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for example,1 as the line representing term one is to the line representing term two, so is the line representing term two to the line representing term three; here the line representing term two is mentioned twice, so that if it be counted twice, there will be four proportionals.)3. [10]

Thus the just also involves four terms at least, and the ratio between the first pair of terms is the same as that between the second pair. For the two lines representing the persons and shares are similarly divided2; 3. [11] then, as the first term is to the second, so is the third to the fourth; and hence, by alternation, as the first is to the third, so is the second to the fourth; and therefore also, as the first is to the second, so is the sum of the first and third to the sum of the second and fourth. Now this is the combination effected by a distribution of shares, and the combination is a just one, if persons and shares are added together in this way. 3. [12] 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,3 since the proportionate is a mean, and the just is the proportionate.3. [13]

(This kind of proportion is termed by mathematicians geometrical proportion4; for a geometrical proportion is one in which the sum of the first and third terms will bear the same ratio to the sum of the second and fourth as one term of either pair bears to the other term.—3. [14] Distributive justice is not a continuous proportion, for its second and third terms, a person and a share, do not constitute a single term.)

The just in this sense is therefore the proportionate, and the unjust is that which violates proportion. The unjust may therefore be either too much or too little; and this is what we find in fact, for when injustice is done, the doer has too much and the sufferer

1 Here the lecturer displayed a diagram.

2 Here was another diagram (one would expect the sentence to run ‘Let two lines representing . . . have been similarly divided’). Two segments, A and B, of one line represented two persons, two segments, C and D, of another their shares. It is shown that, if A:B::C:D, then A+C:B+D::A:B, i.e., if the shares are proportioned to the persons, their relative condition after receiving them will be the same as it was before.

3 i.e., A's just share lies between too large a share and too small a one, too large and too small here meaning more or less than is proportionate to A's claim. Cf. Bk. 2.6.4, third note, and 6.7.

4 We call this a proportion simply: cf. 4.3 and note.

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