We investigate numbers of faces of polytopes. We begin with the face numbers of 3-polytopes. The characterisation of $f$-vectors of $d$-polytopes ($d\ge 4$) is beyond our current means.In view of this, researchers have considered characterisations of the "projections" of the $f$-vectors, namely the proper subsequences of the $f$-vector; we review the existing results. Section 8.2 gives a proof of a theorem of Xue (2021) on the minimum number of faces of $d$-polytopes with at most $2d$ vertices, answering a conjecture of Grunbaum (2003). This is followed by results on the minimum number of faces of $d$-polytopes with more than $2d$ vertices. We then discuss the lower and upper bound theorems for simplicial polytopes, due to Barnette (1973) and McMullen (1970), respectively, and their extensions such as the $g$-conjecture of McMullen (1971), now the $g$-theorem. The proof of the lower bound theorem connects rigidity theory and the combinatorics polytopes. The chapter ends with a discussion of the flag vector of a polytope. This includes a result of Bayer and Billera (1985) on linear equations for flag vectors like the Dehn--Sommerville’s equations for simplicial polytopes.

Studies of agency are crucial if we are to grapple with pressing societal and environmental problems. Relevant conceptual and methodological solutions are needed to make alternative futures possible. This chapter outlines a broad position from which the subsequent contributions to this edited volume depart: one that recognises the urgency of agency and the value of cultural-historical perspectives in breaking away from problematic notions that frame agency as a matter of individuals pitted against the social, or in which individual actions lose their social contingency. Elaborating agency as a matter of struggle where individual and social are in dialectic relations, the chapter focusses on motives, mediation and motion. Within a broader and still-evolving cultural-historical framework, these motifs offer a distinctive way to deal with the challenges of conceptualising and facilitating agency, one which brings alternative futures into the realm of the possible by linking agency with learning and development.

Chapter 4 explores cross-linguistic variation in the language of space along with the potential implications of this variation for the spatial conceptualization of time.

This chapter examines Aquinas’s views on the ontological status of the actualization of an agent’s active power, namely its action, and the patient’s passive power, namely its passion. Aquinas claims that one and the same motion constitutes both an agent’s action and its patient’s passion. This chapter considers Aquinas’s motivations for defending the “action-passion sameness” thesis and his responses to common objections. The chapter also includes a solution to a longstanding interpretive difficulty regarding Aquinas’s views on the ontological status of action. Aquinas claims in some texts that actions are accidents in the agent as subject. This seems to conflict with his standard view that an agent’s action is the motion which it causes in its patient. While advancing a solution to this textual difficulty, the chapter proposes a novel interpretation of Aquinas’s understanding of the relationship between forms and accidents and the metaphysics of inherence.

This chapter locates Spinoza’s scientific interests and contributions in the context of the disciplinary categories of the seventeenth century, investigates the authorship of two small treatises (on the rainbow, and on the calculation of chances) often attributed to him, describes his scientific correspondence, evaluates his strengths and weaknesses as an expositor of Cartesian physics, assesses the role of Cartesian physics in his own philosophy, and explores his conception of methodology in the natural sciences.

This chapter applies the general hylomorphic theory of the human act to one of the two kinds of human acts that Aquinas countenances, namely, bodily human acts. In keeping with the general hylomorphic framework, it argues that a bodily human act is a composite of a volitional act of use and a bodily commanded act. This chapter investigates each of these two components and considers how they form one unified human act. It argues that while use is an immanent act, any bodily commanded is traunseunt. That is to say, use remains in the agent, while the bodily commanded act, which is an exercise of the “motive power” (vis motiva), comes to inhere in a patient external to the human body. What is more, it argues that use is an intrinsically instantaneous act, whereas any bodily commanded act takes time to be completed. On this basis, it concludes that use and the bodily commanded act are inherentially as well as durationally heterogeneous. The last section of the chapter examines whether this dual heterogeneity prevents use and the bodily commanded act from constituting a unified hylomorphic whole. It argue that this is not the case.

Emphasis on the ‘craftsmanlike’ character of creation in the Timaeus can give the impression that the cosmos is no more an ‘animal’ than Dr Frankenstein’s monster. But Middle Platonists took more seriously the biological implications of the claim that the god is the world’s father as well as its maker, implanting a soul in matter which (as in all animals) brings the cosmos to maturity through its own creative agency. Entailments of the view are that the world soul is first and foremost the ‘nutritive’ soul of the cosmos and that the soul must be a structural feature of the cosmic body rather than a distinct substance.

This chapter offers an overview of key points of entry for the study of eighteenth-century science. The first section addresses how seventeenth-century philosophers challenged Aristotelianism and ancient cosmologies. The second details the importance of empiricism in the new study of the natural world. The third focuses on the roles of specific instruments and institutions in natural philosophical inquiry. Sections four and five cover two of the fiercest philosophical debates of the period: first, about gravity and action at a distance, and second, about theories of matter and spirit. The final section examines encyclopaedism and the emergence toward the end of the century of three new sub-disciplines: chemistry, botany, and geology. A distinguishing feature of eighteenth-century science is how closely it was interwoven with theology. Theories about providence and intelligent design were central to nearly every scientific debate because it was assumed that studying the causes of natural phenomena was the best way to understand the Prime Mover’s intentions for humankind. Accordingly, a recurrent theme in this chapter is the interconnection of religion and science.

This chapter tests the claim made by Peter Brook: that through the live practice of drama, the work of Shakespeare offers ‘the greatest school of living’ that we know. Using the resources of philosophy, psychology and neuroscience, it tries to show how the sudden, deep language of Shakespeare, in particular in Macbeth and King Lear, dramatically discloses lost and neglected forms of being, in a primary emotional aliveness that is denied or tamed within more conventional world-orders. Where other writers give us only secondary versions or paraphrases of nature, Shakespeare, said Hazlitt, offers ‘the original text’ of life.

Shakespeare was writing his plays and poems just as the word ‘emotion’ was emerging into common currency. In its first usages, traceable back to the 1590s, the term referred to the general disturbance suggested by the Latin term emovere (to move out), and Shakespeare and his contemporaries indeed often described as motions the impulses that aroused the mind, body and soul. The introduction to Shakespeare and Emotion explains the rationale for giving serious and sustained attention to the emotions as a way of approaching Shakespeare’s works as art from the past, as well as the place of these works in the present. It offers a brief survey of Shakespeare’s classical and early modern sources for his understanding of affect, and an account of how the present-day surge of interest in emotional experience builds on earlier strands of Shakespearean scholarship from the early to mid-twentieth century. The Introduction concludes with a survey of the volume’s chapters, organised around the assumption that emotion offers a deeply promising (and often challenging) prospect for imagining and enacting change.

The laws of treaties are in constant motion and state of flux, which has different connotations. It relates to motion through time and also the manifestation (or lack thereof) of change. The basic principle of the definition of motion, adopted by the authors, is based on the philosophy of Aristotle. He identified six types of motion: creation (genesis), increase (auxesis), diminution (meiosis), alteration (alloiosis), destruction (phthora), and change of place (kata topon metabole). This last type of motion has been amended by the authors to change in space-time (kata topon kai chronon metavole) to reflect our modern scientific understanding of time as a dimension through which motion and change occurs.

This chapters concludes and summarises the main analysis, research and findings presented in the previous chapters.

Fernandez and Mittelmann focus on Aristotle’s analogy between the soul as an efficient cause and a craft. They argue that the acts of a craft are a special kind of vital act of a rational soul and that the continuous unifying activity of the source of motion in those cases, which must be constantly and not episodically at work in and throughout each of its acts, is not an idiosyncratic feature of the rational case, but the rational version of a form of efficient causality that is at work in all those vital phenomena of which the soul is said to be the efficient cause.