taliation). This definition does not adequately represent either distributive or corrective justice; but the just in commerce may be defined as τὸ ἀντιπεπονθός, if by τὸ ἀντιπεπονθός is understood, not ἀντιπεπονθὸς κατ' ισότητα (retaliation), but ἀντιπεπονθὸς κατ ̓ ἀναλογίαν (reciprocal proportion), the formula being A : B :: D: C, which proportion is attained by cross-conjunction (ỷ kaтà diáμeтpov σúšEVģIS).' The following extract from Grant's commentary will serve to recal the usual interpretation of this chapter:

"Now the joining of the diagonal of a square gives us proportionate return.' The joining of the diagonal gives each producer some of the other's work, and thus an exchange is made, but the respective value of the commodities must be first adjusted, else there can be no fair exchange. What, then, is the law of value? It is enunciated a little later (§ 10). deî Toívvv-Tpoonv. 'As an architect (or a farmer it may be) is to a shoemaker, so many shoes must there be to a house or to corn.' That is, the value of the product is determined by the quality of the labour spent upon it. The sort of comparison here made between the quality of farmer and shoemaker seems connected with a Greek notion of personal dignity and a dislike of Bavavoía."

In my opinion ch. 5 should be read in close connection with ch. 2—4, the passage as a whole being an attempt at once to connect and to distinguish three kinds of particular justice. In order to connect these three kinds of particular justice, the author regards them each as áváλoyóv τɩ: in order to distinguish them, he represents each by a special and appropriate kind of ἀναλογία, the word ἀναλοyía being employed in the larger of the two senses recognized by the Greek mathematicians, and therefore including arithmetical proportion which is, strictly speaking, a μeσóτŋs. Cf. Nesselmann die Algebra der Griechen pp. 210-212, where it is shown from Nicomachus Gerasenus and Iamblichus, that, though properly avaλoyía meant geometrical proportion (all other proportions being μeσóτntes), ávaλoyía and μeσórns are frequently used synonymously for any kind of proportion. I shall henceforth use the word proportion as an equivalent for avaλoyía in its extended meaning.

Premising that in the earlier part of ch. 3 particular justice has been made to consist in rò loov, and that it has been afterwards explained that the ἰσότης spoken of is ἰσότης λόγων, or ἀναλογία, § 8, 'between the persons and the things, according to some standard' (Tрós T), $$ 5, 6, I proceed to state as briefly as possible the substance of the investigation of distributive, corrective, and commercial

justice. In the course of my summary, it will, I hope, appear, that the purpose of the author is merely to translate into the language of proportion the following proposition: 'Particular justice is attained. in distribution, correction, and barter, when the parties are, after the transaction, in the same position relatively to one another, as they were before it.' What constitutes identity of relative positions, the author does not ask. The investigation is in fact introduced in order to justify the statement made in 3 § 8, ἔστιν ἄρα τὸ δίκαιον ἀνάλογόν τι, just as the list of virtues is introduced in 11. 7 to justify the definition of virtue. But though the author's principal aim is to show that the just in distribution, in correction, and in commerce is áváλoyov Tɩ, he thinks it worth while to enter into detail and to distinguish them, because Plato had taken one kind of proportion, ἡ ἰσότης ή γεωμετ Tρɩký, as the rule of justice (Gorg. 508 A, Laws 757 A, B: cf. Plutarch Symp. VIII. 2 § 2), whilst the Pythagoreans had endeavoured to reduce all justice to retaliation, тò άνtɩeπovlós, a phrase which may be interpreted by reference to proportion.

I. The first of the three kinds of particular justice, distributive justice, in the distribution of property or honour secures to the individual a share proportioned to his desert. Desert is differently estimated in different cases: for example, in a democracy freedom constitutes desert, in an oligarchy wealth or birth, in an aristocracy άperý.

Thus distributive justice assigns to the persons concerned shares such that the position of the persons relatively to one another is not altered by the distribution, but it does not determine what constitutes alteration of relative position.

Let A, B, C, D be proportionals, so that A: B:: C: D. Hence alternando A: C :: B : D, and componendo A taken together with C: B taken together with D: A: B, which last proportion exactly represents distributive justice as above described. Or, as the author expresses it, distributive justice consists in the conjunction or composition of A and C, B and D, A, B, C, D being proportionals ( ἄρα τοῦ πρώτου ὅρου τῷ τρίτῳ καὶ ἡ τοῦ δευτέρου τῷ τετάρτῳ σύζευξις τὸ év diavoμn díkaιóv éσTɩ 3 § 12), since by such conjunction the position of the two parties, relatively to one another, is not altered, whether, as in democracy, A and B are equal, and therefore C and D, or, as in oligarchy and aristocracy, a difference is assumed between the persons, which therefore necessitates a difference in the shares assigned to them. Distributive justice then may be represented by the formula

A+ C: B+ D :: A : B.

But mathematically when A taken together with C is to B taken together with D as A is to B, A, B, C, D are said to be in geometrical proportion. Hence distributive justice is a geometrical proportion.

At this point I would call attention to 3 §§ 11, 12: WσTe Kai Tò ὅλον πρὸς τὸ ὅλον· ὅπερ ἡ νομὴ συνδυάζει· κἂν οὕτως συντεθῇ, δικαίως συνδυάζει. ἡ ἄρα τοῦ πρώτου ὄρου τῷ τρίτῳ καὶ ἡ τοῦ δευτέρου τῷ τετάρτῳ σύζευξις τὸ ἐν διανομῇ δίκαιόν ἐστι· καὶ μέσον τὸ δίκαιον τοῦτ ̓ ἐστὶ τοῦ παρὰ τὸ ἀνάλογον. Here σúlevέis seems to mean what in the language of proportion is called σúvbeois (cf. Eucl. v. Def. 15), our 'componendo;' the more familiar word being employed in preference to the technical one, because, according to strict usage, σúvOeσis can hardly be applied to the union of persons and things.

2. Corrective justice, the function of which is to remove inequality after it has arisen, deprives the gainer of his unjust gain, and restores to the loser his unjust loss, the words 'gain' and 'loss' being used in an extended sense. The author does not limit this kind of justice to the correction of aκovora ovvadλáyμara, but says expressly, 2 §§ 12, 13, 4 § 1, that it is also concerned with Ekovσia ovvadλáyματα (πρᾶσις, ὠνή, κ.τ.λ.), i. e. with the correction of voluntary transactions in which the balance has been disturbed. Cases of such disturbance will hereafter present themselves.

Now when one man has appropriated what belongs to another, the latter has as much less, as the former has more, than his just right. Hence the former is in excess of the latter by twice the amount by which the former is in excess, or the latter in defect, of his just right. Manifestly justice is attained when the unjust gain of the one is taken from him and restored to the other.

But what we have called the just right of both is an arithmetical mean between the excessive position of the one and the defective position of the other. Corrective justice is therefore represented by an arithmetical proportion in which the positions of the two parties, after the wrong and before the correction of it, are the extremes. Of course, as the author points out in 5 § 4, it may be necessary, in estimating the loss of the injured person, to take into account his superior position. It is not necessary to take into account the wrong done to the state, because we are now considering injustice of the particular kind, which consists in unfairness,—not universal injustice, which consists in the violation of law.

3.

At the beginning of ch. 5 the author criticizes the Pythagorean theory that justice consists in τὸ ἀντιπεπονθός, i. e. τὸ ἀντιπεπονθὸς

tò kat' iσórŋta, or retaliation, and objects that it does not apply either to distributive, or to corrective, justice. In commercial transactions however τὸ ἀντιπεπονθός is the bond of society: but the ἀντιπεπονθός which regulates commercial transactions is, not τὸ ἀντιπεπονθὸς τὸ κατ' ισότητα (retaliation), but τὸ ἀντιπεπονθὸς τὸ κατ ̓ ἀναλογίαν (reciprocal proportion). Now ἡ κατ ̓ ἀναλογίαν ἀντίδοσις is secured by κатà Sιáμεтρоv σúlevέis, i.e. the conjunction of A and D, B and C. For example, let A be a builder, B a shoemaker, C a house, and Da shoe. If A and B agree that a house and a shoe are of equal value, barter may take place without altering the position of A and B relatively to one another: or in the symbolism of ch. 3,

whence

A+D:B + C: A: B,

A B D : C.

:

But as barter does not take place between persons of the same trade, the transaction will be in general more complicated, C and D not being of equal value. In general then B will give to A x shoes in return for his house. Hence commercial justice is represented in general by the proportion

whence as before

A + xD : B + C :: A : B,

A : B :: xD : C.

Now when A: B :: xD : C, A and C, B and xD, are said to be reciprocally proportional (avremenov@évα). Hence commercial justice is represented by reciprocal proportion, τὸ ἀντιπεπονθὸς τὸ κατ ̓ ἀναλογίαν.

It will be observed (1) that in this explanation of ch. 5 I have followed exactly the method of interpretation adopted in ch. 3; (2) that according to my view the author not only limits the application of TÒ ȧVTITETOVós to commercial transactions, but also gives a new meaning to the phrase by the addition of the words τὸ κατ ̓ ἀναλογίαν ; (3) that I conceive the author to say no more than that ‘A and B exchange on equal terms if xD is equivalent to C, x having been determined by the higgling of the market.'

Thus, as I understand the author, he justifies in ch. 3-5 the assertion made in 3 § 8, that τὸ δίκαιον τὸ ἐν μέρει is ἀνάλογόν τι, and assigns kinds of proportion to the several kinds of particular justice. In so doing he shows controversially (1) that the yewμeтρikη iσóτηs of Plato does not include all the varieties of particular justice, and (2) that the Pythagorean theory of Tò άνTITETOVOós (retaliation) is appli

cable only to commercial transactions, and to them only if by To ἀντιπεπονθός is meant τὸ ἀντιπεπονθὸς τὸ κατ' αναλογίαν (reciprocal proportion). On the other hand he has not attempted any investigation of the laws of value, and is wholly innocent of the theory "that the value of the product is determined by the quality of the labour spent upon it." Economically, he contents himself with the statements that barter presumes mutual demand, and that the terms of the barter must be settled before, not after, the needs of the two parties are satisfied.

Before proceeding to comment upon the chapter in detail, it will be convenient to notice some other passages in which Tò avτITEπονθός plays a part.

(1) While in barter A and B exchange on equal terms wares, C and xD, which are equal in value, when proportion is used to express the claims of the superior and the inferior in friendship, A and B, and therefore C and D, would seem to be unequal; but friendship is reduced to a simple case of barter on equal terms, if we assume that the inferior is entitled to the greater amount of assistance, the superior to the greater amount of respect. Thus unequal friends barter assistance and respect, precisely as the shoemaker and the weaver barter wares. N. E. IX. I § 1. VIII. 7 § 2. 8 § I. II S$ 1 sqq. 14 § 2. Cf. Plat. Euthyphr. 15 A.

(2) It follows that a good man will not be on terms of friendship with a superior, unless the superior in rank is also superior in merit, because otherwise the inferior will not feel for the superior that love and regard by which alone he can requite superior services. N. E. VIII. 6 § 6.

(3) As however friendship in general assumes equality of persons, quantitative equality (rò κarà πoσóv) is the primary rule of friendly intercourse, i. e. the same service which A at one time renders to B, B at another time renders to A, proportionate equality (To Kar' díav, cf. Polit. v. 1. p. 195. 8) being of secondary importance. In justice, on the contrary, proportionate equality ranks first, quantitative equality second. N. E. VIII. 7 § 3. (Geometrical proportion is said to be κατὰ ποιότητα, arithmetical proportion κατὰ ποσότητα, cf. Nicomach. Gerasen. II. 21 § 5. Polit. VIII. (v.) 3. p. 198. 3.) Thus arithmetical proportion takes precedence of reciprocal proportion as the rule of friendship, because friends are in general equals and exchange actually equal services: if however the friends are unequal, the rule of friendship is proportionate, qualitative, equality, i. e. that kind of geometrical proportion which is called reciprocal.