Research in the knowledge economy highlights the importance of knowledge circulation in modern knowledge accumulation. This paper contributes by developing a mathematical model of knowledge accumulation based on the *Sphaera* corpus, a comprehensive collection of astronomy textbooks used in European universities from the 15th to 17th centuries. The study uses this dataset to model the spread of scientific knowledge, interpreting this spread as a determinant factor in knowledge accumulation. The research builds on Stephen Cole's conceptual framework, which distinguishes between "core knowledge" (widely accepted fundamental knowledge) and "research-frontier knowledge." The paper focuses on core knowledge, represented by textbook content, and models its homogenization—the increasing similarity of textbook content across Europe—as a proxy for knowledge accumulation and acceptance. The authors leverage epidemic models (SI and Bass models) to analyze the spread of knowledge units (semantically complete text-parts) between textbook editions, with the goal of quantifying the accumulation process and its relationship to knowledge circulation. The *Sphaera* corpus, representing approximately 350,000 books, provides an extensive dataset for this analysis, examining over 76,000 pages of historical material.

The introduction extensively cites prior work emphasizing the link between knowledge circulation and accumulation in the knowledge economy. It references works by Mokyr (2005) on the knowledge economy of modernity, Popper (1972) on evolutionary epistemology, Castells (2010), Clark (1998), Marginson (2016), and Renn (2020) on the importance of knowledge circulation, and Consoli and Patrucco (2003) on the relationship between knowledge circulation and innovation. The paper also directly connects to Stephen Cole's (1983) conceptual framework on scientific core knowledge and its hierarchical organization within disciplines, contrasting their goal of hierarchical analysis with their own focus on quantitative modeling of knowledge accumulation. Finally, the introduction cites existing literature on information diffusion models (Bettencourt et al., 2006; Goffman, 1966; Li et al., 2017; Meel and Vishwakarma, 2020), highlighting the use of epidemic and innovation adoption models (SI and Bass models) in studying information spread (Pastor-Satorras et al., 2015; Bass, 1969, 2004; Kiss et al., 2010).

The study uses a multiplex network to represent the *Sphaera* corpus. The nodes are textbook editions, and the network comprises four layers reflecting different semantic relationships between editions: Same Original Part (SOP), Same Adaptation Part (SAP), Translated Same Original Part (TSOP), and Annotated Same Adaptation Part (ASAP). These layers capture relationships based on shared original text, adaptations (including translations), and commentaries. Each connection is directed, based on the chronological order of publication. A Layer Interaction Network (LIN) analyzes the relationships between the layers, quantifying the degree of overlap and influence between them. The SI and Bass models, adapted from epidemic modeling and innovation adoption theory, are used to model knowledge spread. The SI model considers only the internal transmission rate (β), representing the spread of knowledge between editions. The Bass model introduces an additional parameter (α), representing external influence, reflecting the impact of factors beyond direct interactions between editions. The model fits to the observed spread of knowledge units over time in each layer, allowing estimation of the parameters β and α. The study further explores the aggregated network of the two main layers (SOP and SAP), identifying its major components (clusters of highly interconnected editions) for analysis. The spread of knowledge within these components is also modeled using the Bass model, providing insights into the dynamics of knowledge circulation within distinct communities.

The analysis reveals distinct phases of knowledge circulation and accumulation. The SOP and SAP layers display an S-curve pattern, indicative of a process with slow initial growth, acceleration in the middle period, and eventual saturation. However, TSOP and ASAP exhibit different behaviors: TSOP shows slow growth with a late cutoff, while ASAP starts later and shows rapid initial growth followed by a slowdown. Model fitting using the Bass model reveals varying parameter values for each layer. The SOP layer has a low α (external influence) and high β (internal influence), suggesting endogenous factors drive knowledge spread. The TSOP layer shows a high α, reflecting its sensitivity to external factors due to its focus on translations. The analysis of the aggregated graph (combining SOP and SAP) identified three major components representing distinct phases of knowledge transformation: a smaller initial component (green), a large central component (pink), and a smaller later component (blue). The spread of knowledge in these components is also analyzed using the Bass model. The study finds that the chronologically ordered components represent different phases in the transformation of astronomy from a natural-philosophical discipline to a mathematical one. The pink (largest) component shows a high β and moderate α, while the blue component reveals a high β and a negative α, suggesting strong internal learning and resistance to external influence, possibly reflecting the unique institutional context of the Jesuit order. The paper further attempts to identify the most reliable connections in the network by searching for a network structure whose transmission pattern closely resembles the overall S-curve of knowledge spread. Initial attempts using geographical proximity and book format proved inconclusive. However, three economic-related networks (EC1, EC2, EC3) incorporating information on the lifespan of printers and publishers, publication location, and book type were more successful in approximating the S-curve, highlighting the significance of economic factors in knowledge diffusion. Finally, 'awareness graphs' (AW1, AW2, AW3) focusing on the social and economic awareness between book producers showed convergence to the S-curve, revealing different phases in the role of social and economic relationships in shaping knowledge circulation. Before 1530, author-author and publisher-publisher awareness (AW1, AW2) were crucial, while after 1530, imitation in content and layout (AW3) became dominant, reflecting economic constraints on early modern book production.

The findings demonstrate the effectiveness of using mathematical modeling to analyze historical data and gain insights into the dynamics of knowledge accumulation. The study's results confirm the importance of considering the social, economic, and institutional contexts of scientific production when analyzing knowledge spread. The strong influence of economic constraints on printers and publishers suggests that the homogenization of scientific knowledge was not solely driven by intellectual exchange among scholars but also by market pressures and material conditions. The high transmission rate within the Jesuit component (blue) highlights the significance of institutional structure in knowledge diffusion. The paper suggests that while global book markets facilitated rapid knowledge circulation, local networks and the economic factors influencing book production played a crucial role in shaping the process.

This paper successfully demonstrates the application of epidemiological models to the study of knowledge accumulation in early modern science. The results highlight the crucial role of economic factors in knowledge circulation and accumulation, surpassing the influence of geographic proximity or scholarly networks. The study introduces a new methodological tool for historical research, integrating quantitative modeling with historical analysis to refine our understanding of the past. Future research should explore the integration of semantic interactions and awareness connections as separate network layers to further refine the model.

The study's reliance on the *Sphaera* corpus, focusing on astronomy textbooks, limits the generalizability of its findings to other scientific fields or knowledge domains. The assumption of maximal connection in constructing the initial network might introduce some artificial connections. Further refinement of the model could address this limitation by incorporating more granular historical data. Finally, while the economic factors are prominently highlighted, further investigation into other aspects of the social and institutional context could offer even richer insights.