At the end of the sixth century BCE, Cleisthenes of Athens reorganized the city’s administrative structure. These reforms had a profound impact on the Athenian political system. After the reforms, every Athenian became a member of one of the ten new tribes (phylai). Each tribe consisted of three ‘thirds’ (trittyes) from each of the three regions of Attica: the coast, the city and the inland area. The 30 tribes were further divided into different numbers of demes (demoi), which constituted the smallest organizational units of Cleisthenes’ reforms. Cleisthenes also reorganized the Athenian council (boule). The new council consisted of five hundred members, with fifty representatives from each tribe (Ps.-Arist. Ath. Pol. 21). In recognition of Cleisthenes’ achievements, the historian Herodotus referred to him as the founder of Athenian democracy (Hdt. 6.131).
In their monograph La Cité et le nombre: Cleisthenes d’Athènes, l’arithmétique et l’avènement de la démocratie (2024), Paulin Ismard and Arnaud Macé offer a new perspective on the nature of Cleisthenes’ reforms. The introduction (Prologue, pp. 7–17) outlines the book’s agenda. According to Ismard and Macé, the Athenians accepted Cleisthenes’ measures because his reforms were based on arithmetic operations, such as division and redistribution, that were part of the Greeks’ traditional collective knowledge. This knowledge was passed down from generation to generation within the Greek population, and so was well known among the Athenians.
Chapter 1, ‘L’événement Clisthénien’, discusses the chronological sequence of events, Cleisthenes’ objectives, and the consequences of his reforms. He aimed to mix the Athenian citizen body in order to weaken former local bonds and to create a new civic identity across the regions of Attica. He also encouraged greater participation by the Athenian demos in political and judicial decision-making, thereby fostering Athens’ military strength. Furthermore, Cleisthenes reorganized Athens’ naval forces. According to the fourth-century BCE historian Cleidemus (Cleidemus FGrH 323 F8), Cleisthenes increased the number of so-called naucraries (naukrariai), an early division of Attica’s population related to the organization of the naval forces, to 50. Ismard and Macé argue that this increase indicates that Cleisthenes’ reforms also encompassed the reorganization of the Athenian military forces. However, they acknowledge that there is a lack of ancient evidence for the existence of naucraries after 500 BCE (pp. 41–42).
According to Ismard and Macé, Cleisthenes’ reorganization of Athens mirrored his close association with the Athenian demos. On the basis of the number of new tribes (ten), councilmen representing a tribe in the new Boule (fifty), and trittyes comprising each tribe (three), the authors postulate the high relevance of the numbers three, five, ten, and fifty in what they term Cleisthenes’ ‘universe of numbers.’ Contrary to the views of Pierre Lévêque and Pierre Vidal-Naquet (Clisthène l’Athénien, Essai sur la représentation de l’espace et du temps dans la pensée politique grecque de la fin du VIe siècle à la mort de Platon, Paris, 1964), the authors argue that the numbers prevalent in Cleisthenes’ reforms were not primarily influenced by the elitist philosophy of the Pythagoreans, but were based on arithmetic operations that had been familiar to the Greeks for many generations (pp. 47–49).
The second chapter (‘Notre hypothèse,’) provides an overview of the central hypothesis of the monograph. Cleisthenes’ reforms of the demes, trittyes and tribes were so successful because they were based on the everyday application of simple arithmetic and a small number of key figures. Most Athenians were familiar with this knowledge, as it was tied to important social practices such as the organization of military forces (p. 66). These practices had been handed down among the Greeks for centuries. According to Ismard and Macé, the close relationship between these basic arithmetic operations and military skills can be traced back to the Homeric epics (pp. 69–71).
The third chapter is titled ‘Trois jeux pour construire la cité’. It explores the important role of the numbers three, five and ten in Cleisthenes’ reorganization of Athens, the possible influence of the Pythagorean notion of the tetractys on Cleisthenes’ measures, and how Plato’s philosophical treatises offer valuable insights into the application of fundamental arithmetic operations to the organization of city-states. The number three refers to the three main regions of Attica, which were represented in Cleisthenes’ system of three trittyes per tribe. The number five was present in the fifty naucraries, the fifty prytaneis, and the fifty councilmen, who were elected from each tribe to form the new Boule of five hundred citizens. According to Ismard and Macé, the significance of the number ten in Cleisthenes’ reforms was based on the Pythagorean concept of the tetractys — the sum of the first four numbers (1 to 4), which equals ten. The Pythagoreans considered ten to be a perfect, or even divine number, which formed the basis of their theories on music and geometry (pp. 72–73).
The authors highlight the great importance that Plato places on mathematical education in the military and the organization of the city-state (e.g. Plat. Rep. VII, 521d–527d; Plat. Leg. VII, 819a8–b2). Of particular importance are the passages in Plato’s Laws referring to the ideal city of Magnesia. Ismard and Macé argue that educational games for children described in Plato’s Laws exemplify the mathematical knowledge employed by Cleisthenes in his reforms (pp. 75–77).The authors focus on three types of game: dividing objects into equal-sized groups, dividing objects into groups of different sizes, and distributing and recombining different objects into groups of various sizes. They emphasize that similar arithmetic operations can be found in episodes from the Homeric epics, where they are applied in military contexts. On the basis of these observations, and on the practicality of these basic arithmetic operations in everyday life, Ismard and Macé argue that this knowledge formed part of Greek culture from the time of Homer and was familiar to many Greeks. The authors further emphasize the benefits of selecting 5,040 people as the initial population for the model city of Magnesia in Plato’s Laws. Due to the many ways in which 5,040 citizens can be divided into groups of various sizes — the number 5,040 is divisible by all numbers from one to twelve, except eleven — important organizational measures such as distributing land to citizens, providing food for all households, and recruiting soldiers to defend the city can be implemented without issue (pp. 91–99).
Chapter 4, ‘Recomposer la cité: les expériences des cités archaïques’, examines archaic cities more closely to explore potential role models for Cleisthenes’ reforms. The first example is the reform carried out by Demonax of Mantineia in Cyrene. Demonax divided the Cyreneans into three phylai (tribes) according to their origin: (1) Greeks from Thera and dispossessed Libyans; (2) Greeks from Crete and the Peloponnese; and (3) Greeks from the Aegean Islands (Hdt. 4.161). Like Cleisthenes’ formation of tribes from three trittyes, Demonax’s reorganization illustrates the arithmetic operation of dividing into three equal-sized groups. Aristotle (Arist. Pol. VI, 1319b19–20) also compares the reforms of Demonax and Cleisthenes in his Politics (pp. 104–116).[1]
Corinth is another Greek city that underwent reform. Scarce information from ancient sources suggests a reorganization of the former three Dorian tribes into eight new tribes. Ismard and Macé propose that this reform could have served as a model for Cleisthenes (pp. 116–127). In addition, they present the ‘puzzle of Eretria’ (‘un puzzle eretrien’, pp. 128–134) and argue for a potential link between Cleisthenes’ reorganization of Athens and reform in Eretria, on the island of Euboea. They also suggest that the city of Argos and the territory of Thessaly were reorganized (pp. 134–142).
Chapter 5, ‘Cleisthène le Joueur’ (pp. 143–169) has an unfortunate title. This chapter partly reiterates information that has been presented already in previous chapters. It discusses Greek cities that are said to have been reformed before Cleisthenes (p. 143). This is followed by a more detailed overview of the main aspects of Cleisthenes’ reforms as described in the Pseudo-Aristotelian Athenaion Politeia (Ps.-Arist. Ath. Pol. 21, pp. 144–151). The authors emphasize their hypothesis that Cleisthenes’ reorganization of the Athenian citizen body drew on long-standing Greek knowledge of basic arithmetic operations, and that he had predecessors such as Demonax in Cyrene (pp. 165–169).
A brief epilogue (pp. 171–179) highlights the relevance of Plato’s treatises as ancient sources illustrating the traditional collective knowledge of the Greeks on basic arithmetic operations. Additionally, Aristotle’s Politics is emphasized as this work shows how the various parts of a politeia can be reorganized to achieve different objectives. A glossary of important Greek terms (pp. 181–183) and a bibliography (pp. 185–199), which has a focus on French scholarship, conclude the monograph.
Despite some repetition throughout the book, Paulin Ismard and Arnaud Macé address an important topic. They argue that Cleisthenes’ reforms were widely accepted because they were based on fundamental arithmetic operations which had been part of Greek knowledge since the time of Homer. However, one point in the line of argument is difficult for the reader to resolve. In the first chapters, the authors distance themselves from earlier research into the influence of Pythagorean number theory on Cleisthenes, proposing instead that Cleisthenes used collective knowledge of numbers and arithmetical operations among broad circles of Greeks rather than an elitist Pythagorean theory (see esp. p. 48). Yet, they subsequently describe the Pythagorean tetractys (the number ten as the sum of the numbers one to four) to explain how Cleisthenes increased the number of phylai from four to ten (pp. 72–73).
The question of whether the Homeric epics and Plato’s writings support Ismard and Macé’s hypothesis regarding collective knowledge of arithmetic operations among the Athenians also deserved more attention. After all, both the Homeric epics and Plato’s works are strongly influenced by an elite perspective. It would have been helpful if Ismard and Macé had provided more detail on how games were used for the mathematical education of children from a wider range of social groups, not just elite families. The book’s strength lies in encouraging readers to consider Cleisthenes’ reforms within a broader context, and for this, it is to be praised.
Notes
[1] For Cyrene and Cleithenes’ reforms cf. also Lucia Checchet, Re-Shaping and re-founding citizen bodies, in: Lucia Checchet/Anna Busetto (eds.), Citizens in the Graeco-Roman world. Aspects of citizenship from the Archaic Period to AD 212, Leiden/Boston, 2017, pp. 50–77 (not mentioned in the bibliography of La cité et le nombre).