The aim of this paper is to employ the newly contextualized historiographical category of “premodern algebra” in order to revisit the arguably most controversial topic of the last decades in the field of Greek mathematics, namely the debate on “geometrical algebra.” Within this framework, we shift focus from the discrepancy among the views expressed in the debate to some of the historiographical assumptions and methodological approaches that the opposing sides shared. Moreover, by using a series of propositions related to Elem. II.5 as a case study, we discuss Euclid's geometrical proofs, the so-called “semi-algebraic” alternative demonstrations attributed to Heron of Alexandria, as well as the solutions given by Diophantus, al-Sulamī, and al-Khwārizmī to the corresponding numerical problem. This comparative analysis offers a new reading of Heron's practice, highlights the significance of contextualizing “premodern algebra,” and indicates that the origins of algebraic reasoning should be sought in the problem-solving practice, rather than in the theorem-proving tradition.

The main thesis of this paper is that Copernicus's avoidance of all admission that scripture was contravened in De revolutionibus and his composition of its new Preface in 1542, as well as the non-publication of Rheticus's Treatise on Holy Scripture and the Motion of the Earth, were influenced by the early information they received on the failure of the 1541 Regensburg Protestant-Catholic colloquy, among the major consequences of which were significant increases in the problems concerning publishing works which contravened scripture. This is supported by examining Rheticus's first letter to Paul Eber in conjunction with the documents on the Regensburg colloquy and on censorship in Nuremberg, as well as with the existing literature on Copernicus and his context. In view of the main thesis, Copernicus's apparent dedication of the work to the Pope merits additional explanation, and the second thesis is that components of explanations for several aspects of those parts of the Preface that relate to the Papacy and to theologians can be provided via comparisons with previous diplomacy between Warmia and the Papacy which occurred or was being referred to during Copernicus's time. This is supported by examining these parts of the Preface in the light of a selection of the relevant documents.

Mutuality in “contact zones” has been emphasized in cross-cultural knowledge interaction in re-evaluating power dynamics between centers and peripheries and in showing the hybridity of modern science. This paper proposes an analytical pause on this attempt to better invalidate centers by paying serious attention to the limits of mutuality in transcultural knowledge interaction imposed by asymmetries of power. An unusually reciprocal interaction between a Japanese forester, Ishidoya Tsutomu (1891–1958), at the colonial forestry department, and his Korean subordinate Chung Tyaihyon (1883–1971) is chosen to highlight an inescapable asymmetry induced by the imperial power structure. Ishidoya, positioning himself as a settler expert, as opposed to a scientist in Tokyo, pursued localized knowledge in growing interaction with Chung, resulting in Ishidoya's career change as a herbalist focusing on traditional medicine and Chung's leadership in Korean-only botanizing. However, their mutual transformations, limited by asymmetric constraints on their choices, did not unsettle the imperial power structure or the centrality of centers.

Smallpox inoculation was introduced in Europe in the early eighteenth century and has been considered the first mass treatment of disease based on practical use of probability calculations and mathematical tools of computation. The article argues that these new approaches were deeply entangled with other rationalities, most emphatically that of exemplarity. Changes in inoculation methods around mid-century gradually changed the conceptualization of disease, seeing all cases as fundamentally equal, and thus making it more relevant to count them. Arithmetic changed the ways of thinking about smallpox epidemics, but new ways of conceptualizing disease were vital to making it a matter of arithmetic at all. The article investigates what happened when numbers and figures were introduced into medical matters: Who did the figures really concern, and what types of argument were they fitted into? How were numbers transformed into metaphors, and how did quantitative argument work together with arguments from exemplarity?