Silvia De Marchi (1929) on numerical estimation: A translation and commentary.

Vittorio Benussi (1878-1927) is known for numerous studies on optical illusions, visual and haptic perception, spatial and time perception. In Padova, he had a brilliant student who carefully worked on the topic of how people estimate numerosity, Silvia De Marchi (1897-1936). Her writings have never been translated into English before. Here we comment on her work and life, characterized also by the challenges faced by women in academia. The studies on perception of numerosity from her thesis were published as an article in 1929. We provide a translation from Italian, a redrawing of its 23 illustrations and of the graphs. It shows an original experimental approach and an anticipation of what later became known as magnitude estimation.

Divine mathematics: Leibniz's combinatorial theory of compossibility.

Leibniz's famous proposition that God has created the best of all possible worlds holds a significant place in his philosophical system. However, the precise manner in which God determines which world is the best remains somewhat ambiguous. Leibniz suggests that a form of "Divine mathematics" is employed to construct and evaluate possible worlds. In this paper, I uncover the underlying mechanics of Divine mathematics by formally reconstructing it. I argue that Divine mathematics is a one-player combinatorial game, in which God's goal is to find the best combination among many possibilities. Drawing on the combinatorial theory, I provide new solutions to some puzzles of compossibility.

A tale of a threshing machine: Images of the Voigt-Leibniz mathematical-agricultural machine at the beginning of the 18th century.

This paper examines how a certain threshing machine was developed and improved by Jobst Heinrich Voigt and Gottfried Wilhelm Leibniz between 1699 and 1700. While this machine was based on various mechanical principles and instruments, including the pinned drum mechanism first noted by Georg Philipp Harsdörffer, it was later reconceptualized as a 'mathematical' machine. I claim that such a positioning was not unique to this machine, but part of a wider movement during the 18th century that considered various artisanal instruments as mathematical, as well as agricultural and artisanal knowledge as scientific. Examining the development and subsequent reception of this machine, I show that during the first decades of the 18th century these conceptions gave rise to a double image of this machine, and hence of agricultural knowledge in general: on the one hand, this machine was considered as more efficient and productive (while still in need of improvement); on the other hand, it was viewed, either implicitly or explicitly, as something that should be studied by mathematicians, thus reflecting a changing image of mathematics.

On the relativity of magnitudes: Delboeuf's forgotten contribution to the 19th century problem of space.

Faced with the mathematical possibility of non-Euclidean geometries, 19th Century geometers were tasked with the problem of determining which among the possible geometries corresponds to that of our space. In this context, the contribution of the Belgian philosopher-mathematician, Joseph Delboeuf, has been unduly neglected. The aim of this essay is to situate Delboeuf's ideas within the context of the philosophies of geometry of his contemporaries, such as Helmholtz, Russell and Poincaré. We elucidate the central thesis, according to which Euclidean geometry is given special status on the basis of the relativity of magnitudes, we uncover its hidden history and show that it is defensible within the context of the philosophies of geometry of the epoch. Through this discussion, we also develop various ideas that have some relevance to present-day methods in gravitational physics and cosmology.

New insight into the origins of the calculus war.

The consensus today is that both Newton and Leibniz created calculus independently. Yet, this was not so clear at the beginning of the eighteenth century. A bitter controversy took place at that time, which came to be known as the 'calculus war', probably the greatest clash in the history of science. While it is accepted that the debate started when Fatio de Duillier publicly accused Leibniz of plagiarism in 1699, earlier evidence of its origins can be found in an exchange of letters between Leibniz and Huygens.

Linear Programming from Fibonacci to Farkas.

At the beginning of the 13th century Fibonacci described the rules for making mixtures of all kinds, using the Hindu-Arabic system of arithmetic. His work was repeated in the early printed books of arithmetic, many of which contained chapters on 'alligation', as the subject became known. But the rules were expressed in words, so the subject often appeared difficult, and occasionally mysterious. Some clarity began to appear when Thomas Harriot introduced a modern form of algebraic notation around 1600, and he was almost certainly the first to express the basic rule of alligation in algebraic terms. Thus a link was forged with the work on Diophantine problems that occupied mathematicians like John Pell and John Kersey in the 17th century. Joseph Fourier's work on mechanics led him to suggest a procedure for handling linear inequalities based on a combination of logic and algebra; he also introduced the idea of describing the set of feasible solutions geometrically. In 1898, inspired by Fourier's work, Gyula Farkas proved a fundamental theorem about systems of linear inequalities. This topic eventually found many applications, and it became known as Linear Programming. The theorem of Farkas also plays a significant role in Game Theory.

Stokes' mathematical education.

George Gabriel Stokes won the coveted title of Senior Wrangler in 1841, a year in which the examination papers for the Cambridge Mathematical Tripos were notoriously difficult. Coming top in the Mathematical Tripos was a notable achievement, but for Stokes it was a prize hard won after several years of preparation, and not only years spent at Cambridge. When Stokes arrived at Pembroke College, he had spent the previous two years at Bristol College, a school which prided itself on its success in preparing students for Oxford and Cambridge. This article follows Stokes' path to the senior wranglership, tracing his mathematical journey from his arrival in Bristol to the end of his final year of undergraduate study at Cambridge. This article is part of the theme issue 'Stokes at 200 (Part 1)'.

Sir George Gabriel Stokes in Skreen: how a childhood by the sea influenced a giant in fluid dynamics.

George Gabriel Stokes spent most of his life at the University of Cambridge, where he undertook his undergraduate degree and later became Lucasian Professor of Mathematics and Master of Pembroke College. However, he spent the first 13 years of his life in Skreen, County Sligo, Ireland, a rural area right by the coastline, overlooking the Atlantic Ocean. As this paper will discuss, the time he spent there was short but its influence on him and his research was long reaching, with his childhood activities of walking by and bathing in the sea being credited for first piquing Stokes' interest in ocean waves, which he would go on to write papers about. More generally, it marked the beginning of an interest in fluid dynamics and a curious nature regarding natural phenomena in his surroundings. Stokes held a special affinity for the ocean for the rest of his life, constantly drawing inspiration for it in his mathematical and physical studies and referencing it in his correspondences. This commentary was written to celebrate Stokes' 200th birthday as part of the theme issue of Philosophical Transactions A. This article is part of the theme issue 'Stokes at 200 (Part 1)'.

Early Emotion Knowledge and Later Academic Achievement Among Children of Color in Historically Disinvested Neighborhoods.

This study examined longitudinal relations between emotion knowledge (EK) in pre-kindergarten (pre-K; Mage  = 4.8 years) and math and reading achievement 1 and 3 years later in a sample of 1,050 primarily Black children (over half from immigrant families) living in historically disinvested neighborhoods. Participants were part of a follow-up study of a cluster randomized controlled trial. Controlling for pre-academic skills, other social-emotional skills, sociodemographic characteristics, and school intervention status, higher EK at the end of pre-K predicted higher math and reading achievement test scores in kindergarten and second grade. Moderation analyses suggest that relations were attenuated among children from immigrant families. Findings suggest the importance of enriching pre-K programs for children of color with EK-promotive interventions and strategies.

Between Kepler and Newton: Hooke's 'principles of congruity and incongruity' and the naturalization of mathematics.

Robert Hooke's development of the theory of matter-as-vibration provides coherence to a career in natural philosophy which is commonly perceived as scattered and haphazard. It also highlights aspects of his work for which he is rarely credited: besides the creative speculative imagination and practical-instrumental ingenuity for which he is known, it displays lucid and consistent theoretical thought and mathematical skills. Most generally and importantly, however, Hooke's 'Principles … of Congruity and Incongruity of bodies' represent a uniquely powerful approach to the most pressing challenge of the New Science: legitimizing the application of mathematics to the study of nature. This challenge required reshaping the mathematical practices and procedures; an epistemological framework supporting these practices; and a metaphysics which could make sense of this epistemology. Hooke's 'Uniform Geometrical or Mechanical Method' was a bold attempt to answer the three challenges together, by interweaving mathematics through physics into metaphysics and epistemology. Mathematics, in his rendition, was neither an abstract and ideal structure (as it was for Kepler), nor a wholly-flexible, artificial human tool (as it was for Newton). It drew its power from being contingent on the particularities of the material world.

Visiting Newton's atelier before the Principia, 1679-1684.

The worksheets that presumably contained Newton's early development of the fundamental concepts in his Principia have been lost. A plausible reconstruction of this development is presented based on Newton's exchange of letters with Robert Hooke in 1679, with Edmund Halley in 1686, and on some clues in the diagram associated with Proposition 1 in Book 1 of the Principia that have been ignored in the past. A graphical construction associated with this proposition leads to a rapidly convergent method to obtain orbits for central forces, which elucidates how Newton may have have been led to formulate some of his most fundamental propositions in the Principia.

Karl-Peter Hadeler: His legacy in mathematical biology.

Karl-Peter Hadeler is a first-generation pioneer in mathematical biology. His work inspired the contributions to this special issue. In this preface we give a brief biographical sketch of K.P. Hadelers scientific life and highlight his impact to the field.

Who Discovered the Binary System and Arithmetic? Did Leibniz Plagiarize Caramuel?

Gottfried Wilhelm Leibniz (1646-1716) is the self-proclaimed inventor of the binary system and is considered as such by most historians of mathematics and/or mathematicians. Really though, we owe the groundwork of today's computing not to Leibniz but to the Englishman Thomas Harriot and the Spaniard Juan Caramuel de Lobkowitz (1606-1682), whom Leibniz plagiarized. This plagiarism has been identified on the basis of several facts: Caramuel's work on the binary system is earlier than Leibniz's, Leibniz was acquainted-both directly and indirectly-with Caramuel's work and Leibniz had a natural tendency to plagiarize scientific works.

Instruments of statecraft: Humphrey Cole, Elizabethan economic policy and the rise of practical mathematics.

This paper offers a re-interpretation of the development of practical mathematics in Elizabethan England, placing artisanal know-how and the materials of the discipline at the heart of analysis, and bringing attention to Tudor economic policy by way of historical context. A major new source for the early instrument trade is presented: a manuscript volume of Chancery Court documents c.1565-c.1603, containing details of a patent granting a monopoly on making and selling mathematical instruments, circa 1575, to an unnamed individual, identified here as the instrument maker Humphrey Cole. Drawing on economic and legal history, the paper argues that practical mathematics needs to be understood as one 'project' among many, at a time when monopoly patents were used to advance industry, lower unemployment, secure the realm and reward invention. Drawing on the history and sociology of technology, it argues that the management and control of materials - mathematical instruments themselves, and the local socio-legal context within which they could be made - needs to be understood as prior to and separate from the rhetoric of mathematical authors, which is of interest in its own right but which may not have a direct relationship to mathematical practice.

Newton in China: Translating the Principia into Chinese (c. 1855-2015).

This paper provides an account of Chinese translations of Newton's Principia produced over the past century and a half within the larger context of the dissemination of Newtonian philosophy in China. Given its fundamental importance in the history of science, the Principia, originally penned in Latin, has been translated into a number of other languages. While in all these languages no more than two full translations have appeared, as many as four complete versions in Chinese have been produced since the 1850s, when first attempts were made to translate the Principia in late imperial China. They include a 1931 version in semi-classical Chinese completed during the Republican era and three rival versions in modern Chinese published in contemporary China. This rich history of translating the Principia into Chinese, which remains little known to scholars in the West, is for the first time reconstructed and presented in English. This account is based on a meticulous scrutiny of manuscripts, historical records, secondary literature and interviews with some of the contemporary translators. It demonstrates that Chinese translation of the Principia is a complex process that involves scientific traditions, linguistic peculiarities, translators' subjectivity, readers' expectations and even the role of the market.

Qualitative vs quantitative conceptions of homogeneity in nineteenth century dimensional analysis.

The emergence of dimensional analysis in the early nineteenth century involved a redefinition of the pre-existing concepts of homogeneity and dimensions, which entailed a shift from a qualitative to a quantitative conception of these notions. Prior to the nineteenth century, these concepts had been used as criteria to assess the soundness of operations and relations between geometrical quantities. Notably, the terms in such relations were required to be homogeneous, which meant that they needed to have the same geometrical dimensions. The latter reflected the nature of the quantities in question, such as volume vs area. As natural philosophy came to encompass non-geometrical quantities, the need arose to generalize the concept of homogeneity. In 1822, Jean Baptiste Fourier consequently redefined it to be the condition an equation must satisfy in order to remain valid under a change of units, and the 'dimension' correspondingly became the power of a conversion factor. When these innovations eventually found an echo in France and Great Britain, in the second half of the century, tensions arose between the former, qualitative understanding of dimensions as reflecting the nature of physical quantities, and the new, quantitative conception based on unit conversion and measurement. The emergence of dimensional analysis thus provides a case study of how existing rules and concepts can find themselves redefined in the context of wider conceptual changes; in the present case this redefinition involved a generalization, but also a shift in meaning which led to conceptual tensions.