Polytropy

In the account of justice – the dicaeology – that I looked at last time, equality was a nominal concern. I said it might not be what we mean today by equality before the law. We may come closer to that in the present reading, but I’m not sure.

We have reached the part of the Nicomachean Ethics that I dipped into more than six years ago, when writing what ultimately became a long mathematical and historical paper, “On Commensurability and Symmetry” (Journal of Humanistic Mathematics, Volume 7 Issue 2 [July 2017], pages 90–148, DOI). Back then, I made only a precision raid, as if by helicopter, using coordinates supplied by the LSJ lexicon for the words of interest (σύμμετρος and συμμετρία). Now, in a party, we have been working our way in on foot.

We have reached chapters iv and v of Book V of Aristotle’s Ethics. The mathematics of justice continues to be the theme.

In the first part of Book V, chapters i–iii, we looked at justice in two senses.

1. One justice is virtue itself, although not in “essence.” Considered as
   • habit or condition (hexis), it is virtue;
   • relation among people, it is justice.
2. A particular justice, of the form involved in distributions (εἶδος τὸ ἐν ταῖς διανομαῖς), to be called distributive justice (τὸ διανεμητικὸν δίκαιον) in chapter iv: it concerns honor, money, and anything else that can be shared out equally or unequally. The way these things should be shared out is proportionally, according to the merit of the recipient; at least, that’s what everybody says, although they have different ideas about how merit should actually be determined.

Now we take up two more kinds of justice.

3. Corrective justice (τὸ διορθωτικὸν δίκαιον or τὸ ἐπανορθωτικὸν δίκαιον) governs “transactions,” some of which can be involuntary. Here, justice is assigned “arithmetically.” This seems to mean that, if I unjustly have more than you, then I should split the difference, so as to give you half. However, this is somehow supposed to apply, even if our “transaction” is that I murder you.
4. Reciprocity (τὸ αντιπεπενθός)
   • is not really justice, if the meaning is that of “an eye for an eye”;
   • is precisely what is needed to make a community through the exchange of goods and services.

I am leery of making things too clear, because Aristotle himself is not clear. He may become clear with further study, the way mathematics does – or can do, if one is able to give it attention.

Clarity achieved in mathematics can sometimes be passed along to others. Indeed, this would seem to be what makes progress possible in this field.

In mathematical logic, the Completeness Theorem was first proved by Gödel in 1929; but the proof that one learns, or at least the proof that I learned as a graduate student in David Kueker’s course at the University of Maryland, was the one published by Henkin in 1949. A fellow student asked why we did not learn Gödel’s proof. Answer: “It’s a mess.” I did work through Gödel’s proof anyway, but that was twenty years later.

We can learn from Gödel without reading him ourselves. Is the same true for Aristotle?

The writings of Aristotle that have come down to us may be summaries of his lectures, or notes that he expanded on when actually giving the lectures. He may then have responded to the perceived needs of his students, in the manner of Socrates, as portrayed by Plato – or in the manner of Agnes Callard, by her own account (quoted at greater length in “Badiou, Bloom, Ryle, Shorey”):

Minutes before I walked into the classroom … I had been buried in commentaries and confusion. There was so much I did not understand about Aristotle’s arguments against atomism! But time was up, and I had to get in there and say something. If you were in that class, you probably thought what I said sounded pretty good, pretty coherent. Actually, it was. But that wasn’t all me. I was looking at the students’ faces, noticing how they paid attention when I was making sense, noticing when they didn’t follow. Their interest drew me out.

Possibly Aristotle’s teaching style was more like that of Protagoras, as described or parodied by Plato in the dialogue of that name. It is a dialogue between a companion and Socrates, who describes an encounter with Protagoras at the house of Callias at the crack of dawn (314e-5b):

when we had entered, we came upon Protagoras as he was walking round in the cloister, and close behind him two companies were walking round also … The persons who followed in their rear, listening to what they could of the talk, seemed to be mostly strangers … they follow where the voice sounds, enchanted; and some of our own inhabitants were also dancing attendance. As for me, when I saw their evolutions I was delighted with the admirable care they took not to hinder Protagoras at any moment by getting in front; but whenever the master turned about and those with him, it was fine to see the orderly manner in which his train of listeners split up into two parties on this side and on that, and wheeling round formed up again each time in his rear most admirably.

There are speakers who do not want to look at their audience. I have observed this at mathematics conferences.

---

Aristotle describes money as making things “symmetric”; at least, the Greek adjective is σύμμετρος, but the meaning is expressed today by the corresponding Latin form “commensurable.” Money gives goods a common measure. This makes exchange easier.

Exchange is a two-step process:

1. Determine the ratio of one of your products to somebody else’s.
2. Trade some of your products for some of the other person’s in the inverse ratio.

In Aristotle’s example (§ v.15):

1. The ratio of five to one is that of a house to a couch (or bedstead: κλίνη, cognate with “recline, clinic,” and “clitoris”).
2. Therefore one to five is the ratio to use when exchanging houses for couches.

One way to pass the first step is to know that

• a couch is worth 1 mina;
• a house is worth 5 minae.

We must not get confused, thinking now that five houses should be exchanged for a couch. Such confusion is possible, and Aristotle warns against it. He says (in § v.12) that you have to compute the reciprocal ratio before making an exchange, “or else one extreme will have both excesses.” That is Sachs’s translation of

εἰ δὲ μή, ἀμφοτέρας ἕξει τὰς ὑπεροχὰς τὸ ἕτερον ἄκρον,

but others are similar. What does it all mean?

• For Sachs, if you trade not at the reciprocal ratio, but at the same ratio as the individual products, then the values of the things exchanged will be in the duplicate of that ratio. Five houses are to one bed as five squared to one.
• For Rackham, as part of a trade, if I am supposed to give you a bed, but don’t, then our difference, with respect to beds, is more than it would have been, by two beds. (In place of a bed, Rackham talks about “ten shillings … or something worth that.”)

So Aristotle is ambiguous, at least for us. There are a few possibilities.

• He is confused himself.
• He knows what he means and could explain it in person.
• At least one of our translators is confused.

---

Aristotle says in § iii.12 in the previous reading,

ἡ ἄρα τοῦ πρώτου ὅρου τῷ τρίτῳ καὶ ἡ τοῦ δευτέρον τῷ τετάρτῳ σύζευξις τὸ ἐν διανομῇ δίκαιόν ἐστι, καὶ μέσον τὸ δίκαιον τοῦτ᾽ ἐστί τοῦ παρὰ τὸ ἀνάλογον.

In Rackham’s translation,

The principle of Distributive Justice, therefore, is the conjunction of the first term of a proportion with the third and of the second with the fourth; and the just in this sense is a mean between two extremes that are disproportionate.

Bywater’s text is,

ἡ ἄρα τοῦ α ὅρου τῷ γ καὶ ἡ τοῦ β τῷ δ σύζευξις τὸ ἐν διανομῇ δίκαιόν ἐστι, καὶ μέσον τὸ δίκαιον τοῦτ᾽ ἐστί, 〈τὸ δ᾽ ἄδικον〉 τὸ παρὰ τὸ ἀνάλογον,

which Sachs renders as,

Therefore, the linking together of the A term with the C, and of the B term with the D, is what is just in a distribution, and what is just in this sense is a mean, while what is unjust is what is disproportionate.

The idea seems to be that if

• A is to B as C is to D, and
• A and B are persons, and
• C and D are goods, and
• A possesses C,

then B ought to possess D. Great. However, Aristotle observes in the previous section that if indeed

then, alternately (ἐναλλὰξ), or alternando, as Heath has it in Book V of Euclid’s Elements,

This is Proposition V.16 of the Elements, but it requires A and C to have a ratio in the first place, in the sense of the fourth definition at the head of the book; this means that, for some n,

• if A < C, still nA > C;
• if A > C, still A < nC.

We cannot compare people and goods in this way. The general condition on A and C does not seem to bother Aristotle. Then again, it does not bother Euclid either, when he comes to prove the proposition in question.

If we accept the last proportion, then by Proposition V.18,

If we alternate again,

and therefore also

Perhaps Aristotle does not quite spell out this proportion. He does suggest it, as if to illustrate or justify the principle that if A possesses C, then B ought to possess D.

However, as an example in the Topics, Aristotle observes that a straight line parallel to the sides of a parallelogram cuts the parallelogram in the same ratio that it cuts the base. Thus, if the parts of the parallelogram are now A and B, and of the base, a and b, then

It doesn’t make sense to say

because you cannot add lengths to areas; or if you can, still, the sums A + a and B + b haven’t got a ratio by the definition above.

You can add a wad of banknotes to your wallet. If you do, and I do to mine, does it make sense to talk about whether the ratio of you to me has stayed the same? Aristotle would seem to talk about this ratio.

He may do that, only because other people do, at least implicitly. Perhaps he wants us to figure out that the idea is ultimately incoherent. If so, we haven’t done what he wants, since we continue to talk about proportional justice.

In § v.8 in the present reading, where

if a house is worth n shoes, Rackham says A + nD and B + C “are in ‘arithmetical proportion’ … with the first two terms,” so that “the builder and the shoemaker after the transaction are by an equal amount richer than they were before they began to make the articles.” This is to explain what Aristotle says, starting out with,

Now proportionate requital is effected by diagonal conjunction.

That’s Rackham’s translation of

ποιεῖ δὲ τὴν ἀντίδοσιν τὴν κατ᾽ ἀναλογίαν ἡ κατὰ διάμετρον σύζευξις.

Sachs’s is

And linking up things along the diagonal makes reciprocal exchange be proportionate.

According to his note,

An inverse proportion is involved … Worth of A’s product : Worth of B’s product :: Quantity of B’s product : Quantity of A’s product.

Again we have the ambiguity of whether Aristotle is talking about arithmetical differences or geometrical ratios.

There’s an ambiguity whether Aristotle’s “equality before the law” is ours; but I’ve made notes about that under § iv.3 below. In this context, one should consider § v.4, where “arithmetical” justice would have you strike back at the person who struck you. It could however be that the other person ἀρχὴν ἔχων:

• “has a position of authority” (Sachs);
• “[is] an officer” (Rackham).

In this case, “geometrical” justice would have you punished for striking back. All I can say is that in some ways we are all equal, and in some ways we weren’t, and it continues to be a difficulty to tell the difference.

Contents

• Chapter IV
  • The corrective [just] (τὸ διορθωτικὸν [δίκαιον])
    • involves transaction (τό συνάλλαγμα), be it
      • voluntary or
      • involuntary (§ iv.1);
    • has another form (ἄλλο εἶδος ἔχει) from the earlier kind:
      • the distributive just (τὸ διανεμητικὸν δίκαιον) is proportional (κατὰ τὴν ἀναλογίαν) (§ iv.2);
      • the corrective just is arithmetical (κατὰ τὴν ἀριθμητικήν) (§ iv.3);
    • aims to equalize
      • gain (κέρδος) and
      • loss (ζημία)

      [in each party] (§ iv.4) –

      • those do not always seem to be the right words (§ iv.5), but
      • they are how what is undergone (τὸ πάθος) is measured, and
      • the rectifying just (τὸ ἐπανορθωτικὸν δίκαιον) is their mean (τὸ μέσον) (§ iv.6).
  • The judge (ὁ δικαστής)
    • is therefore called a mediator (μεσίδιος) (§ iv.7);
    • makes sure the line is divided
      • equally (§ iv.8),
      • in arithmetical proportion (κατὰ τὴν ἀριθμητικὴν ἀναλογίαν);
    • is a halver (διχαστής) (§ iv.9).
  • The greater exceeds the less by the excesses, of
    • the greater over the mean and
    • the mean over the less (§ iv.10).
  • This
    • shows how to equalize (§ iv.11).
    • is shown in a diagram (§ iv.12).
  • Loss and gain
    • come from voluntary transactions (§ iv.13).
    • make some sense for involuntary (§ iv.14).
• Chapter V
• Chapter 8
  • The reciprocal (τὸ ἀντιπεπονθός):
    • Pythagoreans and others say it is the just (§ v.1), but
    • it coincides with
      • neither the distributive
      • nor the corrective (§ v.2),

      despite the saying of Rhadamanthus,
      εἴ κε πάθοι τά τ᾽ ἔρεξε, δίκη κ᾽ ἰθεῖα γένοιτο (§ v.3),
      because

      • people are of unequal rank (§ v.4),
      • their interaction may be involuntary (§ v.5).
  • What holds the city together is reciprocal justice,
    which is
    • by “analogy” (κατ᾽ ἀναλογίαν),
    • not by equality (κατ᾽ ἰσότητα) (§ v.6).
  • That’s why there are temples of the Graces (§ v.7).
  • Products are exchanged in numbers reciprocally proportional to their individual worth (§ v.8).
  • Likewise for the arts (§ v.9).
  • Money makes this easier (§ v.10).
  • It’s called money (τὸ νόμισμα),
    because it’s by law or custom (νόμος);
    demand or need (ἡ χρεία) is what really brings us together (§ v.11).
  • The reciprocal proportion is computed before exchange (§ v.12).
  • No demand, no exchange (§ v.13).
  • Money makes future exchange possible (§ v.14).
  • One used to have to establish a ratio for any two products (§§ v.15, 16).
• Chapter 9
  • Justice is what the just person practices by choice (§ v.17).
  • In the crime (ἀδικήμα),
    • the suffering is less,
    • the doing is greater (§ v.18).
  • That’s all § v.19.

[1131^b]

Chapter IV

§ iv.1

τὸ δὲ λοιπὸν ἓν τὸ διορθωτικόν,
ὃ γίνεται ἐν τοῖς συναλλάγμασι

• καὶ τοῖς ἑκουσίοις
• καὶ τοῖς ἀκουσίοις.

§ iv.2

τοῦτο δὲ τὸ δίκαιον
ἄλλο εἶδος ἔχει
τοῦ πρότερον.

• τὸ μὲν γὰρ διανεμητικὸν δίκαιον τῶν κοινῶν
  ἀεὶ κατὰ τὴν ἀναλογίαν ἐστὶ τὴν εἰρημένην·

  καὶ γὰρ ἀπὸ χρημάτων κοινῶν ἐὰν γίνηται ἡ διανομή,
  ἔσται κατὰ τὸν λόγον τὸν αὐτὸν
  ὅνπερ ἔχουσι πρὸς ἄλληλα τὰ εἰσενεχθέντα·

  καὶ τὸ ἄδικον τὸ ἀντικείμενον τῷ δικαίῳ τούτῳ τὸ παρὰ τὸ ἀνάλογόν ἐστιν.

§ iv.3

• τὸ δ᾽ ἐν τοῖς συναλλάγμασι δίκαιον
  • ἐστὶ μὲν ἴσον τι,
  • καὶ τὸ ἄδικον ἄνισον, [1132^a]

ἀλλ᾽

• οὐ κατὰ τὴν ἀναλογίαν ἐκείνην
• ἀλλὰ κατὰ τὴν ἀριθμητικήν.

ἀλλὰ

• πρὸς τοῦ βλάβους τὴν διαφορὰν μόνον βλέπει ὁ νόμος, καὶ
• χρῆται ὡς ἴσοις,
  • εἰ
    • ὃ μὲν ἀδικεῖ
    • ὃ δ᾽ ἀδικεῖται, καὶ
  • εἰ
    • ἔβλαψεν
    • ὃ δὲ βέβλαπται.

§ iv.4

ὥστε τὸ ἄδικον τοῦτο ἄνισον ὂν
ἰσάζειν πειρᾶται ὁ δικαστής·

καὶ γὰρ ὅταν

διῄρηται

• τὸ πάθος καὶ
• ἡ πρᾶξις

εἰς ἄνισα·

ἀλλὰ [πειρᾶται]

§ iv.5

λέγεται γὰρ ὡς ἁπλῶς εἰπεῖν ἐπὶ τοῖς τοιούτοις,
κἂν εἰ μή τισιν οἰκεῖον ὄνομα εἴη,

• τὸ κέρδος,
  οἷον τῷ πατάξαντι, καὶ
• ἡ ζημία τῷ παθόντι·

§ iv.6

ἀλλ᾽ ὅταν γε μετρηθῇ τὸ πάθος,
καλεῖται

• τὸ μὲν ζημία
• τὸ δὲ κέρδος.

ὥστε

• τοῦ μὲν
  • πλείονος καὶ
  • ἐλάττονος

  τὸ ἴσον μέσον,

• τὸ δὲ κέρδος καὶ ἡ ζημία
  • τὸ μὲν πλέον
  • τὸ δ᾽ ἔλαττον

  ἐναντίως,

  • τὸ μὲν
    • τοῦ ἀγαθοῦ πλέον
    • τοῦ κακοῦ δ᾽ ἔλαττον

    κέρδος,

  • τὸ δ᾽ ἐναντίον
    ζημία·

ὧν ἦν μέσον τὸ ἴσον,
ὃ λέγομεν εἶναι δίκαιον·

ὥστε τὸ ἐπανορθωτικὸν δίκαιον ἂν εἴη τὸ μέσον ζημίας καὶ κέρδους.

§ iv.7

διὸ καὶ ὅταν ἀμφισβητῶσιν,
ἐπὶ τὸν δικαστὴν καταφεύγουσιν·

τὸ δ᾽ ἐπὶ τὸν δικαστὴν ἰέναι
ἰέναι ἐστὶν ἐπὶ τὸ δίκαιον·

ὁ γὰρ δικαστὴς βούλεται εἶναι
οἷον δίκαιον ἔμψυχον·

• καὶ ζητοῦσι δικαστὴν μέσον,
• καὶ καλοῦσιν ἔνιοι μεσιδίους,

ὡς

• ἐὰν τοῦ μέσου τύχωσι,
• τοῦ δικαίου τευξόμενοι.

• μέσον ἄρα τι τὸ δίκαιον,
• εἴπερ καὶ ὁ δικαστής.

§ iv.8

ὁ δὲ δικαστὴς ἐπανισοῖ,
καὶ ὥσπερ γραμμῆς εἰς ἄνισα τετμημένης,
ᾧ τὸ μεῖζον τμῆμα τῆς ἡμισείας ὑπερέχει,
τοῦτ᾽ ἀφεῖλε καὶ τῷ ἐλάττονι τμήματι προσέθηκεν.

ὅταν δὲ δίχα διαιρεθῇ τὸ ὅλον,
τότε φασὶν ἔχειν τὸ αὑτοῦ
ὅταν λάβωσι τὸ ἴσον.

§ iv.9

διὰ τοῦτο καὶ ὀνομάζεται δίκαιον,
ὅτι δίχα ἐστίν,
ὥσπερ ἂν εἴ τις εἴποι δίχαιον, καὶ ὁ δικαστὴς διχαστής.

§ iv.10

§ iv.11

τούτῳ ἄρα γνωριοῦμεν

• τί τε ἀφελεῖν δεῖ ἀπὸ τοῦ πλέον ἔχοντος, καὶ
• τί προσθεῖναι τῷ ἔλαττον ἔχοντι·

• ᾧ μὲν γὰρ τὸ μέσον ὑπερέχει,
  τοῦτο προσθεῖναι δεῖ τῷ ἔλαττον ἔχοντι,
• ᾧ δ᾽ ὑπερέχεται,
  ἀφελεῖν ἀπὸ τοῦ μεγίστου.

§ iv.12

[ἔστι δὲ τοῦτο καὶ ἐπὶ τῶν ἄλλων τεχνῶν·

ἀνῃροῦντο γὰρ ἄν,
εἰ μὴ

• ἐποίει τὸ ποιοῦν καὶ ὅσον καὶ οἷον, καὶ
• τὸ πάσχον ἔπασχε τοῦτο καὶ τοσοῦτον καὶ τοιοῦτον.]

§ iv.13

ἐλήλυθε δὲ τὰ ὀνόματα ταῦτα,

• ἥ τε ζημία καὶ
• τὸ κέρδος,

ἐκ τῆς ἑκουσίου ἀλλαγῆς·

• τὸ μὲν γὰρ πλέον ἔχειν ἢ τὰ αὑτοῦ κερδαίνειν λέγεται,
• τὸ δ᾽ ἔλαττον τῶν ἐξ ἀρχῆς ζημιοῦσθαι,

§ iv.14

ὅταν δὲ

• μήτε πλέον
• μήτ᾽ ἔλαττον
• ἀλλ᾽ αὐτὰ

τὰ δι᾽ αὐτῶν γένηται,

• τὰ αὑτῶν φασὶν ἔχειν καὶ
• οὔτε ζημιοῦσθαι
• οὔτε κερδαίνειν.

ὥστε

• κέρδους τινὸς καὶ
• ζημίας

μέσον τὸ δίκαιόν ἐστι
τῶν παρὰ τὸ ἑκούσιον,
τὸ ἴσον ἔχειν

• καὶ πρότερον
• καὶ ὕστερον.

Chapter V

Chapter 8

§ v.1

δοκεῖ δέ τισι καὶ τὸ ἀντιπεπονθὸς εἶναι ἁπλῶς δίκαιον,
ὥσπερ οἱ Πυθαγόρειοι ἔφασαν·

ὡρίζοντο γὰρ ἁπλῶς τὸ δίκαιον τὸ ἀντιπεπονθὸς ἄλλῳ.

§ v.2

§ v.3

—πολλαχοῦ γὰρ διαφωνεῖ·

§ v.4

§ v.5

ἔτι

• τὸ ἑκούσιον καὶ
• τὸ ἀκούσιον

διαφέρει πολύ.

§ v.6

• ἢ γὰρ τὸ κακῶς ζητοῦσιν· [1133^a]

  εἰ δὲ μή,
  δουλεία δοκεῖ εἶναι
  εἰ μὴ ἀντιποιήσει·

• ἢ τὸ εὖ·

  εἰ δὲ μή,
  μετάδοσις οὐ γίνεται,
  τῇ μεταδόσει δὲ συμμένουσιν.

§ v.7

• ἀνθυπηρετῆσαι γὰρ δεῖ τῷ χαρισαμένῳ, καὶ
• πάλιν αὐτὸν ἄρξαι χαριζόμενον.

§ v.8

ποιεῖ δὲ
τὴν ἀντίδοσιν τὴν κατ᾽ ἀναλογίαν
ἡ κατὰ διάμετρον σύζευξις.

• οἰκοδόμος ἐφ᾽ ᾧ Α,
• σκυτοτόμος ἐφ᾽ ᾧ Β,
• οἰκία ἐφ᾽ ᾧ Γ,
• ὑπόδημα ἐφ᾽ ᾧ Δ.

δεῖ οὖν

• λαμβάνειν τὸν οἰκοδόμον παρὰ τοῦ σκυτοτόμου τὸ ἐκείνου ἔργον, καὶ
• αὐτὸν ἐκείνῳ μεταδιδόναι τὸ αὑτοῦ.

ἐὰν οὖν πρῶτον ᾖ τὸ κατὰ τὴν ἀναλογίαν ἴσον,
εἶτα τὸ ἀντιπεπονθὸς γένηται,
ἔσται τὸ λεγόμενον.

εἰ δὲ μή,

• οὐκ ἴσον,
• οὐδὲ συμμένει·

οὐθὲν γὰρ κωλύει κρεῖττον εἶναι
τὸ θατέρου ἔργον
ἢ τὸ θατέρου·

δεῖ οὖν ταῦτα ἰσασθῆναι.

§ v.9

ἔστι δὲ τοῦτο καὶ ἐπὶ τῶν ἄλλων τεχνῶν·

ἀνῃροῦντο γὰρ ἄν, εἰ μὴ

• ὃ ἐποίει τὸ ποιοῦν
  • καὶ ὅσον
  • καὶ οἷον, καὶ
• τὸ πάσχον ἔπασχε τοῦτο
  • καὶ τοσοῦτον
  • καὶ τοιοῦτον.

• οὐ γὰρ ἐκ δύο ἰατρῶν γίνεται κοινωνία,
• ἀλλ᾽
  • ἐξ ἰατροῦ καὶ γεωργοῦ, καὶ
  • ὅλως ἑτέρων καὶ οὐκ ἴσων·

ἀλλὰ τούτους δεῖ ἰσασθῆναι.

§ v.10

διὸ πάντα συμβλητὰ δεῖ πως εἶναι, ὧν ἐστὶν ἀλλαγή.

ἐφ᾽ ὃ τὸ νόμισμ᾽

• ἐλήλυθε, καὶ
• γίνεταί πως μέσον·

πάντα γὰρ μετρεῖ, ὥστε

• καὶ τὴν ὑπεροχὴν
• καὶ τὴν ἔλλειψιν,

πόσα ἄττα δὴ ὑποδήματ᾽ ἴσον

• οἰκίᾳ ἢ
• τροφῇ.

δεῖ τοίνυν

• ὅπερ οἰκοδόμος πρὸς σκυτοτόμον,
• τοσαδὶ ὑποδήματα πρὸς
  • οἰκίαν ἢ
  • τροφήν.

• εἰ γὰρ μὴ τοῦτο,
  • οὐκ ἔσται ἀλλαγὴ
  • οὐδὲ κοινωνία.
• τοῦτο δ᾽,
  εἰ μὴ ἴσα εἴη πως,
  οὐκ ἔσται.

§ v.11

δεῖ ἄρα ἑνί τινι πάντα μετρεῖσθαι,
ὥσπερ ἐλέχθη πρότερον.

τοῦτο δ᾽ ἐστὶ
τῇ μὲν ἀληθείᾳ
ἡ χρεία,
ἣ πάντα συνέχει·

εἰ γὰρ

• μηθὲν δέοιντο ἢ
• μὴ ὁμοίως,

• ἢ οὐκ ἔσται ἀλλαγὴ
• ἢ οὐχ ἡ αὐτή·

οἷον δ᾽ ὑπάλλαγμα τῆς χρείας
τὸ νόμισμα γέγονε
κατὰ συνθήκην·

§ v.12

ἔσται δὴ ἀντιπεπονθός,
ὅταν ἰσασθῇ, ὥστε
ὅπερ

εἰς σχῆμα δ᾽ ἀναλογίας

• οὐ δεῖ ἄγειν, ὅταν ἀλλάξωνται
  (εἰ δὲ μή,
  ἀμφοτέρας ἕξει τὰς ὑπεροχὰς τὸ ἕτερον ἄκρον),
• ἀλλ᾽ ὅταν ἔχωσι τὰ αὑτῶν.

οὕτως

• ἴσοι καὶ
• κοινωνοί,

ὅτι αὕτη ἡ ἰσότης δύναται ἐπ᾽ αὐτῶν γίνεσθαι.

εἰ δ᾽ οὕτω μὴ ἦν ἀντιπεπονθέναι,
οὐκ ἂν ἦν κοινωνία.

§ v.13

ὅτι δ᾽ ἡ χρεία συνέχει ὥσπερ ἕν τι ὄν,
δηλοῖ ὅτι
ὅταν μὴ ἐν χρείᾳ ὦσιν ἀλλήλων,

• ἢ ἀμφότεροι
• ἢ ἅτερος,

οὐκ ἀλλάττονται,

†ὥσπερ ὅταν οὗ ἔχει αὐτὸς δέηταί τις, οἷον οἴνου, διδόντες σίτου ἐξαγωγήν.†

δεῖ ἄρα τοῦτο ἰσασθῆναι.

§ v.14

ὑπὲρ δὲ τῆς μελλούσης ἀλλαγῆς,
εἰ νῦν μηδὲν δεῖται,
ὅτι ἔσται ἂν δεηθῇ,
τὸ νόμισμα οἷον ἐγγυητής ἐσθ᾽ ἡμῖν·

δεῖ γὰρ τοῦτο φέροντι εἶναι λαβεῖν.

πάσχει μὲν οὖν καὶ τοῦτο τὸ αὐτό·

οὐ γὰρ ἀεὶ ἴσον δύναται·

ὅμως δὲ βούλεται μένειν μᾶλλον.

διὸ δεῖ πάντα τετιμῆσθαι·

οὕτω γὰρ ἀεὶ ἔσται ἀλλαγή,
εἰ δὲ τοῦτο, κοινωνία.

τὸ δὴ νόμισμα
ὥσπερ μέτρον
σύμμετρα ποιῆσαν
ἰσάζει·

• οὔτε γὰρ ἂν μὴ οὔσης ἀλλαγῆς κοινωνία ἦν,
• οὔτ᾽ ἀλλαγὴ ἰσότητος μὴ οὔσης,
• οὔτ᾽ ἰσότης μὴ οὔσης συμμετρίας.

• τῇ μὲν οὖν ἀληθείᾳ ἀδύνατον
  τὰ τοσοῦτον διαφέροντα
  σύμμετρα γενέσθαι,
• πρὸς δὲ τὴν χρείαν ἐνδέχεται ἱκανῶς.

§ v.15

ἓν δή τι δεῖ εἶναι,
τοῦτο δ᾽ ἐξ ὑποθέσεως·

διὸ νόμισμα καλεῖται·

τοῦτο γὰρ πάντα ποιεῖ σύμμετρα·

μετρεῖται γὰρ πάντα νομίσματι.

• οἰκία Α,
• μναῖ δέκα Β,
• κλίνη Γ.

• τὸ Α τοῦ Β ἥμισυ,
  εἰ πέντε μνῶν
  • ἀξία ἡ οἰκία, ἢ
  • ἴσον·
• ἡ δὲ κλίνη δέκατον μέρος,
  τὸ Γ τοῦ Β·

δῆλον τοίνυν πόσαι κλῖναι ἴσον οἰκίᾳ,
ὅτι πέντε.

§ v.16

ὅτι δ᾽ οὕτως ἡ ἀλλαγὴ ἦν
πρὶν τὸ νόμισμα εἶναι,
δῆλον·

διαφέρει γὰρ οὐδὲν ἢ κλῖναι

• πέντε ἀντὶ οἰκίας, ἢ
• ὅσου αἱ πέντε κλῖναι.

Chapter 9

§ v.17

• τί μὲν οὖν τὸ ἄδικον καὶ
• τί τὸ δίκαιόν ἐστιν,

εἴρηται.

διωρισμένων δὲ τούτων
δῆλον ὅτι
ἡ δικαιοπραγία μέσον ἐστὶ τοῦ

• ἀδικεῖν καὶ
• ἀδικεῖσθαι·

• τὸ μὲν γὰρ πλέον ἔχειν
• τὸ δ᾽ ἔλαττόν ἐστιν.

• ἡ δὲ δικαιοσύνη μεσότης τίς ἐστιν,
  • οὐ τὸν αὐτὸν δὲ τρόπον ταῖς ἄλλαις ἀρεταῖς,
  • ἀλλ᾽ ὅτι μέσου ἐστίν·
• ἡ δ᾽ ἀδικία τῶν ἄκρων. [1134^a]

καὶ

• ἡ μὲν δικαιοσύνη ἐστὶ καθ᾽ ἣν
  ὁ δίκαιος λέγεται
  • πρακτικὸς κατὰ προαίρεσιν τοῦ δικαίου, καὶ
  • διανεμητικὸς
    • καὶ αὑτῷ πρὸς ἄλλον
    • καὶ ἑτέρῳ πρὸς ἕτερον
       
    • οὐχ οὕτως ὥστε
    • τοῦ μὲν αἱρετοῦ
      • πλέον αὑτῷ
      • ἔλαττον δὲ τῷ πλησίον,
    • τοῦ βλαβεροῦ δ᾽ ἀνάπαλιν,
    • ἀλλὰ τοῦ ἴσου τοῦ κατ᾽ ἀναλογίαν,
      ὁμοίως δὲ καὶ ἄλλῳ πρὸς ἄλλον.

§ v.18

• ἡ δ᾽ ἀδικία τοὐναντίον τοῦ ἀδίκου.
  τοῦτο δ᾽ ἐστὶν
  • ὑπερβολὴ καὶ
  • ἔλλειψις

  τοῦ

  • ὠφελίμου ἢ
  • βλαβεροῦ

  παρὰ τὸ ἀνάλογον.
  διὸ

  • ὑπερβολὴ καὶ
  • ἔλλειψις

  ἡ ἀδικία, ὅτι

  • ὑπερβολῆς καὶ
  • ἐλλείψεώς

  ἐστιν,

  • ἐφ᾽ αὑτοῦ μὲν
    • ὑπερβολῆς μὲν τοῦ ἁπλῶς ὠφελίμου,
    • ἐλλείψεως δὲ τοῦ βλαβεροῦ·
  • ἐπὶ δὲ τῶν ἄλλων
    • τὸ μὲν ὅλον ὁμοίως,
    • τὸ δὲ παρὰ τὸ ἀνάλογον,
      ὁποτέρως ἔτυχεν.

§ v.19

• περὶ μὲν οὖν
  • δικαιοσύνης καὶ
  • ἀδικίας,

  τίς ἑκατέρας ἐστὶν ἡ φύσις,
  εἰρήσθω τοῦτον τὸν τρόπον,

• ὁμοίως δὲ καὶ περὶ
  • δικαίου καὶ
  • ἀδίκου

  καθόλου.