The Philosopher's conception of Mathesis Universalis from Descartes to Leibniz

In Descartes, the concept of a ‘universal science’ differs from that of a ‘mathesis universalis’, in that the latter is simply a general theory of quantities and proportions. Mathesis universalis is closely linked with mathematical analysis; the theorem to be proved is taken as given, and the analyst seeks to discover that from which the theorem follows. Though the analytic method is followed in the Meditations, Descartes is not concerned with a mathematisation of method; mathematics merely provides him with examples. Leibniz, on the other hand, stressed the importance of a calculus as a way of representing and adding to what is known, and tried to construct a ‘universal calculus’ as part of his proposed universal symbolism, his ‘characteristica universalis’. The characteristica universalis was never completed—it proved impossible, for example, to list its basic terms, the ‘alphabet of human thoughts’—but parts of it did come to fruition, in the shape of Leibniz's infinitesimal calculus and his various logical calculi. By his construction of these calculi, Leibniz proved that it is possible to operate with concepts in a purely formal way.     

Maligned for mathematics: Sir Thomas Urquhart and his Trissotetras

Thomas Urquhart (1611–1660), celebrated for his English translation of Rabelais’ Gargantua et Pantagruel, has earned some notoriety for his eccentric, putatively incomprehensible early book on trigonometry The Trissotetras (1645). The Trissotetras was too impractical to succeed in its own day as a textbook, since it lacked both trigonometric tables and sample calculations. But its current bad reputation is based on literary authors’ amplifications of the verdict prefaced to its 19th century reprinting by one mathematician, William Wallace, who lacked the background to appreciate the book’s historical context. Considering that context (including seventeenth century ‘copious’ prose, and medieval logic and ‘art of memory’), the bad reputation is undeserved: the book is mathematically clear, clever (e.g. in superimposing 16 problems into one diagram), and complete. The Trissotetras may thus be viewed as simply one more of Urquhart’s polymathic projects and involvements – which included education, rise of the middle class, religious and class conflicts, development of science and mathematics, search for patronage, universal language construction, and development of English prose – which serve to make him a lively and instructive intellectual Everyman for his time.     

Hugo de Vries and the rediscovery of Mendel's laws

Hugo de Vries claimed that he had discovered Mendel's laws before he found Mendel's paper. De Vries's first ratios, published in 1897, for the second generation of hybrids (F₂) were 2/3:1/3 and 80%:20%. By 1900, both of these ratios had become 3:1. These changing ratios suggest that as late as 1897 de Vries had not discovered the laws, although he asserted, from 1900 on, that he had found the laws in 1896. An Appendix details de Vries's Mendelian experiments as described in the original edition (1903) of volume two of Die Mutationstheorie, but omitted entirely from the English translation (1910).     

Did de Vries discover the law of segregation independently?

It is argued that de Vries did not see Mendel's paper until 1900, and that, while his own theory of inheritance may have incorporated the notion of independent units, this pre-Mendelian formulation was not the same as Mendel's since it did not apply to paired hereditary units. Moreover, the way in which the term ‘segregation’ has been applied in the secondary literature has blurred the distinction between what is explained and the law which facilitates explanation.     

Hobbes on natural philosophy as “True Physics” and mixed mathematics

In this paper, I offer an alternative account of the relationship of Hobbesian geometry to natural philosophy by arguing that mixed mathematics provided Hobbes with a model for thinking about it. In mixed mathematics, one may borrow causal principles from one science and use them in another science without there being a deductive relationship between those two sciences. Natural philosophy for Hobbes is mixed because an explanation may combine observations from experience (the ‘that’) with causal principles from geometry (the ‘why’). My argument shows that Hobbesian natural philosophy relies upon suppositions that bodies plausibly behave according to these borrowed causal principles from geometry, acknowledging that bodies in the world may not actually behave this way. First, I consider Hobbes's relation to Aristotelian mixed mathematics and to Isaac Barrow's broadening of mixed mathematics in Mathematical Lectures (1683). I show that for Hobbes maker's knowledge from geometry provides the ‘why’ in mixed-mathematical explanations. Next, I examine two explanations from De corpore Part IV: (1) the explanation of sense in De corpore 25.1-2; and (2) the explanation of the swelling of parts of the body when they become warm in De corpore 27.3. In both explanations, I show Hobbes borrowing and citing geometrical principles and mixing these principles with appeals to experience.     

The first mite: insect genealogy in Hooke’s Micrographia*

What happens when you take the idea of the biblical Adam—the first human – and apply it to insects? You create an origin story for Nature’s tiniest creatures, one that gives them ‘a Pedigree as ancient as the first creation’. This the naturalist Robert Hooke argued in his treatise, the Micrographia (1665). In what follows, I will retrace how Hooke endeavoured to show that insects—then widely believed to have arisen out of the dirt – were the products of an ancient lineage. These genealogies, while constructed from empirical observation, were conjectures of the imagination. Section 2 shows how Hooke introduced the concept of a ‘prime parent’ (an Adam-insect) to explain the anatomical similarities between ‘mites’. Section 3 demonstrates how Hooke defined the family of “gnats” as tiny machines built from the same components and relates Hookean genealogies to contemporary ideas about Noah’s Ark. Section 4 shows how Hooke outlined the morphology of ‘insects’ (delineating what we now call arthropods). Section 5 explores how Hooke used fossils to study these animals in the distant past. In sum, Hooke was turning natural history – collecting and describing insects – into natural history: reconstructing their origins.     

Rome and Theistic Evolutionism: The Hidden Strategies behind the ‘Dorlodot Affair’, 1920–1926

In 1918, Henry de Dorlodot—priest, theologian, and professor of geology at the University of Louvain (Belgium)—published Le Darwinisme au point de vue de l'Orthodoxie Catholique (translated as Darwinism and Catholic Thought) in which he defended a reconciliation between evolutionary theory and Catholicism with his own particular kind of theistic evolutionism. He subsequently announced a second volume in which he would extend his conclusions to the origin of Man. Traditionalist circles in Rome reacted vehemently. Operating through the Pontifical Biblical Commission, they tried to force Dorlodot to withdraw his book and to publicly disown his ideas by threatening him with an official condemnation, a strategy that had been used against Catholic evolutionists since the late nineteenth century. The archival material on the ‘Dorlodot affair’ shows how this policy ‘worked’ in the early stages of the twentieth century but also how it would eventually reach the end of its logic. The growing popularity of theistic evolutionism among Catholic intellectuals, combined with Dorlodot's refusal to pull back amidst threats, made certain that the traditionalists did not get their way completely, and the affair ended in an uncomfortable status quo. Dorlodot did not receive the official condemnation that had been threatened, nor did he withdraw his theories, although he stopped short on publishing on the subject. With the decline of the traditionalists’ power and authority, the policy of denunciation towards evolutionists made way for a growing tolerance. The ‘Dorlodot affair’—which occurred in a pivotal era in the history of the Church—can be seen as exemplary with regards to the changing attitude of the Roman authorities towards evolutionism in the first half of the twentieth century.     

Re-examining the impact of European astronomy in seventeenth-century China: a study of Xue Fengzuo’s system of thought and his integration of Chinese and Western knowledge

ABSTRACT

During the late Ming and early Qing period, Jesuit missionaries introduced European science into China, and thereby profoundly influenced the later development of Chinese astronomy. Not only did European astronomy become the official system of the Qing dynasty, but the traditional way to ‘attain up above’ by connecting the study of astronomy and Yi learning gradually fell into disuse. However, the astronomers in this period expressed different views on these two processes. As one of the most important early Qing astronomers, Xue Fengzuo’s case presents a distinctive and important example. Firstly, under the influences of both Chinese tradition and European science, Xue Fengzuo rebuilt the way to ‘attain up above’ based on his three-fold ‘calendrical learning’, i.e. calendrical astronomy, astrology and related pragmatic applications, through which he could realize the highest Confucian ideal. Secondly, he integrated Chinese and Western knowledge for all three aspects of his ‘calendrical learning’, instead of ceding the dominant position to Western methods. From Xue Fengzuo’s example, many of the complex effects of the encounter between different cultures and the process of knowledge transfer can be revealed.     

Stanisław Staszic: An Early Surveyor of the Geology of Central and Eastern Europe

In the late eighteenth and early nineteenth centuries the Polish geoscientist, philosopher, and statesman Stanis?aw Staszic (1755–1826) conducted an extensive geological survey of Poland and adjacent areas. In 1815, he completed a book (in Polish), On the geology of the Carpathians and other mountains and lowlands of Poland, complemented by a well-made geological map of Central and Eastern Europe. Early in the nineteenth century, Staszic refined the idea of ‘geological mapping’, though initially he was interested in the exploration of mineral deposits, rock salt, copper and iron ores, and coal. Unlike his predecessors, his book adopted a temporal subdivision of rocks, using a somewhat modified version of Abraham Gottlob Werner's system. He delineated the surface distribution of five rock units and coloured them onto his map. His work gave expression to his view of geological history, and brought the ‘Enlightenment Period’ of geology in Central and Eastern Europe to a close.     

Interprétation chymique de la création et origine corpusculaire de la vie chez Athanasius Kircher

The famous Jesuit father Athanasius Kircher (1602–1680) tried to interpret the Creation of the world and to explain the origin of life in the last book of his geocosmic encyclopedia, Mundus subterraneus (Amsterdam, 1664–1665). His interpretation largely depended on the ‘concept of seeds’ which was derived from the tradition of Renaissance ‘chymical’ (chemical and alchemical) philosophy. The impact of Paracelsianism on his vision of the world is also undeniable. Through this undertaking, Kircher namely developed a corpuscular theory for the spontaneous generation of living beings. The present study examines this theory and its relationship with Kircher's chymical interpretation of the Creation in order to place it in its own intellectual and historical context and will uncover one of its most important sources.     

Tracings of the north of Europe: Robert Chambers in search of the Ice Age

Scottish publisher and naturalist Robert Chambers pursued an amateur interest in geology through much of his life. His early measurements of raised beaches in Scotland earned him membership in the Geological Society of London in 1844, a recognition much appreciated by the anonymous author of the ‘scandalous’ Vestiges published the same year. Although familiar with emerging ice age theories, Chambers remained with most British geologists a sceptic through the 1840s, even after a trip to the glaciers of the Alps in 1848, which nevertheless prepared him for the turning point, which came in 1849 during an extensive field trip in Norway and Sweden. Here a wealth of observations left him in no doubt that vast glaciers had formerly covered Scandinavia, polishing cliffs, scouring striations, depositing old moraines and erratic boulders. This also led him to a new glacial reading of the British landscape, and with the ardent conviction of a fresh convert he became one of the most vocal supporters of glacial theory in Britain in the 1850s at a time when the iceberg drift theory for boulder transport was still favoured by most prominent British geologists. While Chambers through his popular Chambers’s Edinburgh Journal communicated his travels and ice age vision to a wide audience, and also pointed out ice age evidence on guided excursions around Edinburgh, he did not enter this new vision into subsequent editions of Vestiges, probably in order not to reveal its author. This paper explores Chambers’s contributions to the ice age debate, his field trips and the genesis of his convictions, and evaluates his impact on the scientific debate.     

Books received

R. P. de Lamanon was trained in theology and philosophy, but he chose the career of a self-taught geologist/naturalist, later adding experimental physics to his skills. Recommended by Condorcet, Secretary to the Académie Royale des Sciences, for the post of ‘Naturaliste’ on La Pérouse's expedition, he carried out delicate measurements at sea requested by the Académie and made two important discoveries: the barometric tide at the equator, and the variation of magnetic intensity with latitude. Killed by natives of Samoa in 1787, his reports were long delayed in publication, inadequately presented, and some even lost. Except for brief recognition by von Humboldt many years later, Lamanon's pioneering measurements have been largely ignored or forgotten. This paper revives his memory.     

The platinum pyrometers of Louis Bernard Guyton de Morveau,F.R.S. (1737–1816)

As we have seen, it was clearly Guyton's intention, in 1808, to supply details of his improved platinum pyrometer, and he did submit a drawing of the instrument at the meeting of the Class in December 1810. It would seem that on that occasion he did not supply those details which are to be found in the fourth, unpublished, part of the ‘Essay’. The existence of a text fit to be sent to the printer, and the execution of a drawing relating to the improved version of his platinum pyrometer, might be taken as evidence that Guyton intended to publish. His paper of 1810 did not appear until 1814, however, so that publication of the fourth part of the ‘Essay’ could scarcely have occurred until 1815 or later. Guyton probably hoped to be able to read it to the Class of Physical and Mathematical Sciences before publishing it in the Annales de Chimie, but his death on 2 January 1816 robbed him of the opportunity.     

Alexis Fontaine's ‘Fluxio-differential method’ and the origins of the calculus of several variables

Alexis Fontaine des Bertins (1704–1771) was the first French mathematician to introduce what we would now regard as results in the calculus of several variables. One example is Fontaine's theorem nF = (?F/?x)x + (?F/?y)y of 1737 for homogeneous expressions F of degree n in x and y. Many years later Fontaine indicated this particular result to have been ‘a continuation of the method of solution’ introduced by him in 1734 to solve the problem of the tautochrones. It is tempting to disregard this announcement, since the method applied to the tautochrones was a method of variations and not manifestly an exercise in the calculus of several variables. Do we have just another case of a mathematician's confusion about the origins of his earlier work? In this paper I describe Fontaine's possible intentions in his remarks.     

The Replication of Hertz's Cathode Ray Experiments

I reappraise in detail Hertz's cathode ray experiments. I show that, contrary to Buchwald's (1995) evaluation, the core experiment establishing the electrostatic properties of the rays was successfully replicated by Perrin (probably) and Thomson (certainly). Buchwald's discussion of ‘current purification’ is shown to be a red herring. My investigation of the origin of Buchwald's misinterpretation of this episode reveals that he was led astray by a focus on what Hertz ‘could do’—his experimental resources. I argue that one should focus instead on what Hertz wanted to achieve—his experimental goals. Focusing on these goals, I find that his explicit and implicit requirements for a successful investigation of the rays’ properties are met by Perrin and Thomson. Thus, even by Hertz's standards, they did indeed replicate his experiment.