Abstract Aristotle’s analysis of economic exchange in the Nicomachean Ethics involves two paradigms which he addresses separately but then he stresses that there is no difference between them: barter and monetary exchange. Each one of them is rendered here separately but in a mutually consistent way by using geometrical methods which were well established and widely used in Aristotle’s intellectual surroundings. In this framework Aristotle’s ‘monetary equivalence’ in exchange appears as an application of Euclid’s proposition Elements I, 43 about the equality of geometrical complements in a rectangle. Aristotle repeatedly refers to ‘own production’ when mentioning exchange between two artisans, say, ‘builder’ and ‘farmer’. The accounting worth of the quantity of ‘own production’ in terms of money is then Aristotle’s ‘worth’ of an artisan. This interpretation helps to make sense of Aristotle’s statements of the type: ‘as builder to farmer, so food to houses’. We show that this statement is logical and plausible provided that the goods in question are measured as proportions of sales out of own production. This result solves one of the major riddles of Aristotle’s text on exchange. Accounting of exchange should be seen in connection with Aristotle’s critique of the Pythagoreans’ concept of justice. He claims that they wrongly equate justice with ‘reciprocation’. The paper does not speculate about Aristotle’s alternatives. It just shows that his text on ‘reciprocation’ can be interpreted with reference to a consistent and interesting system of geometrical accounting. This system might be Pythagorean in origin, but Aristotle’s writings are its sole literary source. It justifies to list Aristotle’s passages on exchange as being among the most interesting texts of ancient economic analysis. 1. Introduction For some historians of economic thought Aristotle is the first economist of Western culture.1 Others claim that there never was and never can be any economic analysis in anything he wrote.2 The New Palgrave Dictionary of Economics tends nowadays to the latter position and dropped the entry ‘Aristotle’ from its pages.3 This is remarkable in view that in the last two centuries up to the 1987 edition Aristotle was given the honour of having an entry in that illustrious circle (Ritchie,  1923/26; Finley, 1987). In the following it will be argued that at least on account of accounting Aristotle should be acknowledged as being an important author in the history of economics. We will concentrate here on a few lines taken from Aristotle’s Nicomachean Ethics Book V, ch. 5 (‘NE V, 5’ henceforth). This text is generally considered to be extremely unwieldy ‘owing to the use made in it of mathematical formulas which are not always clear’, according to John Burnet (1900, p. xiii). That commentator believes to have found the explanation for these difficulties in the fact that ‘Mathematics was just the one province of human knowledge in which Aristotle did not show himself a master’. But maybe the fault lies not so much with Aristotle himself but rather with some of his commentators. It will be seen below that a fresh look at Aristotle’s ‘formulas’ is possible, namely one which sees them in the context of what some historians of mathematics call the ancient Greek ‘algebra of areas’.4 Under this perspective Aristotle’s ancient text will acquire some new and clearer meaning. In choosing the title for this essay, there is the problem of how assertive one can be in invoking Aristotle’s name. Can we really know the method of geometrical accounting which Aristotle knew and taught? We must admit that nobody can seriously claim today to have knowledge of what Aristotle ‘really’ wanted to say about economic exchange. But we do claim that from the following interpretation there does emerge a fairly consistent picture if the puzzle pieces of Aristotle’s text are put together as here suggested. In this sense we dare to be assertive that this interpretation does bring us nearer to analysis originally contemplated by Aristotle in the context of economic exchange. It is well possible that the formalisms which this analysis implies have their origin not in Aristotle’s own mind-set but that their roots are Pythagorean or that they are Aristotle’s interpretation of a genus of Pythagorean analysis. As Rothbard (1995, p. 16) once famously exclaimed: ‘this particular exercise should be dismissed as an unfortunate example of Pythagorean quantophrenia’.5 But to dismiss this analysis because of its vicinity to Pythagoreanism would itself be unfortunate. Aristotle’s taking up of Pythagorean ideas clearly has not an affirmative intent, but a critical one, directed against too formalistic an approach to reality. This aspect can be treated only briefly in the present context and must be elaborated in a separate treatise (see, however, Section 6.4 below for a brief outline of an interpretation of Aristotle’s critique of the Pythagorean approach to justice). A dismissal of Aristotle’s formal analysis of economic exchange would deprive us of understanding an important basis of his ethical approach. 2. ‘Reciprocation’, ‘valuation’, ‘equivalence’ Aristotle has two distinct paradigms of exchange: barter and monetary exchange. We may call them ‘reciprocation’ resp. ‘valuation’. But then Aristotle stresses that there is ‘no difference’ between the two and there is some confusion what this ‘no difference’ means in terms of accounting. It is therefore important to clarify these points. Aristotle’s first paradigm of exchange is described near the beginning of NE V,5 (1133a 7–14).6 The passage there reads (Rowe and Broadie 2002, p. 165): [‘Reciprocation’:] What brings about proportional reciprocity is the coupling of diametrical opposites. Let A be a builder, B a shoemaker, C a house, D a shoe: the requirement then is that the builder receive from the shoemaker what the shoemaker has produced, and that he himself give the shoemaker a share in his own product. It is clear that Aristotle deals here with the exchange of two things, namely houses and shoes, each being ‘a share’ of the (yearly) ‘own product’ of the respective artisans. Money is not mentioned in this context. The second paradigm—it involves monetary valuation—appears at the end of the same chapter NE V, 5, namely in its §15 (1133b 22–26). It deals with a specific amount of money—ten minae, to be exact. In the original text the amount unfortunately appears with the same symbol which previously was used for the shoemaker. We paraphrase that passage with slight modifications of symbols so that it reads (cf. Rowe and Broadie 2002, p. 167): [‘Valuation’:] Let H be a house, X ten units of currency [M]7, E a bed: house H is half of the amount X [i.e. 1210M=5M], if one house is worth five currency units [ pH=5M], or is equal to that amount, and the bed, E, is a tenth part of X [hence pE=110X=1M]. It is clear, then, how many beds equal the house, i.e. five. Our price symbols pH for one house resp. pE for one bed are not used as such by Aristotle but he describes prices and their significance in words and numbers, as the quoted text clearly shows. Our price symbols are not an extension but a paraphrase of Aristotle’s text. Indeed, a few lines before this example Aristotle expressly emphasizes the importance of prices (1133b 15, Rackham,  2003, p. 287): Hence the proper thing is for all commodities to have their prices fixed … Money then serves as a measure which makes things commensurable and so reduces them to equality. Thus, Aristotle stresses that prices are essential for making ‘things commensurable’, i.e. for making them measurable in terms of money and in order to be able to formulate budgetary ‘equality’. The present valuation example is nowadays best interpreted in terms of algebraic symbols and numbers. We will see presently that Aristotle himself most probably followed quite a different procedure, namely a geometrical one. But in a preliminary attempt to structure the text as just quoted we may first proceed in this algebraic way. We note then that in this example there is a money-value for houses pH×QH=5M, when, as the text states, there is a house-price pH=5M and a house-quantity of QH=1 as also stands in the text. Aristotle concludes in this passage that these 5M of money-value can be seen to be equal to the money-value of a number of beds, namely five, because when the bed-price is pE=1M then it is a bed-quantity of QE=5 which fulfils the equation pE×QE=5M so that the money-value of one house is equal to the money-value of five beds. Aristotle concludes the discussion of this example with the statement that there is ‘no difference’ between monetary equivalence and barter. This is somewhat paradoxical because there is an obvious difference between using money in exchange and not using money, namely in barter. Some modern commentators seem to have difficulties in detecting wherein this ‘no difference’ lies, as we will see.8 The text says (Rowe and Broadie 2002, p. 167): Clearly, that is how exchange took place before currency existed; for it makes no difference whether five beds are exchanged for a house, or currency to the value of five beds. The last subclause clearly describes an equation between ‘currency’ worth five money units and the money-value of five beds. It may be written as 5M=pE×5QE if we follow the argumentation just made (since pE=1). But what is here the formal relation of ‘beds’ and ‘house’ according to Aristotle’s text? According to Meikle’s authoritative book on Aristotle’s Economic Thought (1995),9 the definitive answer to this question is the expression ‘5 beds = 1 house’.10 But if we turn to the Greek text of this passage, then we see that at this particular point (Bekker 1831, 1133, line 27, side b) Aristotle does not write isos for ‘equal’ but we rather find the preposition anti, ‘against’. The best translation of this passage is then that ‘five beds are exchanged against one house’ so that in this particular context the most appropriate symbolic rendering of Aristotle’s statement of the relation between beds and house should not involve an ‘=’ sign but a sign for a proportion or fraction, hence a ‘:’ sign. But if the meaning of Aristotle’s example is: ‘beds exchange against a house at a rate of 5 to 1’, then it should be rendered symbolically as a) QE:QH::5:1 ; b) 5M=pE︸1M/1QE× 5QE=pH︸5M/1QH× 1QH (1) where part (1 a) of this expression shows that Aristotle referred here to an exchange rate in the sense of a specific proportion and not in the sense of an equality.11 But part (1 b) shows that there is indeed also an equality involved, but one of money-values, not one of goods. The braces below the price symbols in (1 b) show that prices are in terms of money per one unit of the good for which the price is noted. With this interpretation of Aristotle’s conception of prices we differ from Meikle (1995, p. 16), who writes that for Aristotle ‘prices … are simply numbers, 1, 5, 100, 11, each representing so much exchange value expressed in money’. In our understanding of Aristotle’s concept of a price the word ‘money’ in this quote must be replaced by the expression ‘money per one unit of the respective good’—as indicated under the braces in eq. (1b).12 It is only in this case that the multiplication of a good with its respective price as ‘money pro good’ makes the dimension ‘good’ (symbol QH or QE) cancel so that all terms become ‘commensurable’ just in money, the latter aspect being stressed repeatedly by Aristotle in sections 14 and 15 of NE V, 5. We quoted above already (from Section 14) that Aristotle stressed that money does make ‘things commensurable’—because ‘all things can be measured by money’, as Aristotle repeats in Section 15 (Rackham,  2003, p. 287). Aristotle’s claim that there is ‘no difference’ between barter exchange and monetary exchange can preliminarily be interpreted as being based on accounting statements which follow from monetary ‘equivalence’—a concept which Aristotle stresses repeatedly in NE V, 5. Thus, generalising eq. (1b) to apply to any two goods with the indices ‘1’ resp. ‘2’, we can write the following monetary equivalence eq. (2a), which states a money-value M as resulting from an evaluation of goods with their respective prices.13 ‘Equivalence’a) M=p1×Q1=p2×Q2 ⇔ b) Q2Q1=p1p2 (2) Rearrangement of terms then gives (2b) (disregarding M) and this expression may be re-transformed again to (2a) so that we may regard the two versions not only as expressions of equivalence but also as equivalent expressions.14 This latter type of equivalence was a major mathematical insight at the time of young Aristotle, as we may gather from Artmann (1985) —a point to which we will return later. The two expressions (2) show the two aspects of exchange, namely monetary equivalence and ratios of barter exchange, but both aspects are just variants of one concept, namely ‘equivalence’, in so far as they stand in a unified framework. It might be argued that the juxtaposition of the barter sphere of exchange as shown by part (2b) against the monetary sphere of exchange as shown by part (2a) is not quite satisfactory here because, after all, we combine the relative money price p1p2 with the supposedly money-less barter ratio of exchange Q2Q1. But for the moment we must leave the matter here as it stands. The separation and unification of the barter sphere and the monetary sphere of exchange will be clarified further in the subsequent geometrical discussion. 3. Reciprocation as a geometrical figure At least since the scholastic middle ages, Aristotle’s paradigm of ‘reciprocation’ has inspired commentators to resort to geometry in the attempt to clarify its meaning. This seems to be due (i) to tradition, (ii) to context, and (iii) to wording. (i) As far as tradition is concerned, even the Arabian and Byzantine comments which predated the scholastic ones seem to have abounded with geometrical illustrations of the passages just quoted. (ii) This is not surprising in so far as the chapters preceding NE V, 5—namely ch. 3 dealing with ‘distributive justice’ and ch. 4 dealing with ‘corrective justice’—all have references to geometrical constructions.15 (iii) The wording of the passage on Aristotle’s ‘reciprocation’ clearly alludes to geometry. If the readers return to the above quote of the first paradigm of exchange, then they will notice the mentioning of ‘diametrical opposites’. This means that there are ‘diameters’ to be drawn, that they connect or separate—in any case they ‘couple’—‘opposites’, and that these geometrical concepts illustrate ‘proportional reciprocity’, i.e. comparisons of relative length of lines in geometrical constructions. A typical example of a medieval illustration of Aristotle’s ‘reciprocation’ is given by Fig. 1(a), which is an anglicised version of a figure with Latin labels as given by Albertus Magnus (, 1972, p. 343). This figure lists Aristotle’s four symbols A, B, C, D at the corners of a rectangle and connects them with two diagonals. Odd Langholm (1979, p. 15), an outstanding scholar of the medieval reception of Aristotle’s text, comments on Aristotle’s original passage and its medieval rendering disparagingly with the remark: Fig. 1. View largeDownload slide A medieval illustration and an alternative figure of Aristotelian barter. Fig. 1. View largeDownload slide A medieval illustration and an alternative figure of Aristotelian barter. With this obscure saying Aristotle introduces a quasi-mathematical argument in four terms, corresponding—according to the Latin tradition—to the angles of a square In short, even for experts of this type of literature the ‘Latin angles’ interpretation of Aristotle’s description of exchange does not appear to be very illuminating. But Langholm (1979, p. 16) acknowledges that there does emerge some interesting perspective from this and similar passages in NE V, 5: what emerges from this obscure passage is a picture composed of … the production and marketing determinants of the exchange rates of commodities. Thus, there is some expert opinion that Aristotle’s intention in this context was the rendering of ‘exchange and production’—and this is by no means uninteresting as a topic for economic analysis. Indeed, a well-known textbook of economics once had several printings under this very title.16 If we accept ‘exchange and production’ as the background of Aristotle’s passages about reciprocation, then an elementary, but appropriate, geometrical rendering should be the one given by Fig. 1(b): It covers both of these aspects. The horizontal line marked with letter A measures the total quantity of good ‘C’, houses, produced by builder A during a year. The quantity of houses given up in exchange may then be represented by a line segment, marked QC, as shown. This notation prefixes the symbol ‘Q’ before Aristotle’s symbol ‘C’, for ‘house’, in order to stress that the measurement is here in quantities and not in money-values or in any other term of valuation. In order to simplify later reference to this geometry it is convenient to have a symbol also for the total production as given by the total length of the line just drawn. Let us refer to it later by using the symbol ‘ QC*’, which stands for the total ‘own production’ of the quantity of good C, produced by builder A. Similarly, the total quantity of product ‘D’, shoes, which are produced by shoemaker B, may be represented as a vertical line running from point B downwards in Fig. 1(b). Its analogous symbol for later reference is QD*. The partial quantity of B’s total own production which is given up in exchange is then given by line segment QD. The activity of house building defines builder A in a qualitative way as a specific artisan. But since it is also a specific number of houses QC* which citizen A produces during one year, A has also a quantitative economic aspect. Likewise, the production of QD* shoes defines the specificity of citizen B in a qualitative sense as a maker of shoes. But this shoe-making has also a quantitative aspect which can be exactly measured, namely QD*, the total quantity of shoes produced during, say, a year. It is important to be clear about the quantitative aspects of house building and shoe making, because in several passages Aristotle refers to his artisans in the context of proportions, using their proper symbols ‘A’ and ‘B’. This is somewhat confusing. As we just quoted Aristotle, it is money which ‘serves as measure’. Thus it is problematic to associate these letters with the quantities of own production in a monetary accounting context. For the moment we therefore use the letters A and B not as standing for quantities or money-values, but just for a connotation of the two different artisans, leaving a further clarification for a later occasion below.17 From the perspective of builder A the ratio of line segments QD/QC measures the amount of shoes which can be obtained per house which is given up in exchange. From the perspective of shoemaker B this ratio measures the number of shoes given up per house obtained. In short, the slope of the diagonal in Fig. 1(b) measures the ‘barter terms of trade’ of the modern theory of bilateral exchange. The point where the two ‘own production’ lines meet may be called the ‘endowment point’ of the two traders, and the whole setup of Fig. 1(b) may henceforth be referred to as ‘production and barter box’. The characterisation of some geometrical properties of our figure by giving them modern names does not affect these properties themselves and their (putative) relevance in an ancient context. The proposed figure is so simple and intuitive that it may well be attributed also to the time of Aristotle in spite of the modernistic touch of the names just mentioned. They allude to the terminology and diagrams of the modern theory of bilateral exchange. As the modern diagrams are meant to be purely descriptive and meant just to coin concepts and to structure them, so does our present interpretation of the ancient Aristotelian ‘reciprocation’. All that the present figure expresses is that there is production and exchange, that the respective quantities can be measured and that these quantitative measures can be set in relations to each other geometrically as lengths of line segments, thus giving a geometrical expression of an exchange ratio. The ‘reciprocation’ passage as quoted at the beginning of the preceding section has the word ‘requirement’. This word might be interpreted in a normative sense. But that is not its meaning in the present context. The ‘requirement’ mentioned here is a logical one: if there is exchange, then it is necessarily the case—i.e., there is the ‘requirement’—that one good is given up in order to obtain another good. Goods could change hands in many other ways, of course. But then we would not have exchange but robbery, extortion, donation, etc. All the latter situations might be relevant topics for philosophical reflection, but they are other topics. The one here under consideration is exchange and that topic has certain logical requirements which can be quite independent of reflections about justice. In view of the last section’s considerations concerning Aristotle’s juxtaposition of barter exchange and monetary exchange, it may be observed that in Fig. 1(b) we now have a ‘clean’ concentration on barter exchange. Money plays no role in this geometrical setup. 4. Monetary values as geometrical figures As Aristotle’s reciprocation can be given a geometrical rendering, so can his second paradigm concerning valuation. This is shown in Fig. 2(a). We see there two hatched rectangles, enveloped by a larger one. The length of the horizontal side of the upper hatched rectangle represents the ‘house’ of Aristotle’s second paradigm of exchange, i.e. it stands for QH=1 of that example. The length of the vertical side of this upper rectangle measures the price pH=5M as discussed above. The hatched area of the upper rectangle then stands for the product pH×QH=5M, given the numbers of that example. Fig. 2. View largeDownload slide The geometry of monetary equivalence. Fig. 2. View largeDownload slide The geometry of monetary equivalence. The upper hatched rectangle has its mirror image in the lower part of Fig. 2(a). There the horizontal line measures the price pE=1 of a bed of Aristotle’s example. It follows then immediately that the other side of the rectangle must have the value QE=5, because only then the area which measures now pE×QE=1×5M=5M will match the former area and hence the former money-value as Aristotle’s example required. Thus Fig. 2(a) represents a case when there is monetary equivalence of one house with five beds—and it also illustrates the nature of this equivalence, namely an equivalence of areas; and these areas are defined by length and width, i.e. by two dimensions. Illustrating now graphically that what was stated verbally above already when relating Aristotle’s accounting exercise, we stress: it is not the length of the QE-line of five beds which is equal to ‘so much money’ (Meikle)18 but the hatched area which is equal to that amount of money. Also, it would be obviously and visibly wrong to say that the ‘five beds line QE’ is equal to the ‘one house line QH’. It is the areas which are constructed on the basis of these lines which give the correct equality. This geometrical figure shows visually that we are only permitted to equate the money-values of house and beds. Aristotle wrote, in symbolic translation, ‘ pH×QH=pE×QE’. It would be a misrepresentation to claim anything else. Figure 2(a) is also helpful for representing the corollary of the equivalence which was stated by Aristotle when he went on to stress the correspondence between barter exchange and monetary exchange as related above and as rendered algebraically in eq. (1). The diagonal through Fig. 2(a) gives two identical ratios. In the lower left context the slope of the diagonal measures the barter terms of trade QE/QH. In the upper right context the slope measures the terms of trade expressed as relative price pH/pE. Thus, this aspect of the figure represents eq. (1b) while the former aspect, focussing the hatched areas, represents eq. (1a). Figure 2(b) shows that the same type of geometrical accounting as given by Fig. 2(a) is also possible with regard to the transactions between the two artisans already mentioned. There, the number of houses QC of builder A’s total own production which is given up in exchange is denoted by the corresponding horizontal line in the hatched lower rectangle. The area of the rectangle measures the total money worth of sales by builder A. These sales must be bought by the rest of society, represented by shoemaker B. But in order that shoemaker B can buy, this household must have money through sales of shoes. The number of shoes which are sold is given by the vertical line QD. Knowledge of the money worth of the sale of shoes requires knowledge of their price pD. The total sales by shoemaker B are now given by the area of the upper rectangle. Since this area is equal to the lower one, it is clear that the accounts match and the respective sales are sufficient for the required purchases. The figure also illustrates the already mentioned equality of ‘price terms of trade’ with the ‘barter terms of trade’, namely along the diagonal drawn. 5. Geometry in the Aristotelian context 5.1 The mundane context The geometrical interpretations presented so far might well appear as being just fanciful inventions which have very little relation to anything Aristotle wrote or was interested in. But this impression should change once we see him in his specific context. Thus, if we focus on his everyday context it is clear that Aristotle was constantly confronted with accounting and geometry. The Athenian community in which he lived for many years was a highly numeric one in which accounts and accounting played a large role in public and in private life (Cuomo 2011). Wealth accounting was primitive by modern standards19 but of utmost importance in public and private life. Isocrates (436–338 BC), an older contemporary of Aristotle, writes that some Athenians bewail ‘their poverty and privation while others deplore the multitude of duties enjoined upon them by the state’. Norlin (1929, pp. 88–89), the translator, comments on this passage and mentions the proviso that if a man thought that ‘another could better afford to stand the expense [of public duties, GMA] he had the right to demand that he do so or else exchange property with him’. This must have resulted in constant comparisons and quantifications among citizens concerning their relative wealth and income.20 But not only in comparisons of wealth among citizens was there ample reason for detailed computation. The finances of the Athens-lead Delian League of the 5th century BC afforded detailed lists of monetary contributions from all over Greece. The war finances, the public works of which the Parthenon temple on Athens’s Akropolis is only a very partial example, the foreign trade—they all required detailed budgets and budgetary control on all levels of society. Accountants were specialists mentioned in a public Athenian document by 426/5 to 423/2 BC.21 But it was not just numbers and calculations of budgets which were all-pervasive, it was geometry and geometric proportions which were particularly present in public life—in the discussion of the geometrical proportions of temples like the Parthenon temple22 or of the patterns of entire cities. By 414 BC the latter point was so pervasive in public life that Aristophanes could make this topic the laughingstock for his audience when in his comedy The Birds he exposed a city planner to ridicule.23 A remarkable manifestation of extraordinary interest in geometrical constructions may be found on contemporary coins. Thus the mathematical historian Benno Artmann referred to the geometrical problem of doubling a square which features at considerable length in Plato’s dialogue Meno. Artmann (1990, p. 44) observes that the corresponding figure was found on a coin from the island of Melos which must have antedated the year 416 BC, and he gives the comment This component of elementary instruction in geometry seems to have been so popular that it found its way into the decoration of coins. But the most striking vestiges of this type are the coins from the island state of Aegina. Since earliest times of coinage the Aeginetans had geometrical patterns on the reverse side of their coins and since about 400 BC until 330 BC the design on all their denominations was virtually the same as the one of our Fig. 2(b). A specimen of such a coin is reproduced here as Fig. 3.24 Fig. 3. View largeDownload slide An Aeginetan coin of ca. 350 BC (cf. Fig. 2(b) above). Fig. 3. View largeDownload slide An Aeginetan coin of ca. 350 BC (cf. Fig. 2(b) above). The only difference between this coin’s geometric design and the geometry which we attribute to Aristotle is that on the coins there are not our letters but different ones. Obviously the coin’s Greek letters refer to its origin, the island of ‘A-IG[INA]’.25 Also, the upper part of the diagonal is missing on the coins’ design. Artmann (1990, p. 47) reproduces such a coin and comments it with the claim that this ‘is precisely the diagram of Euclid’s … geometrical version of the binomial theorem (a+b)2=a2+2ab+b2’ in proposition II.4 of the Elements. But he adds: ‘Similarly subdivided rectangles and parallelograms abound in the Elements.’ Thus, one can invoke a number of other propositions as well in connection with this coin’s design, as will be argued in more detail below. The mathematical details of this type of coin design should be discussed in a different context.26 Here we may just stress that these coins and their design were contemporaries of Aristotle’s almost entire life in Athens. He knew them very well and he expressly referred to Aegina in a monetary context.27 5.2 The Academic context Aristotle was a student of Plato’s Academy for about 20 years. Tradition has it that at its entrance there was the inscription ‘Let no one enter who cannot think geometrically’. Whether this detail is true or not might be left open. But it is generally accepted that mathematics played a very great role at Plato’s Academy.28 Plato’s writings give ample proof that geometrical argumentation was often used in his surroundings. He refers even to his philosophical mentor Socrates in this sense. In Plato’s dialogue Gorgias 508 we have Socrates scolding his dialogue partner Callicles with the words (Jowett 1892, p. 400): you seem to me never to have observed that geometrical equality is mighty, both among gods and men; you think that you ought to cultivate inequality or excess, and do not care about geometry. There is much discussion about the exact meaning of the term ‘geometrical equality’ in this context (see Burkert, 1972, 78, n. 156). Carl Huffman thinks that possibly this passage expresses Plato’s appreciation for his contemporary and friend Archytas of Tarentum. From Archytas we have a fragment praising ‘calculation’, because on this basis redistribution becomes possible: ‘the poor receive from the powerful, and the wealthy give to the needy’, as Archytas claims (Huffman, 2005, p. 183). Burkert (ibid.) thinks that Plato praises here ‘the power of mathematics that governs the world’ not in a specific sense but in a more general one. It is well known, however, that Aristotle did not accept uncritically all of Plato’s teachings. But even if Aristotle should have opposed such high esteem for geometry as we find it in Plato, there can be little doubt that Aristotle was familiar with the application of geometrical argumentation in many areas of science and philosophy (Heath, 1949). He cultivated a synthetic way of argumentation in the sense that he often tried to build on his predecessors’ doctrines. It is imaginable that Aristotle’s accounting is just a rendering of the customary formalisms of his surroundings and not his own invention. But in so far as we have no direct information about those surroundings we must treat his formalisms as being part of his own teachings. A further point should be made when contemplating Aristotle’s Academic context: ‘the banausic anxiety of the ancient upper classes’, as Netz (1999, p. 303) called it, the fear to appear to be as practical as an ordinary craftsman. This explains why (p. 305) Greek mathematics is the product of Greek elite members addressing other elite members. Commercialism is not an issue, of course. These observations characterise a problem and an explanation for Aristotle’s treatment of economic exchange. There is a problem when, in his economic examples, Aristotle goes so deep into the banausic world that he deals at length about craftsmen like shoemaker, farmer, builder. This milieu of practical activity is alien, even anathema, for Aristotle’s elitist audience and colleagues. But these observations offer also an explanation for the specific way of Aristotelian accounting: his reference to geometrical diagrams in this context is totally remote from anything any craftspeople or other accountants were likely to do. Thus Aristotle is saved from the impression of too deep an involvement with mundane banausic matters. 5.3 The general analytical context Aristotle’s geometrical context was not only characterised by the fact that a number of ancient Greek city-states had geometrical motifs on their coins or that Plato had a somewhat eccentric liking for geometry which might have rubbed off in some way on Aristotle also. Greek formal thinking in general was geometry-oriented to an extent which is rarely appreciated in modern philological comments. But there are many helpful indications to this extent given by the mathematical historian Thomas Heath (1921) and by some newer contributions to the history of mathematics. Heath (1921, p. 1) pointed out that ‘pure geometry, supplemented, where necessary, by the ordinary arithmetical operations’ were the ‘only methods’ at the ancient Greeks’ disposal. What nowadays is discussed in terms of algebra was then discussed in terms of ‘geometry of areas’ (Szabó,  1978, passim) or ‘geometrical algebra’ (Heath). Of special importance in that context were propositions on the ‘application of areas’, a geometrical method based ‘on the discoveries of the Muse of the Pythagoreans’ as Heath (1956A, p. 343) relates in connection with El. prop. I,44, quoting a comment by Proclus who in turn quotes the ancient mathematician Eudemus, a student and close collaborator of Aristotle. Thus we have a statement attributed to a contemporary of Aristotle confirming that the propositions of this type of geometry are very old and traceable at least to the beginnings of the Pythagorean school. This type of analytical thinking was old-established by Aristotle’s time. One of Euclid’s propositions in the Elements, I,43, is of particular analytical interest here. In Artmann’s (1999, p. 40) version it reads: In any rectangle the complements of the rectangles about the diagonal are equal to one another. Its illustration as well as the one for El. prop. I,44, the proposition which was just referred to as being paradigmatic for the ancient Greek method of ‘application of areas’ has about the same geometrical appearance29 as that for the ‘binomial’ proposition El. II,4, which Artmann 1990 associated with the Aeginetan coins of the type of our Fig. 3. Thus it could very well have been proposition I,43 rather than II,4 which Aristotle and his disciples might have thought of when seeing the geometrical pattern of the Aeginetan coin just reproduced. In the present context prop. I,43 can be illustrated with reference to our Fig. 2(a) as drawn above. We have there a rectangle which is cut by a diagonal. The proposition then states that the hatched rectangles on the two sides of the diagonal—named ‘complements’ in the above—always have the same area. This equality is obviously and visibly true in the very special case of Fig. 2(a), where one rectangle is supposed to have the dimension ‘ 1×5’, the other the dimension ‘ 5×1’. It is trivial that both have the same area in this special case. But there are cases where such an equality is not that obvious as, e.g., in Fig. 4(b) further down in this text. There the dimensions of complement rectangles are such that they are not symmetrical anymore. It is remarkable that the ancients long before Aristotle have discovered and probably used in mathematical proofs that the equality of complements holds true for any dimensions of complement rectangles. Fig. 4. View largeDownload slide Comparative accounting of income, price, and quantities Fig. 4. View largeDownload slide Comparative accounting of income, price, and quantities The substance of proposition I,43 has an enormous significance in the context of Aristotelian accounting. We have observed already that the rectangles involved in the geometry of monetary exchange can be interpreted as money-values. It follows therefore that the equality of the ‘complements’ of proposition I,43 can be used as a geometrical rendering of Aristotle’s principle of monetary equivalence.30 Thus, when we have modern complaints that Aristotle did not state properly the ‘mathematical formulas’ (Burnet as quoted in the introduction above) which he used, then we can retort that it just was not the case that at his time algebraic relations were rendered in modern-type ‘formulas’. This does not mean that we must not render Aristotle’s accounting in algebraic terms. It means rather that we should first of all interpret Aristotle’s accounting relations on the basis of their most likely geometrical counterparts. 5.4 The specific context of commensurability Several times already we had the occasion to quote Aristotle’s multiple attestations that money ‘makes things commensurable’. Meikle (1995, p. 12) believes that Two-thirds of the chapter [NE V,5, GMA]… are devoted entirely to the problem of explaining how this commensurability is possible.31 Meikle (p. 25) claims, however, that after several abortive beginnings eventually ‘Aristotle is giving up as a bad job the attempt to explain commensurability’. This is a strange and implausible claim. After all, Aristotle went out of his way to stress in several formulations the exact opposite of this claim, namely that money definitely does make ‘things commensurable’. As emphasized above, money-values can be represented geometrically as rectangles and for any rectangle one can always construct a geometrical complement in the sense of Euclid, El. I.43. Such complements are equal by mathematical proof. Therefore, one can safely infer that the areal value of either one such area can be taken as the measure of its complement. Thus these geometrical areas are commensurable and the money-values which they represent are likewise. This was well known to Aristotle. Aristotle certainly was competent to make correct statements about (in)commensurability. He knew perfectly well its mathematical aspects.32Artmann (1999, p. 229) writes and substantiates that for Aristotle incommensurability is ‘the prototype of a scientific discovery’. Heath (1949, p. 22) points out that Aristotle relates the ‘well known’ proof of incommensurability per impossible which later was listed as Euclid’s prop. El. X,117. Meikle’s claim that in Book V of the Nicomachean Ethics Aristotle rambles on inconclusively about problems of commensurability is at variance with Aristotle’s own statements about the appropriate treatment of this topic in the Nicomachean Ethics. In Book III,3 (1112a 22–24) he writes that ‘about eternal things no one deliberates—for example, about … the fact that the diagonal and the side of a square are incommensurable’ (Bartlett and Collins, 2011, p. 48). Aristotle becomes very drastic in this context, writing (ibid.) that only ‘a foolish or mad person’ would engage in such futile deliberations. The statements in Book III refer to incommensurable lines. In economic exchange it is physical ‘things’ which are incommensurable, like the house and the beds of Aristotle’s famous example. But in both cases Aristotle clearly focusses on incommensurability. Immediately before the house and beds example, Aristotle expressly writes (1133b19): ‘Now, in truth, it is impossible for things that differ greatly from one another to become commensurable’ (Bartlett and Collins, 2011, p. 102). We may safely attribute to Aristotle the knowledge that as it is eternally so that lines representing integer numbers are not what we call33 lines that represent irrational numbers in a geometrical context, so it will be eternally true that beds are not houses in a context of economic exchange. It is now most implausible that in Book III of the Nicomachean Ethics Aristotle proclaims that it would be ‘foolish’ to deliberate about these ‘eternal’ aspects of incommensurability of lines, but that in Book V of the self-same treatise he himself should spend ‘two-thirds’ of chapter V,5 in futile deliberations of the analogous aspect of the objects of exchange. As Aristotle knew the incommensurability of sides and diagonals of squares, he also knew that the areas associated with these lines are not affected by this problem. In this context it should be noted that in Book X of the Elements, Euclid refers to some lines as being ‘commensurable in square only’, thus subsuming rational lines representing, say, the number ‘1’ and lines representing, say, what we would now call the ‘irrational’ number ‘ 2’, both under one heading as ‘rational’ lines.34 We may note with Reviel Netz (1999, p. 275) that Aristotle’s ‘use of mathematics betrays an acquaintance with mathematics whose shape is only marginally different from that seen in Euclid’. As a mathematical issue, incommensurability was settled by Aristotle’s time after previous intensive consideration. In geometrical and arithmetical contexts, linear incommensurability was proven to exist so that since well before Plato’s time already philosophers have been thoroughly aware of problems of conceptual incomparability. But simultaneously there has also been awareness of possible solutions, namely by shifting attention from the ‘problematic’ lines to associated unproblematic areas. In our view the geometrical analysis of exchange in NE V,5 has an analogous pattern: economic goods like beds and houses are acknowledged to be incommensurable; but when there is consistent accounting, then the money-value of five beds is very well commensurable with that of one house. Aristotle’s beds and house example illustrates his solution for the acknowledged problem of the lack of direct commensurability of specific physical goods. Commensurability is a necessary prerequisite for meaningful accounting, for interpersonal comparisons and eventually for an evaluation of the fairness of a specific distribution of goods in a society. If we cannot accept that Aristotle did master commensurability in the context of exchange, then it is not conceivable how he could possibly have entered a meaningful discussion of fair distribution of goods in an exchange economy because he would not have had a coherent concept, let alone a measure, of a distributable entity. 6. Comparative geometrical accounting 6.1 synthesis of barter and monetary exchange Returning now to the geometry of exchange as presented so far, we recall that Fig. 2(b) represented the monetary side of the barter exchange. Barter exchange was depicted separately by Fig. 1(b). In the spirit of Aristotle’s unified perspective of these two aspects of exchange, we may combine the two figures as interlocking squares, as shown in the following Fig. 4(a). In Fig. 4(a), we have a replica of Fig. 1(b), which appears here as the square between A and B, marked by four dots.35Fig. 2(b) appears in the lower part of this figure and is marked as the square with the broken circumference. The symbols have the same meaning as in Fig. 2(b), and the two hatched rectangles around the diagonal again represent the fulfilment of monetary equivalence in bilateral trade. The horizontal line from point A to the dot on the diagonal measures again the total ‘own production’ QC* by builder A. Likewise, the vertical line from point B down to the dot on the diagonal measures shoemaker B’s total ‘own production’ of shoes QD*. Aristotle’s term ‘own production’ corresponds in the ‘modern’ Ricardian theory of exchange to the term ‘real income’.36 The upper part of Fig. 4(a) thus represents the ‘production and barter box’ of Aristotle’s exchange model; the lower part represents the corresponding ‘monetary exchange box’. The two ‘boxes’ share the small square in the middle with sides QC and QD. They also share the ‘exchange-line’ giving the barter terms of trade QD/QC as well as the price terms of trade pC/pD. Thus the self-same exchange rate can be seen either in a context of barter (upper ‘parent’ square) or in the context of monetary equivalence (lower ‘parent’ square). This exchange rate between goods is contained in the small sub-square which belongs to both the upper, ‘real’, encompassing square as well as to the lower, broken line, ‘monetary’ encompassing square. Therefore we have here a visual restatement of Aristotle’s remark that it makes ‘no difference’ whether we see exchange in a monetary context or in a real context—the rate of exchange of the two goods is the same, given the same data, of course. 6.2 The case of inequality of incomes Having thus synthesised the geometrical structures of Aristotelian monetary and real accounting, we can now test how far this interpretation fits the text. As quoted above in Section 2 under the heading ‘Reciprocation’, Aristotle presented its general idea by invoking two artisans each giving part of their output in exchange for part of the output of the other. He then continues this characterisation with a number of modifications and explanations. Of these we may quote now his observation that (1133a17–19, Rowe and Broadie, 2002, p. 166): it is not  two doctors that become partners to an exchange, but rather  a doctor and a farmer, and in general people who are of different sorts and not in a relation of equality to each other; Aristotle continues the sentence, however, with: ‘ they therefore have to be equalized’ (numbering added). From this manner of expression we infer that in presenting his thoughts about an exchange economy Aristotle proceeded in three steps: 1. equality; 2. diversity; 3. equality regained through ‘equalisation’. The test of the present rendering of Aristotle’s analysis is now whether Figs 4(a) and (b) can cover these steps. Step 1, equality, was already covered in the last subsection. In the figure discussed there, shoemaker B’s line, representing the ‘own production’ of shoes QD*, if evaluated across the ‘exchange-line’ in terms of A’s product, houses, would cover exactly A’s line, representing the total ‘own production’ QC* of houses, and vice versa. In other words, their ‘real incomes’—to use the term introduced in the last subsection—is in this case the same if evaluated at the prevailing barter rate of exchange. Since along the prevailing ‘exchange-line’ the ratio of real incomes ( QD*/QC*) is the same as the reciprocal relative price ( pC/pD), one could also say that if their real income were expressed in money-values, they would be the same as well, of course. In addition, we have in this situation, as well as in all other situations of bilateral exchange, budgetary equality expressed geometrically by the equality of the hatched rectangles in the lower part of Fig. 4(a). Hence we could give a symbolic characterisation of this situation with the expression ‘step 1’ a) pC×QC*= pD×QD* ; b) pC×QC=pD×QD (3) part a) expressing income equality in accounting terms and part b) expressing budgetary equality in concrete monetary terms of sales and purchases. Assume now that builder A becomes ‘richer’, hence more ‘worth’ in accounting terms,37 because of an increase in the productivity of house-building so that there is a new value for own production QC*+ which is larger than the old quantity QC*, represented in Fig. 4(a) by the horizontal line from the diagonal to point A. The new own production line now extends farther to the left, to the corner marked A+. If now B’s line is projected across the ‘exchange-line’ on the housebuilder’s line, then it would cover only that part which it formerly covered, thus showing that B has now relatively less real income. Although nothing seems to be changed on B’s side, nevertheless the ratio of own production has now shrunk to QD*/QC*+, which is less than the prevailing price ratio along the exchange-line. The symbolic expression for this new situation is now ‘step 2’ a) pC×QC*+>pD×QD* ; b) pC×QC=pD×QD (4) The main symbolic change relative to ‘step 1’ is the ‘+’ sign in (4 a). This sign marks the change from the former relative short line segment QC* in Fig. 4(a) to the longer one in Fig. 4(b), marked now as QC*+. This change in length signifies that now we have a case of higher real income in terms of A’s own production, namely a larger quantity of buildings. At unchanged prices this higher quantity means a higher accounting value in terms of money. In modern parlance this means a higher money-value of income for the housebuilder. Hence the former equal sign between the two incomes is replaced by the greater sign ‘>’. In geometrical terms the only change brought about by ‘step 2’ is that the ‘production and barter box’ which formerly was the square AB changed to the rectangle A+B. One could debate whether the housebuilder’s household is now really better off than before since nothing has changed with respect to exchange. The volume of sales and purchases is unchanged. But due to a windfall increase in house production, this household can indeed enjoy the consumption of more goods: it still has the same amount of shoes but a larger amount of houses and normally (non-satiation assumed) it should be considered that A+ experiences an increase in well-being. Geometrically this level of well-being could be measured in Fig. 4(a) by the area of the rectangle formed by the line segment extending from point A+ to the broken line—measuring the ‘own production’ left for own use, and by the upper part of the broken line—measuring the amount QD of goods obtained through exchange. We will return to this measure when the discussion turns to the satisfaction of need in Section 8 below. 6.3 Price and ‘equalisation’ of incomes We come now to the continuation of the quote at the beginning of the last subsection, namely to Aristotle’s demand in what we called ‘step 3’, that the unequal traders ‘have to be equalized’. This term can mean several things and therefore we put ‘equalisation’ in single quotes.38 Here, we interpret Aristotle’s ‘equalisation’ in this particular quote as an elimination of the difference in income between doctor and farmer, resp., in the present context, between ‘rich’ builder A+ and ‘poor’ shoemaker B. Such an ‘equalisation’ is shown graphically in Fig. 4(b). Here, the ‘production and barter box’ is of the same rectangular shape as it was in ‘step 2’. The main adjustments are in the broken-line ‘monetary exchange box’, which changes from the previous square shape to the present rectangular shape. Formerly it was contained by the two vertical thin lines in Fig. 4(b), which go through the symbols QC+ and pD+. The latter symbol shows that the changes are due to an increase in the price of shoes from pD to pD+ so that the ‘exchange line’ now has the lower slope pC/pD+. It pivots in the ‘endowment point’ where the two ‘own production’ lines meet so as now again to form a diagonal, which is common to the ‘production and barter box’ and to the ‘monetary exchange box’. In the ‘production and barter box’ the vertical side representing B’s total production ( QD*), via the new slope of the transactions line, translates now into the housebuilder’s total production ( QC*) as given by the horizontal line of the box and vice versa. Thus we have again a case of equality of own production similar to the case in ‘step 1’. We have the remarkable result that the real income of the shoemaker increases (in terms of houses) although the shoemaker did not become more productive (the vertical line representing QD* is constant). Again an algebraic version of the case might be helpful in order to check the geometrical analysis. It is given by eq. (5). ‘step3’ a) pC×QC*+=pD+×QD* ; b) pC×QC+=pD+×QD (5) In eq. (5a) we see on the left-hand side again that the increased ‘worth’, making A become A+, was due to an increase of own production from QC* to QC*+. This increase can be matched on the right-hand side of this equation by an increase of the price of shoes from formerly pD to now pD+, leaving the shoemaker’s own production unchanged. As eq. (5b) then shows, monetary equivalence is still maintained, in spite of the higher price pD+ for the shoes, namely by the fact that this enables the shoemaker’s household to translate its higher money-value of sales of the (unchanged) quantity of QD shoes into a higher quantity of houses which it now buys, and which is now marked with symbol QC+ in eq. (5b). This expresses algebraically the geometrical feature of the two double-hatched rectangles in Fig. 4(b). It should be noted that once Aristotle’s ‘own production’ is recognized as real income, then it is easy to formulate nominal income graphically, namely by forming a rectangle with the ‘own production’ line and the relevant price line as its sides. These figures are represented in Fig. 4(b) by the double-lined rectangles on both sides of the ‘exchange line’, the diagonal of the encompassing rectangle. From the viewpoint of the encompassing rectangle with this diagonal the two double-lined rectangles are complements and hence necessarily equal. This is another non-trivial application of prop. I,43—non-trivial because it might not be totally clear just by inspecting the shape of these rectangles whether they are of equal area. As complements they definitely are so and hence income equality is confirmed. Although the above is meant to be a mere exercise in geometrical accounting, we can garnish it with some market analytic arguments and say that the higher accounting value of the income of the builder’s household as introduced in step 2 with the assumption of QC*+>QC* leads to a higher demand by A+ for good D and hence to the higher price pD+>pD of step 3. This price rise triggers price and income effects with the shoemaker, the producer of good D. One effect is that due to the new accounting value pD+QD* the shoemaker’s household rises in ‘accounting worth’ from B to B+. This is, by the way, a pure money-income effect since by assumption and by inspection of B’s ‘own production line’, the vertical side of the ‘production and barter box’ is unchanged. A further aside concerning basic income accounting is that although B’s real income is unchanged in terms of the own ‘own production’ of shoes, in terms of houses it has risen because, when translated across the diagonal, the ‘exchange line’, it has the same barter value as A’s total ‘own production’. Thus, ‘real values’ are, in part at least, an exchange rate phenomenon, and not totally determined by ‘own production’. A further ‘modern’ comment about our ancient representation of bilateral exchange is that—due to what moderns call ‘substitution and income effects’—there might be reactions in household B+ such that in the end B+ is not prepared to sell more of its product so that the quantity of shoes sold, QD, is constant, as it is indeed the case in Fig. 4(b). Thus there are some implicit behavioural assumptions here.39 But in any case, the (by assumption unchanged) own production QD* is now worth more, due to the plausible increase in its price. This might—and in the present context it does—lead to re-establishing equality of money incomes. As far as geometry is concerned, we may note that now we have a non-trivial application of Euclid’s proposition I,43 concerning the equality of complements as mentioned above in section 5.3 when stressing the speciality of the complements of Fig. 2(a). Due to proposition I,43 we know without any detailed measurement that the price change from pD to pD+must be matched by an ‘appropriate’ change of quantities from QC to QC+ where ‘appropriate’ is that quantity which satisfies the equivalence as expressed algebraically by eq. (5b) and geometrically by the hatched complement on the ‘other side’ of the diagonal. Thus this geometrical concept is not only applicable when we have exact numbers as in the case of Aristotle’s ‘beds vs house’-example resp. in the geometrical counterpart of Fig. 2(a), where we operated with the exact numbers ‘1’ and ‘5’. We have here the methodologically interesting case of a rather imprecise verbal qualitative statements like, e.g., ‘there is a somewhat higher price for good D’. In the context of an algebraic equation, we could react to such an imprecise statement by noting in a similarly vague way: ‘whatever the increase on the one side of the equation, there must be the same increase somewhere on the other side of the equation’. In the geometrical context we can now argue similarly: ‘whatever the larger value of the area of one complement expressing the one side of exchange, there must be a matching change for the other complement expressing the other side of the exchange’. Thus we have here the possibility to translate qualitative change in a consistent matching context not only through algebra but also through geometry. 6.4 Accounting and ethics The re-establishment of income equality through price change in the just-discussed ‘step 3’ raises the question whether this step was a moral postulate in Aristotle’s text or whether it was just a computational possibility. The long medieval discussion of ‘just price’ in the reception of Aristotle’s exchange model shows that his text can be read in a moral and prescriptive way. It certainly was intended to be read in some such way. Aristotle’s passages on exchange are part of a book on Ethics, hence the prescriptive nature of this book should be obvious. But the difficult issue is to find out in which way exactly Aristotle’s discussion of ‘reciprocation’ was meant to convey ethical postulates. Aristotle himself seems to have thought that his position on justice in this context was self-evident: it was opposed to the Pythagorean conception which he described—not in these words, but in essence—as being the ancient rule: ‘an eye for an eye’. But what has that to do with economic exchange, the topic of NE V,5? It would go beyond the scope of this paper to probe here for an answer.40 Briefly put, it is the impression of this author that Aristotle claimed that Pythagoreans lacked consideration for equity. In the Nicomachean Ethics (V,10), Aristotle stressed the importance of equity (epieikeia) in the sense of a ‘rectification of legal justice’ as Aristotle wrote,41 or, as put by Höffe (2003, p. 158), as ‘a protection against a precision that could be pedantic or merciless’ in the context of the application of justice.42 This concept of epieikeia was totally absent from the Pythagorean literature concerning justice according to Manthe (1996, p. 30).43 Thus we have here a philological basis for the present interpretation of Aristotle’s criticism of the Pythagoreans as being based on his opposition against their formalistic conception of justice. But it is open to much speculation what Aristotle might have recommended in case the market exchange had results which did not meet the criterium of equity—whether he thought in such circumstances that price controls were called for in the sense of a ‘just’ price or whether he would have called for a redistribution of income.44 Consistent accounting, no matter how precise, is not justice, as Aristotle and his intellectual predecessors were well aware. As Cuomo (2001, p. 41) observed: Aristotle pointed out that one of the conciliatory methods tyrants may adopt in order to secure their power was rendering accounts of receipts and expenditure, thus providing an illusion but not the substance of democracy,45 and Plato depicted a real-life mathematical expert as the embodiment of the dangers of knowledge inappropriately used.46 Aristotle’s entire chapter NE V,5 can be seen as a warning against inappropriate use of formal knowledge, namely as a warning against a misuse of the concept of ‘reciprocation’ in the sense of geometrical and algebraic exercises to which this chapter gives occasion. On the one hand, ‘reciprocation’ is indeed essential for accounting in a society based on exchange, namely for the simple budgetary reason that in economic exchange you must always pay the equivalent of what you get as long as the transaction is one of exchange. But on the other hand, as a concept of justice, ‘reciprocation as such’ is not just according to Aristotle (1132b29; Rackham,  2003, p. 281): ‘For in many cases Reciprocity is at variance with Justice’. Aristotle claims that this point totally escapes the Pythagoreans. As just remarked, such a conceptual discrepancy between Aristotle and the Pythagoreans can be substantiated with regard to the concept of equity / epieikeia but it would surpass the scope of this paper to enlarge now on the difference between the Pythagorean and the Aristotelian concepts of justice.47 It is important to insist here, however, that terms which are used in the context of accounting must not be confused with expressions of justice. The danger of such a confusion becomes apparent when contemplating Aristotle’s expression that citizens ‘have their own’ (1133b3; Rackham, ( 2003, p. 285). In NE V,5 this is used in an accounting context, namely referring to the ‘own production’ of each of the artisans, i.e. to the real income due to the own productive activity of each artisan during a year, as demonstrated above. Since NE V,5 is about ‘exchange and production’, the accounting of ‘own production’ is essential for describing the economic situation. Now Rackham ( 2003, p. 284, note e), in a footnote, refers Aristotle’s mentioning of the term ‘own’ in the sense of ‘own production’ to an ancient Greek proverbial ‘justice own’. Aristotle did indeed mention in the preceding chapter NE V,4 that ‘people say they ‘have their own’, having got what is equal.’ (1132a33; Rackham,  2003, p. 277). But the context of NE V,4 is different from the one of NE V,5. If the separate contexts of the term ‘their own’ are confounded, then neither Aristotelian justice can be properly understood nor can the associated accounting take place on the basis of such a misunderstanding. This is the reason why we insist in this article on a strict separation of the accounting context on the one hand and the context of debating aspects of justice in an Aristotelian analysis of exchange on the other hand. 7. Aristotle’s ‘accounting’ terms of trade There is a particularly strange passage on ‘reciprocity’, as Rowe and Broadie (2002, p. 166) translate Aristotle’s antipeponthos. In 1131b1–4 Aristotle gives the advice But one should not introduce them [artisans A and B, GMA] as terms in a figure of proportion when they are already making the exchange (…), but when they are still in possession of their own products. In his similarly worded translation, Grant (1885, p. 120) refers to this text as a ‘vexed passage’. It offers indeed many puzzles which, however, might be less vexing in the present interpretation than they normally appear to be. In terms of our version of reconstructing Aristotle’s geometrical discussions, we see in this passage a statement that there are two ways of having a ‘figure of proportion’. One version depicts just the ratio of exchange of two goods. That is done above in Fig. 2(b). It will be remembered that in commenting on that figure we have pointed out that the upper part of that figure gives a geometric rendering of what moderns call the ‘barter’ terms of trade while the lower part of the figure gives the corresponding ‘price’ terms of trade. A second version appeared above as Fig. 4(a). There, the lines of the subsquare showing the ‘barter’ aspect of exchange were extended so as to also cover the entire own production of builder A and shoemaker B. We interpret Aristotle’s ‘vexed’ passage as stating that it is the latter version which he advises to use, since in that figure we have A and B while they still have their ‘own products’—a term which is expressed geometrically in Fig. 4(a) by the lines marked with A (resp. A+) and B. In the omitted text contained in the round brackets of the above quote, Aristotle gives a rather strange sounding reason for his advice. It reads: (since otherwise [i.e. if own production is not taken account of, GMA] one of the two terms at the extremes will have both of the excess amounts)48 Finley (1970, p. 9) gives a paraphrase, presumably of this part of Aristotle’s text: ‘In that way, there will be no excess but ‘each will have his own’. ‘But there is no expression anywhere in NE V,5 where Aristotle states that because there is ‘no excess’, therefore artisans have their ‘own’ in some unspecified metaphorical sense. We follow Rowe and Broadie in assuming that what is meant here are the specific ‘own products’ of the respective artisans, as was quoted above. With the hindsight of modern trade theory, one is led to suspect that Aristotle’s term ‘excess’ or, as Grant translates, ‘superiorities’, refers here to what we nowadays call the ‘gains from trade’. It is well known that economists tend to see ‘favourable’ terms of trade as an indicator of welfare gains in trade.49 But there are empirical studies which show that this connection is a rather loose one.50 Aristotle’s reasoning can suggest an interpretation which gives also a methodological reason why terms of trade which focus on trade only and disregard total production might be an unreliable indicator for relative welfare in an exchange economy. In algebraic terms we may express the two conventional measures of the terms of trade by the equation QDQC=pCpD|QC*QD* (6) where the first term gives the ‘barter’ terms of trade and the second term gives the ‘price’ terms of trade.51 The separate ratio in expression (6) which shows the two levels of own production takes up Aristotle’s suggestion to consider these quantities as well. That can be done by multiplying both sides of the equation (6) with this ratio. This gives qDqC=pCpD×QC*QD*=A′B′ (7) where qD is defined as QD/QD* and qC is defined as QC/QC*. In this way the left-hand side of expression (7) takes up the ratio of own production. Symbol qC shows the quantity QC exchanged by artisan A as a ‘share’ of own production QC*. In the same way qD shows the shoemaker’s quantity given up in exchange as a ‘share’. The quotation marks are used here in order to refer back to the quote in Section 2 above concerning Aristotle’s ‘reciprocation’. It is defined there as an exchange of a ‘share’ of respective own production. Thus we consider the left-hand term of expression (7) as being covered by Aristotle’s authentic text.52 The other parts of expression (7) involve a straightforward mathematical multiplication and a subsequent definition: The multiplication of the total own production of buildings times their price gives the accounting value for builder A for which we chose here the symbol A′. As noted above, this value corresponds to the modern concept of total income based on a monetary valuation of own production. In the same way the calculation of the money-value of the total production of shoes by multiplying with the corresponding price gives the shoemaker’s ‘worth’ in terms of money with symbol B′—in other words it gives the calculated money-value of the shoemaker’s income. The transition from expression (6) to (7) shows a simple transformation of the conventional measures of the terms of trade into what we may call an ‘accounting’ version of the terms of trade. This version requires an accountant’s work in so far as it requires somebody to count the total quantity of goods produced by an artisan and then to calculate the total worth of production in terms of money, namely on the basis of a given price per counted piece. As in real life, in the present case an accountant could take several alternative prices for such a calculation: the last market price in the accounting period, or the first price quoted, or an average over time of the market prices for the good in question. There might be administered prices, but the text says nothing about the way in which prices are determined, only that they are required for calculating values in terms of money.53 This interpretation covers in a straightforward way a seemingly mysterious statement of Aristotle’s by which he introduces the ‘vexed passage’ about ‘reciprocity’, namely by claiming (at 1133a33) that reciprocity prevails under the condition that: [A]s farmer [quantified as A′] is to shoemaker [quantified as B′], so the product of the shoemaker [in terms of share qD] is to that of the farmer [in terms of share qC].54 In this passage Aristotle’s original text uses only words for the characterisation of his proportions. We ‘messed up’ his words by inserting the quantity-symbols of the above expression (7). Hopefully it thereby becomes clear that Aristotle’s passage on reciprocal reciprocity is nothing else but a verbal variant of a simple accounting relation. It is not clear, however, by which geometrical construction Aristotle made his argument concerning the unsatisfactory results which might come up if own production is not taken account of in the analysis of economic exchange. But one can trace verbally the problem that even with seemingly ‘favourable’ terms of trade in the sense of few domestic goods buying many foreign goods in exchange, there can be problems concerning relative well-being. If the total production of domestic goods is small, then even a small amount of exports might leave only a small rest of domestic goods for own consumption. Maybe this is sufficiently compensated by the amount of imported goods, but sometimes the net effects might be unfavourable. A detailed tracing of such net effects would go beyond the scope of this paper, however. Even without going into details, this much should follow already from the above considerations: concentrating on regarding just the act of exchange itself (as, e.g. in Fig. 2(b) above) can give biased results concerning the welfare effects of trade. Only the inclusion of total own production gives the opportunity for considering income aspects in the analysis of economic exchange. If this is indeed the message which Aristotle intended to bring across in his ‘vexed passage’, then it is indeed a valid one. The most important result of this section is, however, that if Aristotle’s traders are ‘equalized’ by an analysis in terms of ‘shares’,55 then it can easily be seen that reciprocity in exchange is of reverse type as written in expression (7)—a result which baffled many commentators in the past.56 8. Need and ‘value’ A further difficult passage of the Nicomachean Ethics deals with the satisfaction of need in connection with exchange. There is a long tradition, established already in Plato’s writings,57 that the satisfaction of need is to be considered as the basis and the purpose of economic exchange through markets. It is even the basis of having a polis at all. But it seems that Aristotle goes way beyond that tradition when he writes58 Everything… must be measured by some one thing, as was said before. In truth this one thing is need … But as a kind of substitute for need, convention has brought currency into existence This is a rather complex statement, which Langholm (1979, p. 49) discusses especially under the perspective of the Platonic precedent. He concludes: there is in the end no denying that the Nicomachean Ethics at this one point states that need is a ‘measure’ of commutables. This assures us that the passage has to be read such that it does postulate the existence of quantitative measures of primordial ‘need-values’, as we may call them. Aristotle’s text further claims that on the basis of convention money-values might be considered as standing for ‘need-values’. From this we conclude that in Aristotle’s view the two types of value must have something in common. In any case, ‘in truth’ the basis of all this is the measurement and hence the accounting of (the satisfaction of) need. With ‘in truth’ and with the associated short passage on monetary measurement, Aristotle seems to make an implicit distinction between precise conceptual and precise quantitative measurement: It is an important conceptual clarification to state that the ‘real creator’ of a city-state is ‘need’ (Socrates). But not everybody profits equally from the life in a particular polis. ‘Conventional’ assessment of relative economic position is in terms of money-values. ‘In truth’, however, such measurement should cover the satisfaction of need, but such direct measurement is not possible. Here now the indirect measurement of money-values as geometrical areas comes in. If, as has been seen above, lines standing for incommensurable goods can be transformed into commensurable areas standing for comparable money-values, then it might also be possible to measure the satisfaction of need in an analogous indirect way, namely as a specific geometrical area. Since the method of formal argumentation in Book V of the Nicomachean Ethics relies repeatedly on ‘lettered diagrams’, therefore it is quite plausible that the ‘something in common’ between money-values and (satisfaction of) need-values is the analogous representation by appropriate areas in such diagrams. In order to make this thought more concrete, consider that the satisfaction of need depends on the goods available in the respective household, namely the ‘own production’ which is left in the own household after exchange, and the amounts of those other goods which have been obtained through exchange. In Fig. 4(b), these amounts and the associated areas are marked by the (single and double) hatched rectangles in the upper structure, the ‘production and barter box’. Let us henceforth take rectangular areas with the dimension of available goods in each household as indicative of the level of the satisfaction of need. We may remember now that the term ‘need’ is our translation for Aristotle’s much-debated original term chreia. Since this Greek word begins with the Greek letter ‘ χ’, one may mark areas denoting the satisfaction of need with this symbol (see Fig. 5 below). Fig. 5. View largeDownload slide Accounting concepts in the Nicomachean Ethics Fig. 5. View largeDownload slide Accounting concepts in the Nicomachean Ethics Marking the hatched rectangles in the ‘production and exchange box’ (upper rectangle of Fig. 4 (b)) as representing need-values in the way just described solves our problem of interpretation only partially, however. The now remaining problem is to find the type of money-values which plausibly stand for these need-values, because Aristotle’s text claims that money-values serve as ‘conventional’ substitute for the measure of (the satisfaction of) need. It seems that the best candidates for this task are money-values of income as given by the two rectangles with double-lined circumferences in Fig. 4(b). To check the reasonableness of this suggestion, let us trace the increase of shoemaker B’s satisfaction of need due to the increase in money income, as discussed above in Section 6.3: The original value of B’s need satisfaction is given in Fig. 4(a) by the rectangle formed by the upper part of the ‘shoes’-line which represents what is left of shoes for own consumption after exchange and by the QC-line representing the amount of houses obtained through exchange. In Fig. 4(b), this value is marked by the (single) hatched area in the corresponding upper part of the ‘production and barter box’. The double-hatched extension of this area is then due to the already-discussed increase in the money-value of income due to a price rise of shoes from pD to pD+. In Fig. 4(b), this new accounting value of income is given by the total area of the upper rectangle with double-lined circumference, the old accounting value being measured by this rectangle minus the area to the right of the line going through symbols B+ and pC+. We see now that the larger money-value of income area corresponds with an increase of the measure of the satisfaction of need. But note that in this case the real income in terms of ‘own production’ of shoes stayed constant. Thus real income is not a good proxy for needs satisfaction in this case. Ironically, the better proxy for the ‘real’ consumption of goods is the money-value of income, but it is not perfect. The hatched needs satisfaction rectangle in the upper part of the ‘production and barter box’ of Fig. 4(b) and the adjacent monetary accounting level of income rectangle with the double-lined circumference are not ‘similar’ in the geometrical sense since in general these two types of rectangles don’t have a common proportion of lines. Therefore one cannot infer in a perfect way the change of the needs satisfaction rectangle from knowing the change of the money-value of income rectangle. We conclude: if one wants to judge the satisfaction of need in a perfect way, then, in a geometrical setting, one has to measure the area covered by the goods which are available for household use. ‘Conventionally’ people do not draw geometrical figures for assessing the level of the satisfaction of need, but they think and compare in terms of money-values. The relative values of own production in terms of money might be an acceptable proxy for the relative possibilities to satisfy household needs. We may now link up with the result of the previous section where it was established that the relative money-worth of artisans equals reciprocally the relative shares of own production which is given up in exchange. Thus, if it is more convenient to assess the ratio of these shares, then one can infer from this ratio that of the relative ‘worth’ of artisans. As the present section then suggests, from the ratio of ‘worth’ in terms of money-values one might also infer the relative levels of the satisfaction of household needs. 9. Summary and conclusion This article focussed on Aristotle’s analysis of exchange as we have it in NE V,5. That passage has two levels of analysis, a ‘descriptive’ one, dealing with basic economic concepts and relations, and a related but different ‘prescriptive’ level which deals with justice. Here we concentrate on the first of these two levels, without intending to deny the importance of the second level, of course. The emphasis on this distinction is not customary in Aristotelian literature. But a moment’s reflection should convince the sceptic that before one passes judgements in the context of economic exchange, one must be clear about the categories used in this type of analysis. Thus, it is important to see that Aristotle defines bilateral exchange among his artisans as being on the basis of their ‘own production’, i.e. their respective total traded and not-traded output. Moses Finley (1987), the New Palgrave’s erstwhile expert on Aristotle as mentioned in the introduction above, is a commentator who skips Aristotle’s concept of ‘own production’, and consequently he has no basis to evaluate either the quantity effect on the artisans’ economic ‘worth’59 nor the price effect.60 Finley (ibid.) then wrongly claims that anyhow, the idea of relative worth of artisans is ‘intolerable under conventional economic thinking’, but he never substantiates why that should be the case. In fact, today it is customary to compare people by the accounting value which they stand for,61 and so it was in the Athens at the time of Aristotle.62 ‘Own production’ defines the geometrical dimensions of what we called the ‘production and exchange box’, which translates Aristotle’s verbal presentation of the concept of ‘reciprocation’ into a simple geometrical model descriptive of barter exchange and production. Similarly, a second geometrical model was here presented which concentrates on the quantities exchanged and which describes the economic situation in a ‘monetary exchange box’. Both ‘boxes’ together describe economic production and exchange in a simple, but integrated, context. Fig. 5 below gives a simplified synthesis and summary of the economic categories and their interrelations which appeared in the present text. Fig. 5 shows the ‘production and exchange box’ just mentioned by the upper left square with the thick drawn-out circumference lines depicting the ‘real’—i.e. the non-monetary—sphere of the analysis with the accordingly marked lines representing the total ‘own production’ of houses resp. of shoes. The meeting point of these two lines on the diagonal marks the ‘endowment point’ of the two households with their respective ‘own production’. This point is ‘pivotal’ in the sense that it is the turning point for any changes in relative prices.63 The ‘monetary exchange box’ is represented by the lower right square with broken-line circumference. The transversal lines in this substructure are such that at the point of intersection, the ‘pivotal’ point just mentioned, they have sections marked QC and pD resp. QD and pC and standing for the quantities exchanged and for the prices so that the diagonal through this square depicts two ratios simultaneously: the ‘barter’ terms of trade ( QD/QC) as well as the ‘price’ terms of trade ( pC/pD), as can be read off this figure. In Section 7, above it was argued that Aristotle introduced a third measure for characterising conditions of trade, the ‘accounting’ terms of trade. They state that the ratio of artisans’ (A and B) (monetary) worth of total own production ( A′ and B′) is equal to the reciprocal ratio of the marketed share of their production. Aristotle’s reciprocal reciprocity has baffled many commentators, but in fact it can be seen to be based on simple and plausible calculations, once the concept of ‘own production’ is accounted for using the associated prices. Aristotle famously proclaimed that ‘there would be… no exchange without equality’.64 In the present context this means budgetary equality, the mutual matching of the accounts of the trading partners. This matching of accounts is what the two hatched rectangles marked M express in the ‘monetary exchange box’ of Fig. 5. They are geometrical complements in the sense of Euclid’s proposition El. I,43. Thus, this equality is based on an ‘eternal’, mathematically proven, ‘truth’, to use Aristotle’s terminology of NE III,3 which related to geometrical ‘incommensurability’, as noted briefly above. We stressed that Aristotle, though acknowledging the incommensurability of goods, nevertheless did not consider this as being a problem for monetary accounting, and rightfully so. In NE V,5 Aristotle put much stress on money, on its characteristics in general and on its importance as a medium for transforming incommensurable physical goods into commensurable expressions of economic accounting. Did he mean by this that money is all-important in economic exchange? Here now Aristotle’s ominous measurement ‘in truth’ comes in. In our reading, its context may be put into a paraphrase, reading: ‘In truth the really important thing in society is not money but the satisfaction of need. It is the ultimate concept for valuation’. As money-values in a geometrical context may be represented by areas, so we may also use areal measures for a geometrical representation of the satisfaction of need, i.e. the rectangular area covered by the goods available in each household after completed exchange. This is given by the hatched rectangles in the ‘production and exchange box’ marked as χA and χB in Fig. 5. But such a measure is practicable just in geometry, not in a monetary society. The relative accounting value of income in terms of money may be treated as Aristotle’s ‘conventional’ proxy for the relative possibility of households to satisfy their needs. In Fig. 5 the two rectangles with a double-line circumference represent the accounting value of the respective own production, hence the money income of the artisans—as an accounting magnitude, of course, not as a sum of money. As in expression (7) above, these values are marked A′ and B′ in order to discern the citizens A and B from the accounting value which they stand for due to their respective production as artisans. Such measurement of the relative ‘worth’ of the artisans is central in Aristotle’s rendering of exchange because it is in this way that we can understand his incomprehensible-seeming reciprocal reciprocity between artisans and their products. With this interlocking system of the categories, proportions and geometrical figures associated with economic exchange, we have prepared the conceptual groundwork for a discussion of fairness in this context. But it is still the bare groundwork for such a discussion which we have here because the geometrical figure just summarised is a very schematic one. It is based on the assumption of strict equality of all the relevant categories, like real income (‘own production’), monetary accounting value A′ and B′, their ‘ultimate’ measure of the satisfaction of need ( χA and χB). But, as already quoted, Aristotle was clear that economic exchange involves ‘in general people who are of different sorts and not in a relation of equality’. How and in which way such differences are to be treated under the perspective of justice is a subject matter which goes beyond the scope of this article. This article makes no claims concerning a clarification of Aristotle’s conceptions of justice. Nevertheless, we do claim that it contains a major contribution to an understanding of a difficult part of the Nicomachean Ethics. The present reconstruction of an Aristotelian conception of ‘accounting’ terms of trade in Section 7 implies a new interpretation of Aristotle’s postulate of reciprocal reciprocity in economic exchange. Aristotle’s statements of the type: ‘As builder to shoemaker, so shoes to houses’ have generally been met with a lack of understanding.65 By showing the geometrical, the algebraic and the commonsensical reasoning behind this type of accounting statement, we were able to solve one of the major riddles of the Aristotelian analysis of economic ‘reciprocation’. The author thanks two anonymous referees for their helpful comments. All deficiencies of the text are the sole responsibility of this author. Bibliography Albertus Magnus. , 1972. Opera omnia—Super ethica: commentum et quaestiones Ps. 1, vol. 14, ‘First commentary’, see Langholm (1979) , p. 64, Monasterium Westfalorum Aschendorff: Institutum Alberti Magni Coloniense Alchian A. A. and Allen W. R. 1964. 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Die Ethik des Aristoteles in ihrer systematischen Einheit und in ihrer geschichtlichen Stellung untersucht [The Ethics of Aristotle investigated in its systematic unity and in its historical position] , Regensburg, Manz Footnotes 1 According to Ricardo Crespo (2006, p. 767) Aristotle was ‘the first systematic thinker’ concerning the ‘ontology of the economy’– but it was Plato already who focussed the economic role of the satisfaction of need through market exchange (see note 57 below). See also Dos Santos Ferreira (2002, p. 568) who claims that Aristotle’s passage on bilateral exchange in the Nicomachean Ethics ‘is undoubtedly one of the most influential writings in the whole history of economic thought’. 2 Concerning Aristotle’s economics the renown historian of The Ancient Economy (1973) Moses Finley proclaimed that ‘attempts to interpret his words so as to rescue them as economic analysis are doomed from the outset’ (Finley 1987, p. 113). 3 ‘Your search for “Aristotle” over the article titles within the 2008, 2009, 2010, 2011, 2012 and 2013 editions returned no results.’ The New Palgrave Dictionary of Economics, http://tinyurl.com/Aristotle-Palgrave, last accessed 2 Oct. 2013. 4 Overviews of controversies around ‘algebra of areas’ are, e.g., in Kastanis and Thomaidis (1989) and in Grattan-Guinness (1996). It might also be said that in the present context we have not algebra of areas but rather ‘arithmetic of areas’, as an anonymous referee remarked. Indeed, the following figures with their lines and areas often give specific values and ratios, say, of equality or of discrepancy. But no matter what we call the associated method today, when we see letter-symbols combined with allusion to geometrical figures in Aristotle’s text, then we should be aware of the observation: ‘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, 1999, p. 15). This certainly applies to Aristotle’s A, B, C, D, etc., of economic exchange. 5 See, in this spirit, also note 65 below (last page of the text). 6 The four-digit reference numbers give the pages of the Greek text as published by Bekker 1831. The letters ‘a’ or ‘b’ refer to the left or right column on the page. The last number gives the line number when counting from the top of the column. This is the standard quotation of the original text so that the reader is not tied to a particular translation. 7 Aristotle has the unit ‘mina’ for currency in this place. 8 Aristotle does not write that it makes no difference whether we equate respective money-values of goods or their respective quantities. But see notes 10 and 18 below. 9 Cf. Crisp and Saunders (1999, 140, n.19): ‘For a complete analysis of Aristotle’s economics see Meikle ’. See also Crespo (2011, p. 1): ‘I consider the best book to be by Scott Meikle (1995)’ and Crespo (2013, p. 3): ‘In the field of Aristotelian philosophy of economics, the first contribution that must be mentioned is Scott Meikle’s book Aristotle’s Economic Thought… ’ 10 Meikle (1995) writes ‘5 beds = 1 house’ more than 15 times; see pp. 12, 14 n. 9 (two times), 15 (three times), 22, 24, 24 n. 20, 38, 39, 83, 112, 116, 134 and 183. The locus classicus for this interpretation is Karl Marx ( 1906, p. 68). 11 There are two formal issues here: (i) Replacing Meikle’s ‘=’ sign between beds and house by our ‘:’ sign, thus changing the relation between beds and houses from Meikle’s equality to our proportionality. (ii) The sameness of proportions is, strictly speaking, not the same as the equality of values such as that of rectangular areas or of values in accounting. Therefore Grattan-Guinness (1996, p. 366) has good reason to proclaim: ‘The habit of the geometric algebraist of writing “=” between ratios is inadmissible.’ He advocates to use the sign ‘::’ instead in this case, and credits the 17th-century mathematician William Oughtred with introducing this felicitous convention. We use it here in eq. (1a) also in order to stress the specificity of the present interpretation. But in relation to Aristotle’s original way of expression Oughtred’s ‘::’ is as anachronistic as the ‘=’ sign, the introduction of which being generally attributed to Recorde ( 1969). Thus, since in our following context the distinction between sameness and equality is not essential, therefore we will take the licence to follow the ‘geometric algebraist’ in using the equality sign as well—also when proportions are involved. 12 In the ‘Valuation’ passage quoted above Aristotle writes that ‘one house’ is 5 M ( =pH), that ‘the bed’ is 1 M (= pE), thus the prices are expressly referred to not just as ‘simply numbers’ (Meikle) but as ‘money per unit of the respective good’, as written symbolically under the braces of our eq. (1b). 13 It should be noted that it is only in eq. (2) that we have a proper algebraic form where the symbols stand for variables. In eq. (1) the symbols stand for specific goods and prices. There we have ‘arithmetised’ and ‘pre-Cartesian algebra’ (terms suggested by a comment of one of the anonymous referees), corresponding to a geometry of specific lines and proportions, independent of a Cartesian system of co-ordinates, as will be seen below when discussing the following figures. 14 It is remarkable that such an equivalence is expressly proven in Euclid, Elements, Book VI. See Artmann (1999, p. 141, his brackets): Section 15.4, ‘Book VI…: Proportions and Areas (Products)’. 15 See, e.g., Keyser (1992). 16 See Alchian and Allen (1964). 17 See also note 35 below concerning the contextual usage of the letter-symbols A and B. 18 Meikle (1995, p. 83): ‘The Ethics chapter … is mostly about how goods can possibly be commensurable, or how … 5 beds = 1 house = so much money (1133b 27–28).’ 19 Ste. Croix (1956, p. 34) bemoans that up until the Roman period ‘there is no really significant advance in the system of accounting: capital and income are still not properly separated’. 20 See also Schaps (2008, p. 42): ‘almost certainly by the age of the tragedians [thus well before Aristotle, GMA], Athenians thought of all wealth as being a matter of minae, drachmae, and obols, and regularly expressed themselves that way.’ 21 Cuomo (2001, p. 14) mentions in this context ‘state accountants (logistai), who seem to have had two main tasks: to supervise the accounts of public financial bodies, as exemplified by this inscription, and to supervise the accounts that each public official had to render at the end of his term of service.’ 22 According to the historian of mathematics Benno Artmann (1985, p. 296), the Parthenon has the proportions length:width = width:height (81:36 = 36:16), thus representing the ‘mean proportional’ x in a:x=x:b. The earliest precursor was the temple of Apollo in Corinth (540 BC) with the proportions 25:10 = 10:4. The ‘mean proportional’ is expressly addressed by Aristotle in De anima II.2, 413a 13–20 (ibid., 295) in the geometrical context of ‘squaring a rectangle’. 23 Amati (2010). 24 Source: http://en.wikipedia.org/wiki/File:BMC_193.jpg, accessed on 19 June 2012. 25 The letter ‘A’ can denote the cipher ‘1’ in ancient Greek notation. The alpha is the symbol for ‘the one’ in the Pythagoreans’ Tetraktys, a triangular assembly of alphas which symbolised their oath (van der Waerden, 1979, p. 107; D’ooge et al., 1925, p. 242). ‘It was not uncommon for Greek cities to adopt coin motifs with punning references to their name’, writes Sayles (2007, p. 129). He has a list of fifteen (‘a few’) such cases like Rhodos depicting a rose, or of Delphi depicting a dolphin, etc. Here, too, a pun could have been intended which hinges on the double meaning of ‘A’ as being the first letter of the island’s name and, in ancient Greece, the numeral ‘one’. With their ‘A = 1’ in an isolated upper corner of the coin, the Aeginetans could have been claiming a victory in the game of ‘One-upmanship’. They had reason to be triumphant because their hated rival, neighbouring Athens, was crushed in 404 BC. 26 The coins’ letter ‘A’ in the sense of ‘1’ could also have had an important geometrical meaning, as was argued by Ambrosi (2012). 27 Aristotle mentions Aegina in a monetary context in Metaphysics D.5, where he intends to clarify the word ‘necessary’. He writes that as it is ‘necessary’ to take medicine when one is ill, so it might be ‘necessary’ to go to Aegina for monetary matters: ‘and a man’s sailing to Aegina is necessary in order that he may get his money’ (Ross, 1928, 1015A). At Aristotle’s time all denominations of Aegina’s money always had the reverse side with the geometrical structure of Fig. 3. 28 According to Lasserre (1964, p. 38), the typical curriculum at Plato’s Academy was a course ‘over fifteen years, of which the first ten are devoted to mathematics and the rest to dialectic… Aristotle and most of the philosophers coming from the Academy certainly followed this syllabus from beginning to end’. 29 Artmann (1999, pp. 40–41), Figs 4.18 and 4.19. 30 If the designers of Aegina’s coins did have mathematics in mind, it is far more likely that in the commercial context of the usage of such coins they invoked the exchange of equivalents as just outlined rather than the geometry of (a+b)2 as Artmann believed. It might be noted in this context that ‘the (few and feeble) forces connecting mathematics and economics in antiquity come from economy, not from mathematics’, as Reviel Netz (1999, p. 303) observed. 31 See also note 18 above, where Meikle (1995, p. 83) is quoted as claiming that NE V,5 is ‘mostly’ about commensurability. 32 Mathematical incommensurability can be illustrated with reference to a unit square with sides of one foot each. Then the diagonal is the hypotenuse of a right-angled triangle. By Pythagoras’ Theorem its squared value is 12+12=2 and its un-square value is 2. Thus the diagonal is measured in units of 2 while the side is measured by a real number ‘1’. The proof that these are indeed ‘incommensurable’ entities is a major achievement of Pythagorean mathematics and it was well known to Aristotle. 33 See the following note. 34 Cf. Heath (1956B, p. 11): ‘We should carefully note that the signification of rational in Euclid is wider than in our terminology… a straight line is rational which is commensurable with a rational straight line in square only’; emphasis in the original. 35 Note on the use of symbols A, A+ and B, B+: The reader should take note of their respective context. In Fig. 4(a), they mark ‘own production’ lines. Since prices are not important in this context, the respective lines mark also the relative ‘worth’ of the artisans A and B. In Fig. 4(b), the prices pC and pD+ do come in, namely for calculating money-values. Here the symbols A+ and B+ refer to the two areas with double-lined circumferences, and these specify the respective accounting worth of the artisans. In the algebraic context of eq. (7) below we will use symbol A′ and B′ to denote the accounting value of total own production which characterises quantitatively the producers A and B. 36 Cf. Blaug (1997, p. 110): ‘By “riches” Ricardo means the magnitude of physical output; more riches mean more real income.’ The total A-line ( QC*-line) thus measures the ‘riches’ (real income) of the builder A; the total B-line ( QD*-line) measures the ‘riches’ (real income) of shoemaker B. Ricardo (and Aristotle?) seem to confuse the flow-variable ‘real income’ resp. ‘own production’ with the stock-variable ‘riches’ or ‘worth’. But since the accounting setup is a-temporal, therefore it could be said that the yearly income is a proxy for wealth and that there is in fact no ‘confusion’ involved in the synonymous use of ‘riches’, ‘worth’, and ‘income’. 37 For this rather loose use of the term ‘rich’, see note 36 above. 38 A second meaning of ‘equalisation’ is briefly mentioned below in note 52. 39 It should be noted that the implicit household model is here not the ‘normal’ microeconomic model where a budget is given in terms of money. The budget is given here in terms of ‘own production’. Therefore, when shoes become more expensive, then the shoemaker’s purchasing power does not decline, but it increases because the ‘own production’ of shoes has now a higher accounting value. This is shown geometrically in Fig. 4(b) by the increased value B+. See, e.g., Varian (1992, p. 144), Section 9.1: ‘Endowments in the budget constraint’ for this case in a modern household model. 40 For the juxtaposition of Pythagorean and Aristotelian justice in a geometrical setting, see Ambrosi 2013. 41 NE V, 10 §3; 1137b14; Rackham ( 2003, p. 315) 42 See also Hurri (2013, p. 159): ‘In epieikeia, legality’s ordinary justice appears to be overridden by a godly rumble of cosmic and immemorial justice.’ 43 Even Plato refers to epieikeia not in the Aristotelian sense of ‘equity’ but rather in the sense of ‘leniency’ (‘Billigkeit’ vs. ‘Nachsichtigkeit’ in Wittmann (1920, p. 218)). 44 It was noted above in Section 5.2, that Plato’s friend Archytas of Tarentum—normally listed as Pythagorean—did write that ‘the wealthy give to the needy’ on the basis of ‘calculation’ of the fair share. This suggests that there was at least this Pythagorean who thought in terms of redistribution. But in his extant works Aristotle never listed Archytas as Pythagorean (Huffman 2005, p. 8)—maybe because Archytas’ advocacy of redistribution was so untypical for the ‘genuine’ Pythagoreans. 45 Cuomo’s reference (her note 4, p. 60): “Aristotle, Politics 1314b”. 46 Cuomo’s reference (her note 5, p. 60): “Plato, Lesser Hippias 367a–c, tr. N. D. Smith, Hackett (1992). Cf. also Greater Hippias 281a, 285b–c; Protagoras 318d–e and Xenophon, Memorabilia 4.4.7”. 47 See, however, the already mentioned geometrical approach in Ambrosi (2013). 48 Grant (ibid.) has here the equally unintelligible translation ‘else the one extremity of the figure will have both superiorities assigned to it’. Apostle (1975, p. 88) translates ‘(otherwise, one of the upper terms will have both excesses)’. 49 See, e.g., Diewert and Lawrence (2006, p. 1): ‘Improvements in a country’s terms of trade… raise domestic welfare.’ 50 Diewert and Lawrence (2006, p. 49): ‘The main conclusion emerging from this study is that… changes in the terms of trade have relatively little impact on Australian welfare.’ 51 Equation (6) is, but for the subindexes, the same as the equivalence eq. (2b). 52 It may be noted here that in terms of shares we have for artisan A always qC+(1−qC)=1 and for B likewise qD+(1−qD)=1, i.e. the share of traded and not traded parts of own production always adds up to unity. Thus, in terms of shares both artisans are always equal to the unit value, but the units are ‘total number of good C‘ for artisan A and ‘total number of shoes’ for shoemaker B, thus the unit share is always ‘one’ but in terms of different goods which are not directly comparable. Nevertheless, when, at 1133b5 and in continuation of the ‘vexed passage’, Aristotle writes ‘equality is capable of being produced’, then this might well be a reference to the equality of such unit ‘shares’, and not to an equalisation by an adjustment of relative price, as discussed above in Section 6.3. 53 See Meikle (1995, p. 21, note 16) for a discussion of this point. 54 As one of many confessions of being completely baffled by Aristotle’s calculation of ‘reciprocity’, see Joachim (1951, p. 150): ‘How exactly the values of the producers are to be determined, and what the ratio between them can mean, is, I must confess, in the end unintelligible to me.’ 55 See note 52 in this context. 56 Langholm (1979, p. 52) observes: ‘Six manuscripts of the Translatio Lincolniensis [Bishop Grosseteste’s translation of the Nicomachean Ethics of ca.1250]… have a marginal note at 1133a33, changing the order of the products so as to make it conform to the order of the producers (the direct form).’ 57 Socrates in Plato’s Republic (Book II,11; 369 C–D): the ‘real creator’ of a city-state is ‘need’ and its mutual satisfaction by artisans through economic exchange on markets. (Shorey, 1937, pp. 150–51). 58 1133a26–30; NE V,5 §11; Rowe and Broadie (2002, p. 166)) 59 See the transition from A to A+ in Fig. 4(a). 60 See the transition from B to B+ in Fig. 4(b). 61 See ‘The World’s Richest Billionaires: Full List of the Top 500’, Forbes, 4 March 2013, http://tinyurl.com/super-rich500, last visited 20 April 2016. 62 Concerning ancient Athens, see note 20 above and the text to which it refers. 63 See Fig. 4(b) and recollect the analysis of an increase of the price of shoes from pD to pD+. 64 1133b17–19; NE V,5; Rowe and Broadie (2002, p. 167). 65 Rothbard (1995, p. 16): ‘Aristotle’s famous discussion of reciprocity in exchange in Book V of his Nicomachean Ethics is a prime example of descent into gibberish … How can there possibly be a ratio of “builder” to “shoemaker”? Much less an equating of that ratio to shoes/houses? In what units can men like builders and shoemakers be expressed? The correct answer is that there is no meaning’ (emphasis in the original). © The Author(s) 2017. Published by Oxford University Press on behalf of the Cambridge Political Economy Society. All rights reserved.
Cambridge Journal of Economics – Oxford University Press
Published: Mar 1, 2018
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