Medieval tables can be rich sources of evidence about the practices of the mathematicians and astronomers who used them. This paper analyses an important set of tables, revealing their compiler's learning practices and elucidating a valuable document of inexpert science. Peterhouse, Cambridge MS 75.I, ‘The Equatorie of the Planetis’, is a late-14th-century compilation. It contains a treatise describing the construction and use of an equatorium (an astronomical instrument that computes the positions of the planets), bound with a collection of related astronomical tables. It was long thought to be written by the English poet Geoffrey Chaucer, but has recently been shown to be the work of a Benedictine monk, John Westwyk. This paper reassesses the manuscript as a monastic compilation. Westwyk copied a set of astronomical tables that suited his needs; their use supported and complemented the equatorium he describes in his treatise. He experimented with different techniques, cited astronomers whose work he admired (including Chaucer) and refined his tables in order to obtain the greatest possible precision. By reconstructing Westwyk's mathematical practices in compiling, computing and using tables that required and enabled a range of astronomical techniques, this paper paints a vivid picture of inexpert science in medieval Europe.

This article introduces a writing format, the ‘template table’ (*suanshi*, ) that was designed to guide the process of calendrical astronomical calculations in early modern China. In conjunction with another kind of text, known as ‘detailed procedures’ (*xicao*, ), users could perform calculations easily by operating the ‘template table’ and extracting data from given numerical tables. This method, that not only normalized the use of numerical tables but also linked instructions with the corresponding tables in computational practices, became widespread from the Ming period (1368–1644) onwards. Wanting to acquire this computational regimen, the Joseon court of Korea (1392–1897) even sent skilled officers to China to learn it secretly. The circulation of the template method beyond China suggests its significance. The article also discusses the advantages and disadvantages of using this method.

Manuscript Escorial O II 10 is a late 13th-century document containing a well-known collection of astronomical texts from the arts faculty context. During the first half of the 14th century, this manuscript belonged to John of Murs, an important master of art of the Paris University, responsible, with others for the establishment of the Parisian Alfonsine Tables. John of Murs used the Escorial manuscript to record a wide range of notes over a 20-year period. Among those notes I examine here one concerned with two solar eclipses. Although I will review the relevant information concerning eclipse theory and mathematical practices of European astronomers in the 14th century, this essay will not focus directly on such matters. Rather I am interested in a documentary question: looking at a specific astronomical source I seek clues about the temporal dimensions of a computation as it was recorded in the codex. This focus will help assess the computation practices of John of Murs and will allow an understanding of the meanings such a computational record could have both for its author and in the more general context of early Alfonsine astronomy.

Armillary spheres were part of the Sanskrit astronomical tradition, and had been used for understanding the structure of the heavens. *Goladīpikā* (‘Illumination of the sphere’) is a text in two versions by the same author which deals with structures of the armillary sphere and various related astronomical topics. A close examination of the ways the armillary sphere is described in the two versions of the text will help us understand the main characteristics of the two versions of Parameśvara's *Goladīpikā* and the reasons why the author duplicated his treatise. This case study thus demonstrates how astral sciences sources from the same author may present mathematical practices surrounding the same instrument in contrasting and complementary ways according to intention.

This article examines the case of an observational and a demonstrational armillary sphere confused, one for the other, by fifth-century historians of astronomy He Chengtian and Shen Yue. Seventh-century historian Li Chunfeng dismisses his predecessors as ignorant, and in so doing he supplies the reader with additional evidence. Using their respective histories and what sources for the history of early imperial armillary instruments survive independent thereof, this article tries to explain the mix-up by exploring the ambiguities of ‘observation’ (*guan*) as it was mediated through terminology, text, materiality and mathematics. Reconstructing the material features of the ‘sight’ (*yi*) and ‘effigy’ (*xiang*), the article will reflect upon the mathematics necessary for their operation. The ‘effigy’, as Li Chunfeng defines it, is a substitute for observation; the ‘sight’, however, is so mediated by the material and mathematical sphere as to confound Li's distinction between looking *through* and looking *at*. In the end, however, the difference is moot, since the observational model appears to have played a negligible role in the history of astronomy in first-millennium China, leaving us to wonder what instrument(s) *were* used for observation.

This article suggests that 16th-century sources describing astronomical instruments may be analyzed in terms of ‘geometrical tools’, that is discrete arrangements of lines and curves that solve particular problems. Geometrical tools provided a means for innovation. By playing, literally, with such tools, mathematicians could invent new instruments or add new functions to existing instruments. For a case study of this process, I shall consider the rectangular astrolabe, first proposed in 1515 by Johannes Stabius and reconfigured in several other versions over the course of the 16th century. Geometrical tools, I conclude, are revealed in diagrams found in the sources, not in the accompanying texts.