23 IUSGEN 265

Citation:

Eric Allen Engle, THE GENERAL PRINCIPLE OF PROPORTIONALITY AND ARISTOTLE, 23 IUS Gentium 265 (2013).

15.1 Introduction

This article proposes a brief history of the concept of proportionality in law in order to understand the rule, found around the world, that state action must be a rational means to a permissible end which does not invade protected human rights unless strictly compelled by necessity. Although the concept of fundamental human rights is a very modern one, dating from the Scottish enlightenment (circa 1776) [FN1] the concept of proportionality, which is used to adjudicate conflicts between fundamental rights, is much older.

The idea of distributive justice as proportionality is first seen in Book V of Aristotle's Nicomachean Ethics. [FN2] Proportionality as commutative justice (“an eye for an eye” lex talionis) is even older and can be traced to the Code of Hammurabi. *266 The proportionality principle in law, though evolving, has shown remarkable continuity over centuries. The theory of distributive justice as geometric proportionality seems to spring forth from Aristotle's brain like Athena, fully formed and crying “Victory!” [FN3] However, the practice of proportionality in law, i.e. the implementation of the general concept in legal practice, occurred over several centuries. Athena's cry echoes to this day: a law must be a rational means to a permissible end; the punishment must fit the crime; the use of force in self-defence must be necessary, i.e. unavoidable, and limited to only that violence which is needed to extinguish the threat. The abstract proportionality concept became more precisely defined as a rule of law through historical experience.

The general principle of proportionality is now a world-wide principle of law [FN4] found in common law, civil law (Poto 2007) and international law (Andenas and Zleptnig 2007). The proportionality principle is a key organizing principle of contemporary legal thought, converging civil law and common law to a global uniform ius commune which hybridizes aspects of common law (binding inductive case law) alongside civil law (deductive general principles of civil law and common law fundamental rights). This hybridized ius commune in turn converges substantive rules of national law toward uniform global rules. Norm convergence arises due to intensified trade, vastly improved communication and transit, and machine translation. Base and superstructure here both reach toward the same goal, the withering of the state and its replacement by civil society through peaceful trade to replace war as the principal mode of state interactions. Contemporary global law is guided by a teleology of peace through interdependence and protection of human rights. This teleology seeks to replace conflict oriented negative and zero sum state interactions with positive sum private law interactions. War is to be replaced by law, and law in turn is to be replaced by voluntary market transactions. Thus, proportionality as an adjudicative principle which determines the outcomes of conflicts between fundamental rights is central to contemporary law. Legal theories are also converging globally. At the broadest level, conceptual jurisprudence (Begriffsjurisprudenz) is linked to legal process interest balancing (Interessenjurisprudenz). Conceptual jurisprudence is then applied to natural law, and legal process interest balancing is applied to positive law.

Proportionality, first clearly elucidated by Aristotle, has become the rational principle which adjudicates conflicts between state power and fundamental individual rights as well as conflicts between competing fundamental human rights. Aristotle did not however analyse law in terms of conflicts involving fundamental individual or human rights. The idea of individual fundamental rights as abstract deductive general principles of law which are hierarchically superior to ordinary laws only arose with the Scottish enlightenment circa 1776, which recognized *267 individuals as having inalienable fundamental rights. However, the early modern idea of inalienable rights is linked to the pre-modern idea of natural right. Natural right is universal because it is an inevitable consequence of the nature of things. Modernity's inalienable human rights are also universal, and universality links them to the pre-modern concept of international law; antiquity saw international law as those laws which were true in all countries due to the nature of things. Modernity's fundamental rights are also natural in that they are inevitably linked to the attainment of Aristotle's idea of the good life in political society. Because fundamental rights are universal and hierarchically superior to ordinary laws they are deductive general principles of the law of reason (Vernunftrecht). The idea of inalienable individual rights which are inviolable and hierarchically superior to ordinary laws is clearly an early modern idea, yet the early modern idea of human rights is intellectually coherent with pre-modern thought. The modern idea that inalienable individual rights are hierarchically superior to ordinary law also coheres with the idea of antiquity that law is subordinate to justice - that an unjust law is not law (lex mala, lex nulla). As fundamental rights were increasingly recognized, they also became relativized against each other in late modernity. Thus, the idea of the just as recta ratio and law as recta ratio naturae congruens, pre-modern concepts, became applied in late modernity to adjudicate conflicts between inalienable fundamental human rights, an early modern concept. Unlike inalienable fundamental human rights, which inhere by nature and are universal, economic rights are alienable, positive, and vary with time and place. Market rights may be bought and sold. Thus, they are positive and subject to economic analyses (interest evaluation and balancing) and are hierarchically subordinate to inalienable human rights.

To understand the global rise and success as well as the contours of this general principle of (constitutional) law and how proportionality serves the constitutionalization and globalization of law, we examine the history of the concept and then its legal practice.

15.2 Proportionality in Antiquity

The idea of justice as proportionality appears first and clearly in Aristotle's Nicomachean Ethics Book V, and also Book III Ch. 10-12. To understand Aristotle's idea of proportionality in law, and to see how it is the root of contemporary proportionality analysis, we must first understand Aristotle's theory of law and justice.

15.2.1 Aristotle's Theory of Law

Aristotle's rightly distinguishes two types of law: nomos and dike (EN V.7). Nomoi, positive laws, are established by convention and like all man-made things (technê - EN 1140 a 1) vary from place to place. Dike in contrast is the idea of universal right *268 which is natural (physis) - that which is natural cannot be otherwise (EN 1139b20-23). Aristotle saw the co-existence of a universal natural law, valid in all places and times, alongside positive national laws which would hold true in one land, but not in another (EN 1134b18-20). Aristotle rightly regarded what we call positive law (nomoi) and natural justice (dike) as complementary, (EN V III.4) not conflicting (EN V.7).

15.2.2 Aristotle's Theory of Justice

Aristotle identifies several forms of justice (EN 1129a27). As a moral virtue, justice addresses the idea of men who are just and acts which are just (EN 1129a31-l 129b7, 1136a25-3, 1135b25, 1135a16, 1133b29-1134al). As a theory of law, Aristotle distinguishes commutative justice and social distributive justice (EN 1131b9-20). Distributive justice is positive, not natural, and may vary from one land to another and follows a geometric proportion. Commutative justice in contrast follows arithmetic proportionality (EN 1131b32-1132al). Equity, in Aristotle's thinking, acting ex aequo et bono, is the means whereby Judges correct errors which result in applying laws ex post to situations legislated ex ante (EN 1137b10-14).

15.2.3 Aristotle's Theory of Proportionality

Proportionately measures distributive justice (Hanau 2004) in Aristotle's schema, which appears to be the earliest known historical source of the contemporary general principle of proportionality in law (Radbruch 2003, 122, nt.). [FN5] Distributive justice addresses public law. [FN6] It is conventional: different societies have different distributive principles. [FN7]

In Aristotle, proportionality is justice as the right ratio - the relationship between a distributive principle and the shares apportioned thereby. Aristotle's idea has since evolved in theory and practice to become more precisely defined. However, the essence of the principle of proportionality in law is clearly seen in Aristotle's idea of distributive justice as a rational principle (recta ratio) which determines the attribution of rights between State and citizen as well as the rule of law. [FN8] The idea of *269 proportionality as a specific rule of law emerged obliquely from Aristotle's thought as an abstract general principle and became an increasingly concrete and definite proposition of the law of self-defense in Cicero, [FN9] Justinian (Digest 43.16.3.9), [FN10] Augustine (City of God, 1998, ch. 7), [FN11] and Aquinas (Sum. Theol. 2a2ae 90-97 esp. 95/3, 96/1). [FN12]

15.2.4 Other Pre-modern Theorists on Proportionality

The well-defined abstract theoretical principle became concretized and realized by Cicero into positive law (Jolowicz and Nichols 1972, 104-5; Schiller 1978, 374-5; Jackson 1915). Cicero (De Republica) describes law as the recta ratio naturae congruens the right ratio, i.e. the proper proportion. This concretization (realization) was further refined by Aquinas in the law of self-defence of states (Sum. Theol. 2a2ae 40). Aquinas then presented the first decomposition of Aristotle's concept into the now known multi-step proportionality procedure (Sum. Theol. 2a2ae 90-97). In the law of self-defence, Aquinas argued that there are conditions which must exist for the use of force to be just (necessity); that force, when used, must not be excessive (proportionality), and that must be exercised by the sovereign according to rules. Aquinas' theory on proportional self-defence, in turn, became seen as a general principle of law by Grotius (War and Peace, 1901, introd. par. 62). [FN13] The principle would apply not just to states in their mutual relations but also to individuals *270 in their mutual relations. Grotius transitions the concept into modernity and links the idea of justice as proportion (ratio) to the idea of interest balancing as a method for dispute resolution. Grotius unites the ancient concept of justice as ratio, the medieval concept of proportional self-defence, and the modern concept of balancing interests. [FN14] Modern proportionality thus emerged as a general principle of law. This legal principle of proportional self defence, first articulated in the law of nations (also known as jus gentium - public international law) was increasingly applied in cases of self-defence (Totten 2007, nt. 36) not only of states, but also of the person, and then in national police and then administrative law: the right to self defence must be exercised in proportion to the threat; punishments should be proportional to crimes; the administration must not act excessively. Rational principle rules - not caprice nor Diktat; and it rules with laws - laws which serve justice.

15.3 Proportionality in Early Modern Law

15.3.1 German Law

This proportionality principle, brought into national law as the right to proportional individual self defence and the duty of the state to punish crime only proportionally, became a heavily litigated aspect of German national administrative law (Sweet and Mathews 2008). In German law, the principle, as Wieacker (1979) noted, is rooted in antiquity. The concept evolved from a prohibition of disproportionality (Uebermassverbot) [FN15] (the state must not act too broadly) toward a more clearly defined and restrictive principle that the state must use proportional means to legitimate ends (Verhaeltnismaessigkeit) in the post-war era. This became a key principle of German constitutional law (Baer 1999). The idea is so popular that it even found expression in East German law, (Mampel 1982, 731-743) evidence of a broader thesis that Socialist law is a variant of Western law, albeit organized by the general principle of equality rather than the general principle of liberty.

15.3.2 Proportionality in Anglo-American Common Law

Proportionality, as a general principle of international law likewise found its way into Anglo-American common law (Gray 1763, 844-847). Proportionality in common *271 law is rooted in Magna Charta, which can be seen as the legal source of the principle of proportionality in British [FN16] and U.S. common law. [FN17] Thus, the Eighth amendment of the U.S. Constitution permits only proportional punishments. [FN18] Just as in German law, in common law the principle of proportionality found its earliest expression in the areas of police powers [FN19] Despite controversy (Pillai 2002), it is clear that punishment must be proportional to the crime. [FN20] The parallel evolution of the proportionality principle in German law and common law is likely due to the common connected conceptual roots of the principle in the thought of Aristotle, Cicero, Aquinas, and Grotius.

Early modernity developed two concepts in parallel:

1. Interest balancing (often seen as political and thus non-justiciable) [FN21]

2. Proportionality (legal and thus justiciable)

Grotius links interest balancing and proportionality (Ch XXIV). He appears to be the source of the fusion of interest balancing and the inquiry into whether law is proportional, i.e. a rational means to a permissible end. However, interest balancing and means-end review address two different types of rights: positive economic rights (interest balancing) and natural human rights (means-end review). The fusion of these two ideas does not appear in the works of the pre-modern primary sources surveyed here.

As we see first in Grotius, the general principle of proportionality (means-end review) and economic interest balancing are sometimes joined together. Their fusion is an error because they address two different categories of rights and are mathematically distinct. That which in Europe is referred to as proportionality analysis is known in *272 U.S. courts, unequivocally, as means-end rational review with strict scrutiny for fundamental rights and suspect classes. Proportionality in Aristotle is expressed as a geometric ratio (A:B::C:D) which may be continuous or discontinuous (Aristotle). Interest-balancing, in contrast, is economic cost/benefit analysis - it is a simple inequality (A>B). To avoid confusing cost/benefit analyses of alienable economic interests with proportionality analysis of conflicting inalienable constitutional rights I advise avoiding the use of the term “balancing” entirely. Interest analysis is an unequivocal alternative term for interest balancing (Interessenjurisprudenz). It is especially important to avoid confusing interest balancing and proportionality because early modernity regarded interests as political and non-justiciable. The dichotomy of non-justiciable political interests versus justiciable legal rights subsists to this day in some views of international law and in the U.S. constitutional ““political question” doctrine.

15.3.3 Proportionality in Contemporary Law (Late Modernity)

In EU law the proportionality test (means-end rational review) is very well worked out. Most recently it was reiterated in Viking and Laval before the ECJ (Case C-438/05). According to the EU Treaty, an interference with a fundamental right “is warranted only if [1] it pursues a legitimate objective compatible with the Treaty [permissible end] and [2] is justified by overriding reasons of public interest [necessary means]; if this is the case, it must be [3] suitable for securing the attainment of the objective which it pursues [rational means to the legitimate end] and [4] not go beyond what is necessary in order to attain it” [least restrictive means; braces supplied by author] (Case C-341/05). “In simplest terms, the proportionality principle requires some articulable relationship between means and ends, specifically that the means chosen by an administration be suitable or appropriate, and no more restrictive than necessary to achieve a lawful end. The principle operates in each of the Member-State's domestic jurisprudence, though not always in hic verba, and it qualifies as a peremptory norm of international law.” (Steinhardt 1994)

As already mentioned, people sometimes wrongly equate the last step in proportionality - the scrutiny as to whether the invasion of the fundamental right is as non-invasive as possible - with (economic) interest balancing. The proportionality inquiry goes to the determination of the right relationship between the means and ends of state action with respect to private rights in accord with the nature of things - it is expressed by Cicero's as recta ratio naturae congruens in De Republica (Commonwealth). Distributive proportionality does not concern the evaluation and comparison of costs and benefits or alienable economic interests. The last step in proportionality analysis is correctly referred to as proportionality strictu sensu (EU) or as “strict scrutiny” (US), and it applies respectively to fundamental rights and to “suspect classes”. “Suspect classes” are discrete and insular politically powerless minorities who have suffered a history of invidious discrimination. Proportionality strictu sensu in the EU is also called proportionality in the narrower sense. Strict *273 scrutiny in the US is also called “least restrictive means analysis”. [FN22] One divergence in proportionality analysis is that the EU does not apply proportionality strictu sensu to govern minority/majority relations (suspect classes). It does however require strict necessity when the state invades a fundamental right. Regardless of this minor transatlantic rift, proportionality strictu sensu (strict scrutiny) is decidedly not the balancing of competing (economic) interests. Proportionality strictu sensu is the inquiry into whether state power is exercised in the least invasive manner possible. Any divergence between the ideas of strict scrutiny for suspect classes, least restrictive means analysis, and proportionality strictu sensu - which are all rooted in the same concept, necessity - would be found in the question of how majorities and minorities relate to each other. Proportionality strictu sensu (strict scrutiny) is not the question of identifying the interested parties, defining their competing interests, assigning weights to the interest, and then deciding the issue in favor of maximizing social wealth - which is the method known as legal process interest balancing (Interessenjurisprudenz).

The term “balancing” in law is used to indicate several different things, namely:

1. Commutative justice - after application, the scales of justice are restored to their ex ante balance, e.g., lex talionis.

2. Cost-benefit analysis (CBA) - we weigh the costs and benefits of a policy against the costs and benefits of another policy and decide for the one that generates the most social wealth. Cost-benefit analysis is the proper contemporary use of the term balancing, but that use rightly applies only to alienable economic rights, not to inalienable (fundamental, universal, human) rights. CBA is positive law, not natural justice.

*274 3. Proportionality in the narrower sense i.e. proportionality strictu sensu.

The use of the term “balancing” to indicate proportionality in any sense should simply be avoided and allowed to fall into disuse to prevent confusion. ““Proportionality in the narrower sense”, “proportionality strictu sensu” or even “strict scrutiny”.are all more exact terms for the last step in proportionality analysis and avoid confusing different concepts by referring to them with the same term. Multi-factor Interest balancing is the determination of interested parties, of their competing interests, and their evaluation against each other: its roots are in the free-law school (freie Rechtslehre), a forerunner of critical legal studies, and its methodology is known as Interessenjurisprudenz - interest analysis. Interest balancing is the comparison of costs and benefits associated with competing interests and is decided by the principle of social wealth maximization. Thus, interest balancing logically refers to alienable economic rights rather than inalienable fundamental human rights.

15.4 Conclusions

This historical overview shows that proportionality as a principle of law arose out of the Aristotelian concept of justice. This general theoretical fact partly explains the worldwide success of the concept, since it has deep, global common roots. After Aristotle, the principle was refined by Cicero, Aquinas and Grotius and instantiated into law by Justinian, Magna Charta, the US Constitution, and the Prussian Civil Code (ALR). Proportionality in law first appeared as a principle of the law of war, then in national police law and then administrative law from whence it evolved into a principle of constitutional law as a tool for the adjudication of conflicting constitutional rights. The principle has become a method for the formation of a transnational ius commune, a hybrid of common law (inductive binding case law) and civil law (deductive general principles). Because the proportionality principle is one of the most important methods of converging global norms, future developments of the general principle of proportionality should seek:

1. to develop a universally coherent terminology which avoids confusion. Meansend review with strict scrutiny for suspect classes and proportionality are methodologically synonymous and unequivocal. Interest balancing, in contrast, is a much broader term and is equivocal or at least polysemious.

2. to clearly delineate the positive law (economic interest balancing) versus the natural right aspects (means-end rational review) of proportionality discourse. Economic interest balancing through cost/benefit analysis and similar economic tests are inappropriate for adjudication of fundamental inalienable rights.

Aristotle's thought about law and justice structures jural relations to this day. [FN23] Proportionality, like causation (Met. 1.3.11) in tort (Engle 2009) is one more example *275 of Aristotle's influence on contemporary law. True, Aristotle was sexist (Pol. 1254b 10-15,1259b 2) racist (Pol. 1252 b 7, 1237 b 23-32) and homophobic. However, his contributions to justice through law are the greater and better part of his work. Aristotle's schema of justice, brilliantly and pithily exposed in Book V of The Nicomachen Ethics(*) (see also EN III.10-12) holds true to this day (Engle 2008) as seen in concepts such as equity jurisprudence (ex aequo et bono), proportionality (law as right ratio) and causation in tort. Aristotle's distinction between commutative and distributive justice is tenable and useful. Aristotle serves as a good common universal starting point for contemporary thought about proportionality. This brief historical overview provides a synopsis of the history of proportionality discourse so that contemporary jurists can continue to develop the rule of law in the most rational way to serve justice.

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[FNa1]. Humbolt Universität, Berlin, Germany

[FN1]. As legal concepts the idea of inalienable individual rights is first seen in the founding US constitutional documents such as the US Declaration of Independence (1776 “all men are created equal; they are endowed by their creator with certain inalienable rights”) and the US Constitution. The first modern legal use of the exact term “human rights” is the French Déclaration des Droits de l'Homme of 1789 (lit.: declaration of the rights of man).

[FN2]. Aristotle's idea is that distributive justice can be expressed as ratio, i.e. proportionality, is one form of justice. As one of my anonymous peer reviewers rightly pointed out, Aristotle also discusses justice in the sense of a personal virtue - a virtuous mean. (EN 1106b28) Thus, Aristotle's theory of justice also connects to his theory of moral virtue as the prudent mean between opposite extremes of vice. (EN 1107a2) As one of the reviewers noted, adjudication involves the judicial prudential determination of the mean between the opposing positions of the plaintiff and defendant. Adjudication is an act of practical reasoning - Aristotle's phronêsis. Aristotle's ideas of justice as ratio and virtue as mean explain the application of the proportionality to distributive and commutative justice - respectively, social justice (proportional shares in the constitution of the Polis, i.e. the State) on the one hand and proportional punishment of crimes on the other.

[FN3]. Id.

[FN4]. “From German origins, proportionality analysis spread across Europe, into Commonwealth systems (Canada, New Zealand, South Africa), and Israel; it has also migrated to treaty-based regimes, including the European Union, the European Convention on Human Rights, and the World Trade Organization” (Sweet 2008).

[FN5]. See also Bydlinski (1991, 339); Engisch (1971, 162, 222, 229).

[FN6]. Aristotle clearly intends his concept of distributive justice (the just as ratio, i.e. proportionality) as governing public-private relations. “Of particular justice and that which is just in the corresponding sense, (A) one kind is that which is manifested in distributions of honour or money or the other things that fall to be divided among those who have a share in the constitution” (EN, Book V Ch. II v. 8 (emphasis added).

[FN7]. EN 1131 a 24-28.

[FN8]. “We do not allow a man to rule, but rational principle, because a man behaves thus in his own interests and becomes a tyrant” (EN V, VI, 5).

[FN9]. “No war can be undertaken by a just and wise state, unless for faith or self-defence. This self-defense of the state is enough to ensure its perpetuity, and this perpetuity is what all patriots desire. Those afflictions which even the hardiest spirits smart under poverty, exile, prison, and torment private individuals seek to escape from by an instantaneous death. But for states, the greatest calamity of all is that death, which to individuals appears a refuge. A state should be so constituted as to live for ever. For a commonwealth, there is no natural dissolution, as there is for a man, to whom death not only becomes necessary, but often desirable. And when a state once decays and falls, it is so utterly revolutionized, that if we may compare great things with small, it resembles the final wreck of the universe. All wars, undertaken without a proper motive, are unjust. And no war can be reputed just, unless it be duly announced and proclaimed, and if it be not preceded by a rational demand for restitution.

Our Roman Commonwealth, by defending its allies, has got possession of the world” (Treatise on the Commonwealth, trans. Barham 1841-42, emphasis supplied).

[FN10]. “Those who do damage because they cannot otherwise defend themselves are blameless ... It is permitted only to use force against an attacker and even then only so far as is necessary for self-defense.”

[FN11]. Augustine discusses just war theory but doesn't use the term proportionality (between force and threat). He does however use the term “just war”. This seems to be the first use of the signifier “just war” (certainly one of the earliest).

[FN12]. Also see Aquinas Sum. Theol. 2a2ae 40.

[FN13]. “The Law of Nations does not consist, therefore, of a mere body of deductions derived from general principles of justice, for there is also a body of doctrine” (trans. Campbell).

[FN14]. Chapter XXIV: Precautions Against Rashly Engaging in War, Even Upon Just Grounds. “In all cases of deliberation, not only the ultimate but the intermediate objects leading to the principal ends are to be considered. The final object is always some good, or at least the evasion of some evil, which amounts to the same. The means are never to be considered by themselves, but only as they have a tendency to the proposed end. Wherefore in all cases of deliberation, the proportion, which the means and the end bear to each other, is to be duly weighed, by comparing them.”

[FN15]. Remmert(1995).

[FN16]. Hodges v. Humkin (1615). “By the seventeenth century, England had extended this principle to punishments that called for incarceration. In one case, the King's Court ruled that “imprisonment ought always to be according to the quality of the offence”.

[FN17]. See Note The Eighth Amendment, Proportionality, And The Changing Meaning Of Punishments (2009). “In Solem v. Helm, Justice Powell traced the history of the Cruel and Unusual Punishments Clause back to the Magna Carta and the English Bill of Rights of 1689, which he found to have embodied a strong principle of proportional punishment.” See also Harmelin v. Michigan (1991).

[FN18]. Coker v. Georgia (1977), Eberheart v. Georgia (1977). “The Eighth Amendment requires that every punishment imposed by the government be commensurate with the offense committed by the defendant. Punishments that are disproportionately harsh will be overturned on appeal. Examples of punishments that have been overturned for being unreasonable are two Georgia statutes that prescribed the death penalty for rape and kidnapping”.

[FN19]. U.S. Const, Amdt. VIII. See also Solem v. Helm (1983), overruled by Harmelin v. Michigan (1991). The proportionality of punishment to crime is, however, differently tested than the proportionality of means to ends. In Solem, the Court determined that objective criteria should guide the proportionality analysis. (463 U.S. at 292) The objective criteria considered by the Court were “(i) the gravity of the offense and the harshness of the penalty; (ii) the sentences imposed on other criminals in the same jurisdiction; and (iii) the sentences imposed for commission of the same crime in other jurisdictions” (Id. at 292).

[FN20]. Coker v. Georgia (1977); Eberheart v. Georgia (1977) (State death penalty for rape and kidnapping unconstitutional as disproportionate).

[FN21]. See the Federalist Papers (1961); also Montesquieu's (1914), Spirit of the Laws.

[FN22]. Aristotle (EN V.3.) “The just, then, is a species of the proportionate (proportion being not a property only of the kind of number which consists of abstract units, but of number in general). For proportion” is equality of ratios; and involves four terms at least (that discrete proportion involves four terms is plain, but so does continuous proportion, for it uses one term as two and mentions it twice; e.g. ‘as the line A is to the line B, so is the line B to the line C’; the line B, then, has been mentioned twice, so that if the line B be assumed twice, the proportional terms will be four); and the just, too, involves at least four terms, and the ratio between one pair is the same as that between the other pair; for there is a similar distinction between the persons and between the things. As the term A, then, is to B, so will C be to D, and therefore, alternando, as A is to C, B will be to D. Therefore also the whole is in the same ratio to the whole; and this coupling the distribution effects, and, if the terms are so combined, effects justly. The conjunction, then, of the term A with C and of B with D is what is just in distribution, and this species of the just is intermediate, and the unjust is what violates the proportion; for the proportional is intermediate, and the just is proportional. (Mathematicians call this kind of proportion geometrical; for it is in geometrical proportion that it follows that the whole is to the whole as either part is to the corresponding part.) This proportion is not continuous; for we cannot get a single term standing for a person and a thing.” (trans. Thomson, emphasis added: Geometric justice is distributive it is the even handed application of a positive general principle to apportion shares of social goods to the citizens who then may interchange their goods according to the principles of commutative, i.e. arithmetic, transactional justice).

[FN23]. One can for example trace the dichotomies of universal international law / particular national law, public law / private law and voluntary obligations (contract) / involuntary obligations (tort) to Aristotle (See, e.g. EN 1130 b 30-1131 a 8).