Relativity of motion: From Occam to Galileo

Early observations of the southern celestial sky were reported in many sixteenth-century books and compilations of voyages of discovery. Here we analyse these accounts in order to find out what was really seen and reported by the first navigators. Our analysis had resulted in new interpretations of the phenomena reported by Amerigo Vespucci and Andreas Corsali. Thus, a reassessment of the discovery of the Coalsack Nebula, the Magellanic Clouds, and the Southern Cross can be made. From a comparative review of the observations of the latter constellation as published between 1500 and 1600, we demonstrate that only questionable records found their way to contemporary compilations of voyages of discovery, and that as a result public knowledge about this constellation at the end of the sixteenth century was entirely unreliable. Another problem we discuss is that although the stars of the Southern Cross were the first to be discovered, and were observed again and again by many navigators, it was not until 1678 that their proper positions were found in stellar atlases and star catalogues accessible to astronomers. We explain how negligence of and subsequently confidence in Ptolemy's astronomy by, respectively, the early navigators and cartographers, were at the root of this amazingly long-lasting gap in the knowledge of the southern celestial sky.     

Galileo and prior philosophy

Galileo claimed inconsistency in the Aristotelian dogma concerning falling bodies and stated that all bodies must fall at the same rate. However, there is an empirical situation where the speeds of falling bodies are proportional to their weights; and even in vacuo all bodies do not fall at the same rate under terrestrial conditions. The reason for the deficiency of Galileo’s reasoning is analyzed, and various physical scenarios are described in which Aristotle’s claim is closer to the truth than is Galileo’s. The purpose is not to reinstate Aristotelian physics at the expense of Galileo and Newton, but rather to provide evidence in support of the verdict that empirical knowledge does not come from prior philosophy.     

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.     

Galileo and the problem of infinity

During the period before the Greek revolution of 1821, and especially during the years between 1750 and 1821, there were two ways in which European scientific thought was propagated in Greece. The first is traditional. It comes from ancient Greece and, through Byzantium, reaches the period before the Greek revolution. It makes known the thought of Aristotle, Democrititus, and others on ‘natural philosophy’. The second way comes from Europe. The Greek scholars of the period before the Greek revolution, and especially at the end of the eighteenth century, tried to bring to and propagate in Greece the spirit ofthe European Enlightenment, They tried to make known to the Greek people the scientific achievements of Newton, Descartes, Lavoisier, and Laplace. Scientific knowledge is an important weapon against superstition, and Greek students had to learn about science to become free persons in an independent Greek state.     

Galileo and the problem of concentric circles

In 1892, Eliakim Hastings Moore accepted the task of building a mathematics department at the University of Chicago. Working in close conjuction with the other original department members, Oskar Bolza and Heinrich Maschke, Moore established a stimulating mathematical environment not only at the University of Chicago, but also in the Midwest region and in the United States in general. In 1893, he helped organize an international congress of mathematicians. He followed this in 1896 with the organization of the Midwest Section of the New York City-based American Mathematical Society. He became the first editor-in-chief of the Society's Transactions in 1899, and rose to the presidency of the Society in 1901.     

Leibniz and the post-Copernican universe. Koyré revisited

This paper employs the revised conception of Leibniz emerging from recent research to reassess critically the ‘radical spiritual revolution’ which, according to Alexandre Koyré’s landmark book, From the closed world to the infinite universe (1957) was precipitated in the seventeenth century by the revolutions in physics, astronomy, and cosmology. While conceding that the cosmological revolution necessitated a reassessment of the place of value-concepts within cosmology, it argues that this reassessment did not entail a spiritual revolution of the kind assumed by Koyré, in which ‘value-concepts, such as perfection, harmony, meaning and aim’ were shed from the conception of the structure of the universe altogether. On the contrary, thanks to his pioneering intuition of the distinction between physical and metaphysical levels of explanation, Leibniz saw with great clarity that a scientific explanation of the universe which rejected the ‘closed world’ typical of Aristotelian cosmology did not necessarily require the abandonment of key metaphysical doctrines underlying the Aristotelian conception of the universe. Indeed the canon of value-concepts mentioned by Koyré—meaning, aim, perfection and harmony—reads like a list of the most important concepts underlying the Leibnizian conception of the metaphysical structure of the universe. Moreover, Leibniz’s universe, far from being a universe without God—because, as Clarke insinuated, it does not need intervention from God—is a universe which in its deepest ontological fabric is interwoven with the presence of God.     

Some observations of Leonardo,Galileo, Mariotte and others relative to size effect

A letter in which astronomer John Flamsteed expounded his unusual views about the causes of earthquakes survives in a number of drafts and copies. Though it was compiled in response to shocks felt in England in 1692 and Sicily in 1693, its relationship to the wide range of comparable theories current in the later seventeenth century must be considered. Flamsteed's suggestion that an ‘earthquake’ might be an explosion in the air was linked with contemporary thinking about the roles of sulphur and nitre in earthquakes underground, and in combustion, respiration, and other processes. It reveals his concern with subjects other than astronomy and the influence of his continuing contact with members of the Royal Society; it also offers an early example of how seventeenth-century work on sulphur and nitre prepared the way for ‘airquake’ and electrical theories associated with the London earthquake of 1750.