Fifteenth-century Italian Symbolic Algebraic Calculation with Four and Five Unknowns

The present article continues an earlier analysis of occurrences of two algebraic unknowns in the writings of Fibonacci, Antonio de’ Mazzinghi, an anonymous Florentine abbacus writer from around 1400, Benedetto da Firenze, and another anonymous Florentine writing some five years before Benedetto, and Luca Pacioli. The following article investigates how Benedetto da Firenze explores in 1463 the use of four or five algebraic unknowns in symbolic calculations, describing it afterwards in rhetorical algebra; in this way he thus provides a complete parallel to what was so far only known from Johannes Buteo’s Logistica from 1559. It also discusses why Benedetto may have seen his innovation as a merely marginal improvement compared to techniques known from Fibonacci’s Liber abbaci, therefore omitting to make explicit that he has created something new

From Hesiod to Saussure, from Hippocrates to Jevons: An Introduction to the History of Scientific Thought between Iran and the Atlantic

This work offers an introduction to the history of scientific thought in the region between Iran and the Atlantic from the beginnings of the Bronze Age until 1900 CE—a “science” that can be understood more or less as a German Wissenschaft: a coherent body of knowledge carried by a socially organized group or profession. It thus deals with the social and human as well as medical and natural sciences and, in earlier times, even such topics as astrology and exorcism. It discusses eight periods or knowledge cultures: Ancient Mesopotamia – classical Antiquity – Islamic Middle Ages – Latin Middle Ages – Western Europe 1400–1600 – 17th century – 18th century – 19th century. For each period, a general description of scientific thought is offered, embedded within its social context, together with a number of shorter or longer commented extracts from original works in English translation

Peeping into Fibonacci’s Study Room

The following collects observations I made during the reading of Fibonacci’s Liber abbaci in connection with a larger project, “abbacus mathematics analyzed and situated historically between Fibonacci and Stifel”. It shows how attention to the details allow us to learn much about Fibonacci’s way of working. In many respects it depends critically upon the critical edition of the Liber abbaci prepared by Enrico Giusti and upon his separate edition of an earlier version of its chapter 12 – not least on the critical apparatus of both. This, and more than three decades of esteem and friendship, explain the dedication

Otto Neugebauer and the Exploration of Ancient Near Eastern Mathematics

The exploration of Mesopotamian mathematics took its beginning together with thedecipherment of the cuneiform script around 1850. Until the 1920s, “mathematics in use” (number systems, metrology, tables and some practical calculations of areas) was the object of study – only very few texts dealing with more advanced matters were approached before 1929, and with quite limited results. That this situation changed was due to Otto Neugebauer – but even his first steps in 1927–28 were in the prevailing style of the epoch, so to speak “pre-Neugebauer”. They can be seen, however, to have pushed him toward the three initiatives which opened the “Neugebauer era” in 1929: The launching of Quellen und Studien, the organization of a seminar for the study of Babylonian mathematics, and the start of the work on the Mathematische Keilschrift-Texte. After a couple of years François Thureau-Dangin (since the late 1890s the leading figure in the exploration of basic mathematics) joined in. At first Thureau-Dangin supposed Neugebauer to take care of mathematical substance, and he himself to cover the philology of the matter. Very soon, however, both were engaged in substance as well as philology, working in competitive parallel until both stopped this work in 1937–38. Neugebauer then turned to astronomy, while Thureau-Dangin, apart from continuing with other Assyriological matters, undertook to draw the consequences of what was now known about Babylonian mathematics for the history of mathematics in general

The symbolic model for algebra : functions and mechanisms

The symbolic mode of reasoning in algebra, as it emerged during the sixteenth century, can be considered as a form of model-based reasoning. In this paper we will discuss the functions and mechanisms of this model and show how the model relates to its arithmetical basis. We will argue that the symbolic model was made possible by the epistemic justification of the basic operations of algebra as practiced within the abbaco tradition. We will also show that this form of model-based reasoning facilitated the expansion of the number concept from Renaissance interpretations of number to the full notion of algebraic numbers

Innovation by Experimenting in Public Services

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