A New Translation of and Guide to Newton's Principia

This paper discusses the emergence of new medical experimental specialties at the Medical School of Surgery (Escola Médico-Cirúrgica) and the Faculty of Medicine of Lisbon University (Faculdade de Medicina da Universidade de Lisboa) between 1897 and 1946, as a result of the activities of Marck Athias's (1875–1946) histophysiology research school. In 1897, Marck Athias, a Portuguese physician who had graduated from the Faculty of Medicine in Paris, founded a research school in Lisbon along the lines of Michael Foster's physiology research school in England and Franz Hofmeister's physiological chemistry school in Germany. His research programme was highly innovative in Portugal. Not only did it bring together many disciples and co-workers, but it branched out and created new medical specialties within Portuguese medical science. These new disciplinary areas grew out of the study of the histology of the nervous system but eventually expanded into normal and pathological histophysiology, physiological chemistry and experimental endocrinology. The esprit de corps that existed between research school members ensured the school's success and influence in various fields social and political as well as scientific. Athias's school was strongly influenced by positivist ideals and promoted a teaching and research style that sought inspiration in Humboldt's university model, thus helping to bring about a change in the dominant scientific ethos and to modernize scientific research in Portugal during the first half of the twentieth century.     

<Emphasis Type="Italic">Newton's Argument for Proposition 1 of the</Emphasis> Principia

The first proposition of the Principia records two fundamental properties of an orbital motion: the Fixed Plane Property (that the orbit lies in a fixed plane) and the Area Property (that the radius sweeps out equal areas in equal times). Taking at the start the traditional view, that by an orbital motion Newton means a centripetal motion – this is a motion ``continually deflected from the tangent toward a fixed center' – we describe two serious flaws in the Principia's argument for Proposition 1, an argument based on a polygonal impulse approximation. First, the persuasiveness of the argument depends crucially on the validity of the Impulse Assumption: that every centripetal motion can be represented as a limit of polygonal impulse motions. Yet Newton tacitly takes the Impulse Assumption for granted. The resulting gap in the argument for Proposition 1 is serious, for only a nontrivial analysis, involving the careful estimation of accumulating local errors, verifies the Impulse Assumption. Second, Newton's polygonal approximation scheme has an inherent and ultimately fatal disability: it does not establish nor can it be adapted to establish the Fixed Plane Property. Taking then a different view of what Newton means by an orbital motion – namely that an orbital motion is by definition a limit of polygonal impulse motions – we show in this case that polygonal approximation can be used to establish both the fixed plane and area properties without too much trouble, but that Newton's own argument still has flaws. Moreover, a crucial question, haunted by error accumulation and planarity problems, now arises: How plentiful are these differently defined orbital motions? Returning to the traditional view, that Newton's orbital motions are by definition centripetal motions, we go on to give three proofs of the Area Property which Newton ``could have given' – two using polygonal approximation and a third using curvature – as well as a proof of the Fixed Plane Property which he ``almost could have given.' (Received August 14, 2002) Published online March 26, 2003 Communicated by G. Smith     

The Integrability of Ovals: Newton's Lemma 28 and Its Counterexamples

Principia (Book 1, Sect. 6), Newton's Lemma 28 on the algebraic nonintegrability of ovals has had an unusually mixed reception. Beginning in 1691 with Jakob Bernoulli (who accepted the lemma) and Huygens and Leibniz (who rejected it and offered counterexamples), Lemma 28 has a history of eliciting seemingly contradictory reactions. In more recent times, D.T. Whiteside in 1974 gave an “unchallengeable counterexample,” while the mathematician V.I. Arnol'd in 1987 sided with Bernoulli and called Newton's argument an “astonishingly modern topological proof.” This disagreement mostly stems, we argue, from Newton's vague statement of the lemma. Indeed, we identify several different interpretations of Lemma 28, any one of which Newton may have been intending to assert, and we then test a number of proposed counterexamples to see which, if any, are true counterexamples to one or more of these versions of the lemma. Following this, we study Newton's argument for the lemma to see whether and where it fails to be convincing. In the end, our study of Newton's Lemma 28 provides an answer to the question, Who is right: Huygens, Leibniz, Whiteside and the others who reject the lemma, or Bernoulli, Arnol'd, and the others who accept it? (Received November 6, 2000)     

Editing Newton in Geneva and Rome: The Annotated Edition of the Principia by Calandrini,Le Seur and Jacquier

This contribution examines the circumstances of composition of the annotated edition of Newton's Principia that was printed in Geneva in 1739–1742, which ran to several editions and was still in print in Britain in the mid-nineteenth century. This edition was the work of the Genevan Professor of Mathematics, Jean Louis Calandrini, and of two Minim friars based in Rome, Thomas Le Seur and François Jacquier. The study of the context in which this edition was conceived sheds light on the early reception of Newtonianism in Geneva and Rome. By taking into consideration the careers of Calandrini, Le Seur and Jacquier, as authors, lecturers and leading characters of Genevan and Roman cultural life, I will show that their involvement in the enterprise of annotating Newton's Principia answered specific needs of Genevan and Roman culture. The publication and reception of the Genevan annotated edition has also a broader European dimension. Both Calandrini and Jacquier were in touch with the French république des lettres, most notably with Clairaut and Du Châtelet, and with the Bernoulli family in Basel. Therefore, this study is also relevant for the understanding of the dissemination of Newton's ideas in Europe.     

Accuracy in forecasting: A survey

In this study, the author provides a brief and concise summary of empirical investigations pertaining to forecasting with special reference to the accuracy of different forecasting techniques. The study mainly focuses on comparisons of the accuracy of these techniques. The comparisons cover both quantitative and qualitative methods. In addition the summary includes studies seeking to test or improve accuracy by combining forecasting techniques.     

The rule of law: Natural,human, and divine

In this paper we compare two different contexts—the legal and the scientific—in which the concept of law is prominent. We argue that the acute philosophical awareness, in the early modern period, of the difficulties surrounding the law concept in the scientific context, and the various responses to these difficulties, are rooted in an earlier tradition of jurisprudential concerns over the concept of natural law in its legal sense. We seek to show, further, that each one of the various philosophical accounts of the concept of natural law (in both of its senses) is embedded in a metaphysical and theological context, so that different visions of God yield different accounts of the meaning of the natural law idiom in science as well as legal theory.     

Kant’s third law of mechanics: The long shadow of Leibniz

This paper examines the origin, range and meaning of the Principle of Action and Reaction in Kant’s mechanics. On the received view, it is a version of Newton’s Third Law. I argue that Kant meant his principle as foundation for a Leibnizian mechanics. To find a ‘Newtonian’ law of action and reaction, we must look to Kant’s ‘dynamics,’ or theory of matter.