Contrary to what modern observers might have you believe, tax dodging during the 1950s and 1960s was more about tax cuts than tax increases. Faced with a high tax rate it did not support, but, for political reasons, it could not lower, Congress did the next best thing. It riddled the tax laws with “leaks, loopholes, exemptions, and preferences,” while looking the other way at much of the widespread “income-tax chiseling” in American society and only occasionally passing watered down legislation targeting the more high-profile tax dodging schemes.1 In effect, it cut the tax rates implicitly, rather than explicitly, which amounted to a tax cut of the worst kind. It was not transparent, it was not evenly distributed among the taxpayers or even targeted to achieve any policy objective in some cases, and, because it was too unpredictable for taxpayers to rely upon for planning purposes, it was inefficient.

In this chapter we return to the topic of Chapter 3, (small) sieves, which we now treat, at least initially, in some generality. However our objective is to give nothing more than an introduction and some applications to what has become a vast and complex subject. Readers who wish to see the many aspects of the subject in more detail are advised to consult the standard reference on the subject, Friedlander & Iwaniec (2010).

Minkowski’s convex body theorem provides a fundamental bridge between algebraic number theory and Euclidean geometry. The theorem asserts that any sufficiently large, centrally symmetric, convex set in n-dimensional real space must contain a nonzero lattice point.

Classical applications include representing integers as sums of two or four squares, where the existence of lattice points in two- and four-dimensional balls yields concrete arithmetic solutions.

The theorem’s deeper significance appears in the context of number fields, which are finite extensions of the rational numbers. These fields can be embedded into Euclidean space, turning arithmetic problems into geometric ones. The ring of integers and its ideals can be regarded as lattices, making Minkowski’s theorem directly applicable.

This geometric viewpoint leads to central results in algebraic number theory, such as the finiteness of the class group and explicit bounds on its size.

Likewise, the group of units can be regarded as a lattice in Euclidean space via the logarithmic embedding, and Minkowski’s theorem is instrumental in determining its structure (Dirichlet’s unit theorem).

These ideas come together in the analytic class number formula, which relates the size of the class group to the residue of the Dedekind zeta function at s = 1 and to the regulator.

This book comes in two parts; the first, consisting of §§1–7, offers an informal axiomatic introduction to the basics of set theory, including a thorough discussion of the axiom of choice and some of its equivalents. The second part, consisting of §§8–14, is written at a somewhat more advanced level, and treats selected topics in transfinite algebra; that is, algebraic themes where the axiom of choice, in one form or another, is useful or even indispensable.

Talking with women in the pre-conception or perinatal periods about psychotropic medication is an essential, sometimes difficult, part of the work of the perinatal psychiatrist. Understanding the current evidence base; knowing how and when to acknowledge the uncertainty inherent in current knowledge and how that translates to the individual woman; balancing risks of medication with risks of not treating and benefits of treating; sharing decision-making while not putting all of the responsibility on the woman; communicating with the woman, her partner, other professionals, services and agencies; and knowing when and how to seek further help or advice, are all essential components of good practice when prescribing in pregnancy and breastfeeding.

The expansion of the Roman Empire into the Mediterranean in the early second century BCE represented a gradual diminution of the independence and autonomy of the Greek cities. At the same time, processes internal to the poleis were moving them in a more elitist direction, as the “big benefactors,” ultrawealthy men who bestowed ever-greater favors on their cities, moved toward monopolizing participation in civic magistracies. The council and other political bodies became off-limits to citizens who were not among the euergetistic elite. Still, democratic institutions and ideas of the previous period persisted, especially in the popular assembly. Christianity, the centralization of administrative power in the Roman Empire under Constantine, and various crises combined to deprive the cities of the last vestiges of dēmokratia in the fourth century CE, when popular assemblies largely disappear from the poleis.

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.

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.