Hipparchus realised that for calculation to be effective and useful it must proceed from good observational practice. Without accuracy in the latter, the former, however sophisticated, cannot provide usable answers. Accordingly, he would have made many observations of astronomical facts and phenomena for himself, and for this he would have needed instrumentation. Very little is known for sure about what tools he used. The Antikythera Mechanism, which is roughly contemporary with Hipparchus, demonstrates the engineering skills that were available in his time, and on the reasonable assumption that such skills did not advance much by the time of Ptolemy, the latter’s description of his own instruments provides a basis for discussion. Also important in this context are notions of accuracy, precision and units of measurement. Evidence is brought to bear on these issues in relation to Hipparchus, and how they should be interpreted in the context of his works, using information gleaned from both him and from Ptolemy.

This chapter explores the application of large language models (LLMs) in empirical legal studies, with a focus on their potential to advance research on EU law at scale. The chapter provides a non-technical introduction to LLMs and the role they can play in legal information retrieval, including the classification of case characteristics and outcomes, which constitutes one of the most common research tasks in legal scholarship. The chapter stresses the importance of validation – researchers cannot treat the output of LLMs as automatically correct and instead must demonstrate the relevance and reliability of measures and results obtained through the use of LLMs in the context of their research topic. While LLMs are capable of significantly reducing the cost of doing legal research, their use will place growing demands on scholars to ensure the integrity of their findings. The chapter also reflects on the distinction between closed- and open-source models and how ethical and replicability imperatives might influence model choices in an increasingly crowded field.

Not everything we know about numbers can be interpreted as true of what they represent, so this chapter explores how far we can go in applying the numbers to real problems, and how we can be sure they are meaningful.

Empirical legal studies in EU law routinely, if not inevitably, engage with text. From the decisions of national courts applying EU law, applicants’ case filings, to the Court’s own jurisprudence, these texts are an invaluable source of information for researchers seeking to understand the dynamics involved in the shaping of EU law and its broader societal impact. Distilling relevant information from legal texts, however, is anything but trivial. Intended to serve as a reference manual, the chapter offers detailed guidelines to researchers of both law and political science interested in employing a text-as-data approach to the study of EU law. To this end, we elaborate on how to conceptualise real-life phenomena in a way that renders them conducive to measurement, providing practical guidance on hand-coding and the use of deep learning classifiers. Further, we address potential challenges arising in the specific context of EU law. This includes limitations to access to relevant documents, as well as ensuring inter-coder reliability in data collection efforts that require specialised legal expertise.

Typologies are well-established analytic tools in the social sciences. They can be “put to work” in forming concepts, refining measurement, exploring dimensionality, and organizing explanatory claims. Yet some critics, basing their arguments on what they believe are relevant norms of quantitative measurement, consider typologies old-fashioned and unsophisticated. This critique is methodologically unsound, and research based on typologies can and should proceed according to high standards of rigor and careful measurement. These standards are summarized in guidelines for careful work with typologies, and an illustrative inventory of over 100 typologies is included at the end of the chapter.

The challenge of finding appropriate tools for measurement validation is an abiding concern in political science. This chapter considers four traditions of validation, using examples from cross-national research on democracy: the levels-of-measurement approach, structural-equation modeling with latent variables, the pragmatic tradition, and the case-based method. Methodologists have sharply disputed the merits of alternative traditions. The chapter encourages scholars – and certainly analysts of democracy – to pay more attention to these disputes and to consider strengths and weaknesses in the validation tools they adopt. An appendix summarizes the evaluation of six democracy data sets from the perspective of alternative approaches to validation.

This chapter focuses on teaching the formation and analysis of concepts in research, emphasizing the importance of clarity in defining and measuring concepts. It presents a structured approach to conceptual thinking, outlining four steps: formulation, contextualization, operationalization, and measurement. Bussell highlights the role of examples and typologies to deepen understanding and discusses challenges in concept analysis, using “democracy” and “corruption” as primary examples. The chapter underscores the iterative process of concept development and its role in fostering rigorous academic research.

It is well past time to take scholarly disputes about words seriously, for where there’s smoke, there’s fire. Conceptual disagreement is a manifestation of disagreements about ideas. Yet no attempt has been made to measure the degree of conceptual disagreement that exists or to track those concepts identified as essentially contested. Accordingly, it is unclear how one might distinguish contested from uncontested concepts or test propositions about the causes of contestation. This chapter begins by introducing an approach to measuring conceptual contestation within social science. Next, the chapter explore factors that may help to explain variation in conceptual contestation. The characteristics of concepts – their value, abstraction, and normativity – explain most of the variability in conceptual contestation.

This chapter presents key quantum mechanics principles essential for understanding quantum computation. The postulates of quantum mechanics, mixed states, and density matrices are introduced, along with the Stern–Gerlach experiment’s role in illustrating quantum behavior. Topics such as quantum coherence, entanglement, and the EPR paradox are covered to clarify the fundamental distinctions between classical and quantum systems. Measurement is explored with an emphasis on positive operator-valued measures (POVM), a key concept in understanding quantum state collapse. These principles provide a foundation for studying quantum computation and are essential for understanding qubit behavior, quantum information processing, and subsequent algorithmic structures.

The chapter examines how moral rhetoric is used in party communication using dictionary-based text analysis of party manifestos. The main data include 158 manifestos from six English-speaking democracies (Australia, Canada, Ireland, New Zealand, the United Kingdom, the United States) across thirty-four elections. I first measure moral rhetoric in the aggregate. The measure captures the overall level of moral rhetoric used by a party in its campaign. I show that there is variation in moral rhetoric across countries and within countries. I also show the validity of the measurement approach and its robustness to alternatives. Overall, we learn that moral rhetoric is a distinct aspect of party messaging. I then explore patterns in more disaggregate measures of moral rhetoric. Analyses reveal that there are more commonalities in the ways that parties use moral rhetoric than one might expect. Building on the framework of the Moral Foundations Theory, I find that differences in the moral palettes of the left and the right that we can expect based on prior work are more nuanced and not as stark when we examine specific moral foundations separately and when we examine appeals at the level of issues.

Critics have routinely voiced their frustrations with William Carlos Williams’s term ‘measure’. But from the late 1930s onwards, he compared his idea of ‘measure’ to the science of measurement. This chapter suggests, first, that to fully appreciate Williams’s measure, one must understand how the science of measurement frequently appeared in the vocabulary of a variety of contemporaneous critics of poetry. In so doing, it sketches a lineage of scientific criticism that began in the late nineteenth century and that shaped modernist theories of prosody. Second, by close reading Williams’s long poem Paterson (1963), it suggests that by rejecting the term ‘rhythm’ and reprising ‘measure’, Williams was attempting to define the knowledge practices proper to poetry in an era where to measure was to know.