An Aristotelian Geometry of
Just Distributon
Gerhard Michael Ambrosi
University of Trier; ambrosi@uni-trier.de
October 13, 2022
Contents
1 Introduction
2
2 The text
3
3 The geometry of just distribution
5
4 Concluding remarks
9
References
10
List of Figures
1
The geometry of the just in distribution . . . . . . . . . . . . . .
6
List of Tables
Abstract
There is much musing in the modern literature why Euclid did not arrange differently the subject matters of the Elements. We do not question the fruitfulness
of such considerations. But we argue here for taking also a phenomenological
approach. The ancient treatments of reciprocity can be seen as “phenomena” of
ancient geometrical and also of ethical discussion (Aristotle). Ours is an attempt to
have a new look at these old phenomena.
1
1 Introduction
In Aristotle’s Nicomachean Ethics the concept of “just distribution” seems to change
its appearance from being intuitively intelligible to being not knowable. In chapter 3 of Book V (NE V,3 henceforth) one reads at 1131a25 (Bekker (1831)-line):
“all agree that what is just in distributions ought to accord with a certain merit.”1 In
this passage it is taken for granted that “all agree” about what is just in distribution.
At this point this topic appears as being intuitively clear.
Subsequently, at 1131b10f., and after some elementary exercises with ratios and
proportions, Aristotle writes:
Therefore, the combination of term A with C and of term B with D is
what is just in the distribution; and the just here is a middle term,2 for
the proportion is a middle term and the just is a proportion.
Hardie (1980, p. 189) comments that this statement cannot be understood as
being based on something knowable in a mathematical sense:
There is no question here of arriving at a knowledge of the mean by
a mathematical calculation, or quasi-mathematical quasi-calculation,
which starts from a knowledge of extremes.
There seems to be a strange contradiction here between Aristotle’s repeated reference to mathematical expressions on the one hand and what sensibly can be expressed with such expressions on the other hand.
In the context of Hardie’s remark one is inclined to think of John Burnet’s (1900,
p. v) comment that “Book V is notoriously difficult owing to the use made in it of
mathematical formulas. . . which seem to give the writer almost as much trouble as
they have given to his editors.”
In view of these difficulties we suggest to reconsider Aristotle’s “formulas”.
In this we follow a remark by Reviel Netz (1998, p. 37): “The metonym of ancient science is a diagram, a visual representation.” For moderns the corresponding
metonym – the mental image – of science is “a formula, a symbolic text”.3 Aristotle’s “formulas” should then be seen in a different way than modern ones.
In a scientific or otherwise systematic deductive context – philosophical considerations about ethics in the present case – Aristotle’s letter symbols are most likely
to refer to a “lettered diagram” as discussed by Netz (1999) in a treatise about ancient Greek mathematical deduction. Rackham (2003 [1926], p. 270) already has
two footnotes that diagrams were displayed in the discussion of proportions and
of just distribution. No diagram has survived in connection with Aristotle’s text.
1 This
and the next translation by Bartlett and Collins (2011, p. 95). At p.... the authors have an
interesting comment concerning the term “the just”:
2 At this point Bartlett and Collins’ translation adds:“whereas the unjust is what is contrary to the
proportion”. This mentioning of “unjust”, adikon, is not in Bekker (1831), only as an insert in
Bywater (1890, p. 95).
3 Netz, ibid. He suggests the formula E = mc2 as “symbol of Einstein”, the modern scientist.
2
Neither Rackham nor any other well known translator and / or commentator has
made the attempt to reconstruct a diagram for just distribution as a “middle term”.
There is an interesting proposal by T.R. Keyser (1992) for a diagram in connection with NE V,3. But that proposal does not address the problematic passage
about the proportion being “a middle term”. Thus it is still a desideratum to find an
“Aristotelian” lettered diagram which deals with this problem. This desideratum
will be addressed in the following.
2 The text
After having stated that “all agree” about just distribution, Aristotle engages in
some symbolic and argumentative exercises dealing with ratios and proportions in
a just distribution. This passage addresses its readers in a remarkably elementary
mathematical style. In commenting Aristotle’s treatment of just distribution, Keyt
(1991) refers several times to Euclid’s Elements (Prop. V. 11, 16, 18, and the
definition for similarity at VI, Def. 1.).
Before and after presenting his specific proportions for a just distribution, Aristotle has some general remarks about proportions. Thomas Heath (1956, p. 131)
invokes them in his influential English translation of the Elements. He observes
that at 1131a31ff. Aristotle writes that “proportion is in four terms at least”. This,
Heath (p. 131) points out, anticipates Euclid, V, Def. 8 (“A proportion in three
terms is least.”).
Aristotle explicitly addresses the seeming difference between his statement and
Euclid’s definition – which is, of course, written many years later. In essence he
points out that a “discrete proportion” has four letters (A:B::C:D), but a “continuous proportion” has three (A:B::B:C). Since the latter has also four terms, since
one letter appears twice, the difference between the two statements is only an apparent one. Heath comments (ibid.) that Aristotle’s “distinction between discrete
and continuous seems to have been Pythagorean”. It is not clear, however, what
this means in terms of the history of such formulations.
According to Zhmud (2012, p. 265) there is a “plausible” (his term) report by
Nicomachus of Gerasa (fl. ca. 100 AD) that “the arithmetic, geometric, and harmonic proportions came down from Pythagoras to Plato and Aristotle”. Whether
Pythagoras himself was the inventor of these proportions is doubtful. But scholars
At 1131b15, after having made his formal point about the proportion of a just
distribution, Aristotle returns to this distinction, again in a very basic way by remarking that “we cannot get a single term standing for a person and a thing”, and
hence a just distribution “is not continuous”. No ethical conclusion are drawn from
the latter remark.
The details of a proportion for a just distribution are developed in Aristotle’s text
between lines 1131b5-10:4
4 The quotes in this section follow the translation by Ross (1925) as they appear at the Bekker-lines
just listed.
3
[i]As the term A, then, is to B [the persons], so will C be to D [the shares],
[ii] and therefore, alternando, as A is to C , B will be to D. Therefore also
[iii] the whole is in the same ratio to the whole [A+C:B+D::A:B];. . .
[iv] The conjunction, then, of the term A with C and of B with D is what is just in
distribution, [A+C:B+D::C:D]. . .
It is maybe of some interest to check the mathematical soundness of these items
and to list their relation to corresponding passages in Euclid’s Elements.5 But, as
already briefly mentioned, the narrow correspondence between Aristotle’s items
and the corresponding passages in the Elements should not be understood as an
expression of Aristotelian prescience. It is rather an expression that these matters were known long before Aristotle and probably were transmitted either from
Pythagoras (Zhmud) or from younger Pythagorean authors.
Item [i]: El., V, Def. 6, “Let magnitudes having the same ratio be called proportional.”
Item [ii]: El., V, Def. 12, “Alternate ratio is taking the antecedent to the antecedent
and consequent to the consequent.” (ibid.) Correspondingly, there is Prop. V,16:
“If four magnitudes are proportional, they will also be alternately proportional.”
Items [iii] and [iv]: El., Prop. V,18, “If divided magnitudes are proportional, they
will also be proportional when compounded.”
In the latter context a further relevant text item may be listed
[v] it is in geometrical proportion that it follows that the whole is to the whole as
either part is to the corresponding part.)
1131b Perseus
http://data.perseus.org/citations/urn:cts:greekLit:tlg0086.
tlg010.perseus-eng1:1131b
"The principle of Distributive Justice, therefore, is the conjunction of the first
term [A] of a proportion with the third [C] and of the second [B] with the fourth [D];
and the just in this sense is a mean between two extremes that are disproportionate,
since the proportionate is a mean, and the just is the proportionate."
Hardie
commenting on 1131b9-10
“The just share is a mean in the sense that it could fail to be according to the
proportion, i.e. could fail to be just, either by being too large or by being too
small. There is no question here of arriving at a knowledge of the mean by a mathematical calculation, or quasi-mathematical quasi-calculation, which starts from a
knowledge of extremes.”
It is naively formalistic, of course, when, after some general remarks about proportions and numbers, Aristotle writes (1131b4ff. B-C, p. 96) that the just “is
5 The
references to the Elements follow the translations in Mueller (1981, pp. 317–370): “Appendix 4, The Contents of the Elements”
4
divided into at least four terms”, and that in just distribution “the ratio is the same,
for it is divided similarly between the persons and the things involved.”6
This is then followed by several letter symbols describing proportions and their
variations in ways which anticipate some items in Euclid’s Elements (Roman numbering by the present author, other inserts by the translators):
“[i] Therefore, as the term A is to B [the persons], so also C is to D [the things];7
[ii] and so too alternately, as A is to C , so B is to D.8
[iii] So too, as a result, is the whole to the whole [(A+C):(B+D)::(A:B)]. . . 9
[iv] Therefore, the combination of term A with C and of term B with D is
what is just in the distribution;
[v] and the just here is a middle term
Some commentators, e.g. Rackham (2003 [1926], 274, n. c), are convinced that
this text is matched by geometrical diagrams.10 No diagrams have survived together with Aristotle’s text. Few commentators have endeavored to reconstruct the
associated geometrical diagrams.11 But it seems to be clear that, in the above list,
item [i] corresponds to figure 1 (a) and item [iv] corresponds to 1 (b), as will be
briefly argued below.
3 The geometry of just distribution
There is a frequently quoted author who traces the correspondance between Aristotle and the Elements.
These proportions were used in the Pythagorean harmonics, and there are
Aristotle’s presentation of a “refined” doctrine about just distribution is often
treated as if it were his very own teaching. It is in this sense that, e.g., under the
title of “Aristotle’s doctrine of justice” Kelsen (1957, p. 128) writes:
Aristotle’s definition of distributive justice is but a mathematical formulation of the well-known principle suum cuique. . . But this tautology [is]. . . legitimizing the positive law . . .
6 This
comes after Aristotle’s textbook remark at 1131a31 that “proportion is an equality of ratios,
and it involves at least four terms.” Heath (1956, p. 119) observes that with this line Aristotle
“appears to be quoting from the Pythagoreans”.
7 El., V, Def. 6: “Let magnitudes having the same ratio be called proportional.” (Mueller, p.331)
8 El., V, Def. 12: “Alternate ratio is taking the antecedent to the antecedent and consequent to
the consequent.” (ibid.) Prop. V,16: “If four magnitudes are proportional, they will also be
alternately proportional.” (Mueller, p. 333)
9 El., Prop. V,18: “If divided magnitudes are proportional, they will also be proportional when
compounded.” (Mueller, p. 333)
10 In more general terms see Netz (1999, p. 15): “Aristotle used the lettered diagram in his lectures.
The letters in the text would make sense if they refer to diagrams – which is asserted in a few
places.” Netz (n. 12) gives as example Meteor. 363a25-6, but not the Nicomachean Ethics.
11 See, however, Keyser (1992) for a longer and interesting discussion of this issue. His figure 2 is
similar to our figure 1 (a), except that he introduces “minuscule letters εζϑ for ease of reference”
to auxiliary lines (p. 142). The parallel lines of his construction are analogous to the preset lines
“A” and “B” and no additional letter-symbols are needed here.
5
(a) Proportion; Euclid,
El., Def. V,6; Prop. V,12
(b)The just as “middle term”;
cf. Euclid, El., Prop. V,18.
Figure 1: The geometry of the just in distribution
We propose, however, to see Aristotle’s presentation of this matter as part of a
dialectical argumentation which involves the stock-taking of the “reputable beliefs”
of others. In that case the following formulations express not necessarily his own
conviction, but they are candidates for later puzzlement over their implications
and / or about their limited validity.
Strictly speaking, in order to illustrate distribution according to merit as a proportion, only figure 1 (a) is needed: the ratio of merits, represented by the parallel
lines A resp. B, is equal to the ratio of just rewards which are represented by the
sections C resp. D on the perpendicular line connecting the matching endpoints of
the parallels so that two similar triangles are formed by a line connecting the outer
endpoints of the lines marked A and B.12 Above, the second item in the quote,
[ii], describes the same diagram as item [i], with specific passages in Euclid’s Elements applying to this case of “alternation”, as is indicated by a footnote attached
to item [ii].
Figure 1 (b) is described by items [iii] and [iv] above. It represents a geometrical
addition of lines C resp. D to lines A resp. B by forming two isosceles triangles
in addition to the former ones. The figure is made symmetrical by choosing the
appropriate scaling of the lines, their absolute lengths being not relevant since the
discussion is only in terms of ratios.
The symmetrical form of figure 1 (b) corresponds in appearance to an observation by Saito and Sidoli (2012, p. 140) in an article about “Diagrams and arguments
in ancient Greek mathematics”. Under the section heading “Overspecification” the
authors write (p. 140f.):
One of the most pervasive features of the manuscript figures is the tendency to represent more regularity among the geometric objects than
is demanded by the argument.
12 Since A:B must be equal to C:D it seems to be irritating that just before this list of ratios Aristotle
writes that these two ratios are similarly divided , not using the word “equal”. This expression
might be due to the geometrical solution of fig. 1 (a) producing two similar triangles, and this
might explain Aristotle’s choice of words. For his definition of geometrical similarity see Posterior Analytics. 99a12-14.
6
In many manuscripts they find “symmetry in the figure where none is required
by the text.” This tendency towards overspecification “is so prevalent . . . that it
almost certainly reflects ancient practice.” The background of this practice is that
for reasons to be discovered by specialist scholars, the ancient geometers were
prepared to adjust their scaling accordingly.
The lines of fig. 1 (b) are “overspecified” by having their respective units defined for a symmetrical appearance. The axis of symmetry is the line with the two
segments depicting “just distribution” (ratio of C:D). Although “just distribution”
gives vertical line segments which typically are not equal (typically C̸= D), the
horizontal lines to its left and right are of equal length. Thus one has here not
only an equality of ratios, their single elements typically being not equal to each
other, but also a symmetrical horizontal equality of lines, giving a visual expression of equality and of a mean between left and right. This is what is called for by
Aristotle’s text marked as item [v] above.
If one goes by fig. 1 (b) one can interpret the last aspect of Aristotle’s text fairly
easily: The line which gives the just distribution is the axis of symmetry with
sections C:D. This line is a “middle term” in so far as it stands in the middle of the
figure. But it is not the line itself which depicts the just distribution. It is rather the
ratio of the two sections on this line.
This ratio of just distribution can be stated verbally or by letter symbols as one
has it in the surviving text of the Nicomachean Ethics. In principle, any geometrical
construction could have accompanied that text as long as the issue is simply to
represent that ratio – be it now our fig. 1 (a), or any one of the two figures offered
by Keyser (1992). It is, however, only the present fig. 1 (b) which justifies the
“bloating” of the simple proportion of just distribution by invoking – or rather: by
anticipating – the Euclidean Prop. V,18 about “combining” terms of a proportion
while maintaining the original, the just, ratio, namely by having the line of the just
ratio as a middle term.
A further aspect
Without the geometry of fig. 1 (b) it is difficult to give a plausible reason why
in the text on 1131b9-10 Aristotle combines “A with C” and “B with D” and then
he claims that “the just here is a middle term”. This does not make any mathematical sense as far as arithmetics is concerned, as Hardie (1980, p. 189) observes in
commenting this passage:
There is no question here of arriving at a knowledge of the mean by
a mathematical calculation, or quasi-mathematical quasi-calculation,
which starts from a knowledge of extremes.
What is the point of Aristotle’s characterising a just distribution by equating
“the whole to the whole” to “A:B”? Commentators have great difficulties to give a
plausible reason.13 It is most unlikely that Aristotle himself “wants” this formula.14
13 See,
e.g., Keyt (1991, p. 241) who claims: Aristotle “wants his formula to display the yoking
together (hê suzeuxis)” – but why does Aristotle want that?
14 See the antecedent footnote.
7
He rather has it in order to expose a “reputable belief”. Aristotle is well aware of
the problem of commensurability of magnitudes. This appears a few lines further
down in chapter NE V,5 at line 1133b19 where he stresses: “it is impossible for
things that differ greatly from one another to become commensurable”.
Certainly the measurement of the worth of a person differs greatly from the
measurement of a piece of land.15 It is true that the just quoted line continues that
“with respect to need” relative measurement sufficiently possible. This is practiced
daily in market exchange, establishing, e.g., an exchange rate of “house : beds”
as being “1:5” (example given by Aristotle in NE V,5 on lines ) to the problem of
“homogeneity”.
In the context of tracing a dialectical argument the answer is that here the
As indicated by the captions, part (a)
the first, only in a bloated way. It might appear as being either an unnecessary
Pythagoreanism, in any case it is an anticipation of Euclid, Elements, Proposition
V,12.16
The last quoted item reveals a possible purpose of “bloating” the basic result.
By choosing an appropriate scale for the good which has to be passed out, the
“just” point can be made to appear as the intersection of two 45-degree lines which
intersect at right angles. The result has an appealingly symmetrical appearance
with a middle line depicting the result of a “just” distribution. Item three of the
quote can be interpreted as having been written for the purpose of giving a verbal
expression for the visual appearance of fig. 1 (b). This particular shape of the figure
depends on choosing the unit of accounting in a very particular way. An unfriendly
characterisation of this type of measurement might call it to be “quantophrenic”.
Its geometrical version is given by fig. 1 (b).
One link to a Pythagorean mathematical tradition comes via Aristotle’s argumentation which involves proportions. In his text the just mentioned “accord” between
merit and just distribution is determined by proportionality. This motivates Aristotle to resort to a very basic recapitulation of definitions and postulates which clearly
agree with Pythagorean doctrine. Thomas Heath (p. 119), for example, notes that
in the (present) chapter on distributive justice, line 1131a31, Aristotle writes that
“proportion is an equality of ratios, and it involves at least four terms”. In this line
“he appears to be quoting from the Pythagoreans”.17
Not only Aristotle’s formal discussion of proportionality agrees with a tradition
which “came down from Pythagoras” (Zhmud). The presentation of this tradition
and its ethical application by Aristotle seems to be a reception – maybe a caricature – of Pythagorean doctrine on “particular justice”. It is remarkably basic and
bloated.18
15 This
is the example used by Keyt, as referred to in the two preceding footnotes.
p. 160: Euclid’s theorem V,12 “is quoted [sic] by Aristotle, Eth. Nic. . . . 1131b14”. Artmann (1999, p. 125) sees “Prop. V, 12 as a simple arithmetical or geometrical consequence:
a : b = c : d ⇒ (a + c) : (b + d) = a : b ”.
17 Compare Zhmud (2012, p. 265). He considers it as being “plausible” that “the arithmetic, geometric, and harmonic proportions came down from Pythagoras to Plato and Aristotle”.
18 See above, n. ??: Aristotle (!) produces “pedestrian,. . . pompous common sense”.
16 Heath,
8
4 Concluding remarks
When Aristotle writes that in the case of a just distribution “the proportion is a
middle term”, then this should not be understood as being one of the “mathematical formulas” about which Burnet complained. It should be rather understood as
referring to the middle term in a lettered diagram, the latter being a specimen of a
means of deduction which is frequently used by Aristotle according to Netz. The
obvious problem is that in
Indeed, in recent years several authors have put enhanced emphasis on the importance of geometry in ancient scientific deduction.19 Although Netz (1999, p. 15)
does not refer to the Nicomachean Ethics in this context, it is relevant here that he
observes: “Aristotle used the lettered diagram in his lectures. The letters in the text
would make sense if they refer to diagrams”.
The issue
19 Saito
and Sidoli (2012). Netz (1998, p. 37): “The metonym of ancient science is a diagram, a
visual representation.”
9
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