The Astronomical Book of Enoch describes the passage of the moon through the gates of heaven, which stand at the edges of the earth. In doing so, the book describes the position of the rising and setting of the moon on the horizon. Otto Neugebauer, the historian of ancient science, suggested using the detailed tables found in later Ethiopic texts in order to reconstruct the path of the moon through the gates. This paper offers a new examination of earlier versions of the Astronomical Book, using a mathematical analysis of the figures and astronomical theories presented throughout the Aramaic Astronomical Book; the results fit both the data preserved in the scrolls and the mathematical approach and religious ideology of the scroll's authors better than the details found in the late Ethiopic texts. Among other new insights, this alternate theory also teaches about the process of the composition of the Astronomical Book in the first centuries of its composition.

From the time of al-Khwārizmī in the ninth century to the beginning of the sixteenth century algebraists did not allow irrational numbers to serve as coefficients. To multiply $\sqrt {18} $ by x, for instance, the result was expressed as the rhetorical equivalent of $\sqrt {18{x^2}} $. The reason for this practice has to do with the premodern concept of a monomial. The coefficient, or “number,” of a term was thought of as how many of that term are present, and not as the scalar multiple that we work with today. Then, in sixteenth-century Europe, a few algebraists began to allow for irrational coefficients in their notation. Christoff Rudolff (1525) was the first to admit them in special cases, and subsequently they appear more liberally in Cardano (1539), Scheubel (1550), Bombelli (1572), and others, though most algebraists continued to ban them. We survey this development by examining the texts that show irrational coefficients and those that argue against them. We show that the debate took place entirely in the conceptual context of premodern, “cossic” algebra, and persisted in the sixteenth century independent of the development of the new algebra of Viète, Decartes, and Fermat. This was a formal innovation violating prevailing concepts that we propose could only be introduced because of the growing autonomy of notation from rhetorical text.

Metrological techniques to establish shared quantitative measures have often been seen as signs of rational modernization. The cases considered here show instead the close relation of such techniques with antiquarian and revivalist programs under imperial regimes. Enterprises in survey sciences in Egypt in the wake of the French invasion of 1798 and in India during the East India Company's revenue surveys involved the promotion of a new kind of oriental metrology designed to represent colonizers’ measures as restorations of ancient values to be applied to current systems of survey and measurement. Surveyors’ practice and hardware help clarify the significance of the complex historical and political functions of scientific standards. The balance of the paper discusses the survey work of later nineteenth-century indigenous Egyptian astronomers at a conjuncture of major economic and political dislocation to explore the various versions of antiquity at stake in these metrological programs.