, and you think that all things are commensurable, what is your position in regard to them?
CLEINIAS: Clearly, far from good.
ATHENIAN: Concerning length and breadth when compared with depth, or breadth and length when compared with one another, are not all the Hellenes agreed that these are commensurable with one another in some way?
CLEINIAS: Quite true.
ATHENIAN: But if they are absolutely incommensurable, and yet all of us regard them as commensurable, have we not reason to be ashamed of our compatriots; and might we not say to them: O ye best of Hellenes, is not this one of the things of which we were saying that not to know them is disgraceful, and of which to have a bare knowledge only is no great distinction?
CLEINIAS: Certainly.
ATHENIAN: And there are other things akin to these, in which there spring up other errors of the same family.
CLEINIAS: What are they?
ATHENIAN: The natures of commensurable and incommensurable quantities in their relation to one another. A man who is good for anything ought to be able, when he thinks, to distinguish them; and different persons should compete with one another in asking questions, which will be a far better and more graceful way of passing their time than the old man’s game of draughts.
CLEINIAS: I dare say; and these pastimes are not so very unlike a game of draughts.
ATHENIAN: And these, as I maintain, Cleinias, are the studies which our youth ought to learn, for they are innocent and not difficult; the learning of them will be an amusement, and they will benefit the state. If any one is of another mind, let him say what he has to say.
CLEINIAS: Certainly.
ATHENIAN: Then if these studies are such as we maintain, we will include them; if not, they shall be excluded.