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Nicomachean Ethics   

what is equal. The equal is intermediate between the greater and the
lesser line according to arithmetical proportion. It is for this
reason also that it is called just (sikaion), because it is a division
into two equal parts (sicha), just as if one were to call it sichaion;
and the judge (sikastes) is one who bisects (sichastes). For when
something is subtracted from one of two equals and added to the other,
the other is in excess by these two; since if what was taken from the
one had not been added to the other, the latter would have been in
excess by one only. It therefore exceeds the intermediate by one, and
the intermediate exceeds by one that from which something was taken.
By this, then, we shall recognize both what we must subtract from that
which has more, and what we must add to that which has less; we must
add to the latter that by which the intermediate exceeds it, and
subtract from the greatest that by which it exceeds the intermediate.
Let the lines AA', BB', CC' be equal to one another; from the line AA'
let the segment AE have been subtracted, and to the line CC' let the
segment Cd have been added, so that the whole line DCC' exceeds the
line EA' by the segment CD and the segment CF; therefore it exceeds
the line Bb' by the segment CD. (See diagram.)
These names, both loss and gain, have come from voluntary exchange;
for to have more than one's own is called gaining, and to have less
than one's original share is called losing, e.g. in buying and selling
and in all other matters in which the law has left people free to make
their own terms; but when they get neither more nor less but just what
belongs to themselves, they say that they have their own and that they
neither lose nor gain.
Therefore the just is intermediate between a sort of gain and a sort
of loss, viz. those which are involuntary; it consists in having an
equal amount before and after the transaction.
Some think that reciprocity is without qualification just, as the
Pythagoreans said; for they defined justice without qualification as
reciprocity. Now 'reciprocity' fits neither distributive nor
rectificatory justice-yet people want even the justice of Rhadamanthus
to mean this:
Should a man suffer what he did, right justice would be done -for in
many cases reciprocity and rectificatory justice are not in accord;
e.g. (1) if an official has inflicted a wound, he should not be
wounded in return, and if some one has wounded an official, he ought
not to be wounded only but punished in addition. Further (2) there is
a great difference between a voluntary and an involuntary act. But in
associations for exchange this sort of justice does hold men
together-reciprocity in accordance with a proportion and not on the
basis of precisely equal return. For it is by proportionate requital
that the city holds together. Men seek to return either evil for
evil-and if they cana not do so, think their position mere slavery-or
good for good-and if they cannot do so there is no exchange, but it is
by exchange that they hold together. This is why they give a prominent
place to the temple of the Graces-to promote the requital of services;
for this is characteristic of grace-we should serve in return one who
has shown grace to us, and should another time take the initiative in
showing it.
Now proportionate return is secured by cross-conjunction. Let A be a
builder, B a shoemaker, C a house, D a shoe. The builder, then, must
get from the shoemaker the latter's work, and must himself give him in
return his own. If, then, first there is proportionate equality of
goods, and then reciprocal action takes place, the result we mention
will be effected. If not, the bargain is not equal, and does not hold;
for there is nothing to prevent the work of the one being better than
that of the other; they must therefore be equated. (And this is true
of the other arts also; for they would have been destroyed if what the
patient suffered had not been just what the agent did, and of the same
amount and kind.) For it is not two doctors that associate for
exchange, but a doctor and a farmer, or in general people who are

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