This chapter is devoted to continued fractions, a classical yet often overlooked subject. Continued fractions provide an efficient representation of real numbers, expressing them as sequences of integers through iterative fraction expansions. Every real number has a continued fraction expansion – finite for rationals and infinite for irrationals – with convergents that yield optimal rational approximations. This property makes continued fractions invaluable in Diophantine approximation, where one seeks to approximate irrational numbers by rationals with minimal denominators.

The theory of continued fractions is deeply connected to Möbius transformations, which describe their recursive structure. The action of the group of 2 × 2 invertible integer matrices on quadratic irrationals via Möbius transformations leads to periodicity in their expansions – a characteristic property of quadratic irrationals. This periodicity is intimately related to solutions of Pell’s equation and to units in quadratic rings.

Among the applications of continued fractions are the characterization of integers that are sums of two squares and the solution of some quadratic Diophantine equations (Cornacchia’s algorithm). Beyond number theory, continued fractions have practical applications such as Wiener’s attack on RSA, which exploits the continued fraction expansion of ratios related to the private exponent to break poorly chosen keys.

Genocide is a contested concept with normative, legal, political, and empirical dimensions, each pulling in different directions. The chapter emphasizes that, in addition, a fundamental contradiction exists in the concept: The legal definition, anchored in an international treaty, dominates the social sciences, even though the legal definition is problematic for the specification of a particular type of violence. The concept is thus perpetually trapped between legal and empirical imperatives, rendering it unwieldy for the social science objective of isolating a particular type of phenomenon. The chapter rejects the solution of jettisoning the concept, instead proposing a set of solutions for how to retain the concept and separate it from its legal origins. The chapter concludes by arguing that such solutions are unlikely to carry the day, but that social scientists should still seek to develop a rigorous conceptualization of the term that allows for the identification of specific types of political violence. Finally, a postscript offers an interim assessment – from the perspective of this chapter – of the war in the Middle East that began on October 7, 2023.

This chapter traces the way that employers used the expense account and the deduction for entertainment, meal, and travel expenses to facilitate a rapid expansion of tax dodging. Employers were able to divert taxable compensation that would have been subject to the steep post-war marginal rates to business expenses that the employer could deduct and the employee could exclude from his individual income tax return. People of modest means were able to live lavishly, encouraging the development of country clubs, dinner clubs, expensive restaurants, and post-dinner entertainment options such as theaters and operas, to service this new demand. Resorts also began to host business conferences to attempt to make vacations tax deductible. Moreover, the rise in conspicuous consumption appeared to induce others to seek out similar opportunities to finance personal consumption with tax-deductible dollars.

This chapter emphasizes the growth and consolidation of democratic regimes across the ancient Greek world beginning in the fourth century BCE. The conquests of Alexander the Great and the policies of his successors did not spell the death of ancient Greek democracies but instead may have actually increased their number. Highly participatory institutions, including assemblies, magistracies, and law courts, can be found across the Mediterranean and beyond. Democracies shared best practices in this period on how to fend off tyranny, oligarchy, excessive demagoguery, and other threats to the democratic constitution. Festivals, monumental art, religious cult, and coinage contributed to a specifically democratic culture. At the same time, democracies settled on a general paradigm in which citizen women, while enjoying certain civic privileges, were excluded from political decision-making; democratic stability also depended on the presence of an enslaved class.

Chapter 3 explores the representation of the royal body—both literal and metaphorical—in early eighteenth-century New Spain. It argues that references to the body of the king, the body politic, and the body of Christ were used strategically to affirm loyalty and unity under Bourbon rule. Drawing on sermons, ephemeral art, and royal portraiture, this chapter shows how the perceived contrast between the frail Carlos II and the youthful, fertile Felipe V was mobilized to reinforce a narrative of imperial resurrection. Through symbolic condensation, sacred imagery such as Lazarus, the Eucharist, and the phoenix converged to suggest the rebirth of the monarchy. The chapter also introduces the idea of ‘composite identities,’ in which loyalty to king, kingdom, and locality were expressed through overlapping symbolic systems. It highlights how cultural producers in New Spain helped local subjects imagine themselves as part of a revitalized imperial body.

Pregnancy induces vast physiological shifts within several systems of the expectant mother’s body, including haemodynamic, haematological, renal, endocrine and metabolic functions. These are necessary to adjust conditions for optimal growth and development of the fetus but are finely tuned to avoid subsequent compromise to maternal health; an imbalance of which may present with medical complications to both mother and baby.

It is essential to understand normal physiological changes in pregnancy in order to appreciate pathology, which may arise. This chapter aims to outline the main changes that occur, and further, to delineate a few common obstetric emergencies and complications that may develop during pregnancy.

An anonymous and rarely cited text, Fe política de un colombiano was consulted by the editors at the John Carter Brown Library, whose catalogue attributes it to Eloy Valenzuela (1756–1834), a Catholic priest from Santander, Colombia. However, it is highly unlikely that Valenzuela, who was close to Simón Bolívar at the time of publication, penned a pamphlet which appeared to be criticizing (though not in explicit terms) the Convención de Ocaña, convened by Bolívar to replace the existing 1821 Constitution. The main message of the Fe política was against the concentration of power in the executive, then perceived to be Bolívar’s purpose in convening the Ocaña Convention. On April 1, 1826, the Gaceta de Colombia registered its publication stating that “the author … was born in one of the departments of the antigua Venezuela.” Its author might have been Javier Francisco Yanes (1776–1846), particularly as in his Manual político del venezolano, Yanes used the Fe política without attribution, in a book that is seemingly generous in acknowledging the work of others. But if the Gaceta de Colombia was right, this would rule out Yanes’ authorship as he was not born in Venezuela but in Cuba.

Cyclotomic fields, fields obtained by adjoining the nth roots of unity to the rational numbers, play a pivotal role in number theory as they contain all abelian extensions of the field of rational numbers according to the Kronecker–Weber theorem. They also offer insights into reciprocity laws and certain Diophantine equations, such as the Fermat equation.

In this chapter, we offer a second proof of the quadratic reciprocity law, which shows how Gauss sums over roots of unity encode quadratic residue behavior. This approach reveals the intrinsic connection between quadratic fields and cyclotomic fields.

We prove the particular case of the Kronecker–Weber theorem asserting that a quadratic field of discriminant D is contained minimally in the cyclotomic field of |D|th roots of unity.

Of great historical relevance is Kummer’s proof of the first case of Fermat’s Last Theorem for regular primes, which uses the unique factorization of ideals in cyclotomic fields. Kummer introduced ideal theory to overcome limitations posed by the non-unique factorization of elements in attempted proofs of FLT. His work laid the foundation for modern algebraic number theory, illustrating the profound utility of cyclotomic methods in solving classical problems and establishing these fields as central objects in arithmetic geometry.

The pioneering role of the Netherlands in shaping European law, particularly through its early engagement with the European Court of Justice is well known. Dutch courts, businesses, and legal scholars played a central role in contributing to the breakthrough of the constitutional interpretation of European law, most notably through the landmark Van Gend en Loos (1963) ruling. The chapter explores why the Netherlands emerged as a key actor in European legal integration. It attributes this to the Dutch legal system’s openness to international law and a dominant self-image of the Netherlands as defender of international law, amongst others. However, the conventional narrative that Dutch integration with European law was smooth and unproblematic is also challenged. It reveals that the Netherlands’ role as a “pioneer” of European law was as much a product of strategic national interests as it was of ideological commitment to legal integration.

Chapter 6 (Mishnah ‘Abodah Zarah 3:4 and the Bad Faith Argument): In this chapter, I analyze the existing scholarship on the much-discussed narrative of Rabban Gamliel in the Bathhouse of Aphrodite in m. ‘Abodah Zarah 3:4. I then propose that Rabban Gamliel’s response to Proclus within the story is, in fact, best understood as a deployment of the “Bad Faith Argument.”

This chapter reflects on the book’s exploration of drama and performance that reveal the entanglements of natural phenomena and human behaviours in society, each shaping one another in dynamic interplay. The book concludes by considering how artists continue to want to make work about anthropogenic ecological change. We suggest here that performative actions of dissent can offer shared optimistic moments. This chapter looks at the tactics of tactics of artists and activists who perform actions to physically resist, slow or block ecological damage, attempting to hold big corporations and politicians to account and keep urgent ecological issues under public scrutiny.