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This paper investigates the relationship between Helmholtz's formulation of the principle of the conservation of force and the two principles well known in rational mechanics as the principle of vis viva and the principle of the conservation of vis viva. An examination of the relevant literature from Leibniz to Duhamel reveals both Helmholtz's indebtedness to that tradition and his creative refashioning of it as he endeavoured to craft an argument that would both prohibit the construction of a perpetuum mobile and the efficacy of a supposed vital force and demonstrate the necessity of an ontology of mass points subject to attractive and repulsive central forces depending solely on distance.

This paper provides a detailed account of the period of the complex history of British algebra and geometry between the publication of George Peacock's Treatise on Algebra in 1830 and William Rowan Hamilton's paper on quaternions of 1843. During these years, Duncan Farquharson Gregory and William Walton published several contributions on 'algebraical geometry' and 'geometrical algebra' in the Cambridge Mathematical Journal. These contributions enabled them not only to generalize Peacock's symbolical algebra on the basis of geometrical considerations, but also to initiate the attempts to question the status of Euclidean space as the arbiter of valid geometrical interpretations. At the same time, Gregory and Walton were bound by the limits of symbolical algebra that they themselves made explicit; their work was not and could not be the 'abstract algebra' and 'abstract geometry' of figures such as Hamilton and Cayley. The central argument of the paper is that an understanding of the contributions to 'algebraical geometry' and 'geometrical algebra' of the second generation of 'scientific' symbolical algebraists is essential for a satisfactory explanation of the radical transition from symbolical to abstract algebra that took place in British mathematics in the 1830s-1840s.

Summary This paper will consider the range of British participation in mathematics internationally during the nineteenth century through an analysis of British mathematical contributions to scientific journals outside of Britain. Viewing scientific papers contained in journals as significant indicators of research, we consider scientists who authored or read and responded in print to papers in a given area within a given group of journals to constitute a publication community. The extent of publication by British mathematicians in these journals can help characterize the role of foreign publication in nineteenth-century British mathematics. Moreover, the isolation of educational, societal, and personal circumstances which motivated British mathematicians to present their work to foreign journals highlights limited but concentrated groups of mathematicians committed to developing and strengthening international mathematical ties with Britain.

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.

Summary This paper is concerned with the introduction of statistical concepts in the molecular theory of heat. In particular, we analyse the arguments invoked by Clausius between 1857 and 1862, and the motivation presented by Maxwell in 1860 to introduce his distribution of velocities. We first show that Maxwell's great insight seems to have been the recognition that the dynamical laws of molecular collision and thermal equilibrium could be made compatible only if the theory of heat became statistical. As for Clausius, historians of science have observed that in 1857 he used probabilities to bring regularity to the theory, but have not satisfactorily analysed important features of his use of probabilities in the context of the theory of heat. We show that only in 1862, in order to answer a criticism to his paper of 1858, Clausius gave a meaning to random motion, along the lines proposed by Maxwell, in his paper of 1860.

In this paper I will probe into Herman Boerhaave's (1668-1738) appropriation of Isaac Newton's natural philosophy. It will be shown that Newton's work served multiple purposes in Boerhaave's oeuvre, for he appropriated Newton's work differently in different contexts and in different episodes in his career. Three important episodes in, and contexts of, Boerhaave's appropriation of Newton's natural philosophical ideas and methods will be considered: 1710-11, the time of his often neglected lectures on the place of physics in medicine; 1715, when he delivered his most famous rectorial address; and, finally, 1731/2, in publishing his Elementa chemiae. Along the way, I will spell out the implications of Boerhaave's case for our understanding of the reception, or use, of Newton's ideas more generally.

In this paper we give a full account of the work of William Thomson (Lord Kelvin) on absolute temperature, which to this day provides the theoretical underpinnings for the most rigorous measurements of temperature. When Thomson fashioned his concepts of 'absolute' temperature, his main concern was to make the definition of temperature independent of the properties of particular thermometric substances (rather than to count temperature from an absolute zero). He tried out a succession of definitions based on the thermodynamics of ideal heat engines; most notably, in 1854 he gave the ratio of two temperatures as the ratio of quantities of heat taken in and given out at those temperatures in a Carnot cycle. But there were difficulties with using such definitions for experimental work, since it was not possible even to approximate an ideal Carnot engine in reality. More generally, it is not trivial to connect an abstract concept with concrete operations in order to make physical measurements possible. In the end, Thomson argued that an ideal gas thermometer would indicate his absolute temperature, and that the deviation of actual gas thermometers from the ideal could be estimated by means of the Joule-Thomson effect. However, the measurement of the Joule-Thomson effect itself required measurements of temperature, so there was a problem of circularity.

In this paper I illustrate the evolution of series theory from Leibniz and Newton to the first decades of the eighteenth century. Although mathematicians used convergent series to solve geometric problems, they manipulated series by a mere extension of the rules valid for finite series, without considering convergence as a preliminary condition. Further, they conceived of a power series as a result of a process of the expansion of a finite analytical expression and thought that the link between series and analytical expression was not restricted to the interval of convergence. This gave rise to formal aspects that became progressively more evident while the complexity of the theory increased. Asymptotic and recurrence series emerged as the result of the natural evolution of theory, without the difference between these infinite processes with respect to ordinary series being highlighted. The various aspects of dealing with series can be viewed today as derived from different definitions of the sum. However...

Summary This paper aims, on the one hand, to determine the valid span of the diagram of unequal hours and, on the other, to find a mathematical expression for the error. It is found that the diagram is valid for the two days of the equinoxes and for the times when the sun is on the horizon or on the meridian. This subject has previously been treated by Delambre in 1819 and Drecker in 1925, but not comprehensively.

Summary Alexis Fontaine des Bertins (1704–1771) was the first French mathematician to make use of the calculus of several variables in the integration of ordinary differential equations (1738). In this paper I argue that this usage evolved from Fontaine's ‘fluxio-differential method’ of the early 1730s. In this way I extend the thesis enunciated in an earlier paper in this journal.