The calculus according to S. F. Lacroix (1765-1843)

Silvestre François Lacroix (Paris. 1765 - ibid., 1843) was not a prominent mathematical researcher, but he was certainly a most influential mathematical book author. His most famous book is a monumental Traité du calcul différentiel et du calcul intégral (three large volumes, 1797-1800; a second edition appeared in 1810-1819) - an encyclopaedic appraisal of 18th-century calculus. He also published many textbooks, one of which is closely associated to this large Traité: the Traité élémentaire du calcul différentiel et du calcul intégral (first edition in 1802; four more editions in Lacroix's lifetime; four posthumous editions). Although most historians acknowledge the great influence of Lacroix's large Traité in early 19th-century mathematics it has not been thoroughly studied. This thesis is a contribution for correcting this omission. The focus is on its first edition, but the second edition and the Traité élémentaire, are also addressed. The thesis starts with a short biography of Lacroix, followed by an overview of the first edition of the large Traité. Next corne five chapters where particular aspects are analyzed in detail: the foundations of the calculus, analytic and differential geometry, approximate integration and conceptions of the integral, types of solutions of differential equations (singular/complete/general integrals, geometrical interpretations, and generality of arbitrary functions), and three aspects related to finite differences and series (the use of subscript indices, types of solutions of finite difference equations, and mixed difference equations); for all these aspects Lacroix's treatment is compared to the 18th-century background, and to his likely sources. Then we examine how the large Traité was adapted to a textbook - the Traité élémentaire, we take a look at the second edition of the large Traité, and conclude the body of the thesis with some final remarks

The calculus according to S. F. Lacroix (1765-1843)

Silvestre François Lacroix (Paris. 1765 - ibid., 1843) was not a prominent mathematical researcher, but he was certainly a most influential mathematical book author. His most famous book is a monumental Traité du calcul différentiel et du calcul intégral (three large volumes, 1797-1800; a second edition appeared in 1810-1819) - an encyclopaedic appraisal of 18th-century calculus. He also published many textbooks, one of which is closely associated to this large Traité: the Traité élémentaire du calcul différentiel et du calcul intégral (first edition in 1802; four more editions in Lacroix's lifetime; four posthumous editions). Although most historians acknowledge the great influence of Lacroix's large Traité in early 19th-century mathematics it has not been thoroughly studied. This thesis is a contribution for correcting this omission. The focus is on its first edition, but the second edition and the Traité élémentaire, are also addressed. The thesis starts with a short biography of Lacroix, followed by an overview of the first edition of the large Traité. Next corne five chapters where particular aspects are analyzed in detail: the foundations of the calculus, analytic and differential geometry, approximate integration and conceptions of the integral, types of solutions of differential equations (singular/complete/general integrals, geometrical interpretations, and generality of arbitrary functions), and three aspects related to finite differences and series (the use of subscript indices, types of solutions of finite difference equations, and mixed difference equations); for all these aspects Lacroix's treatment is compared to the 18th-century background, and to his likely sources. Then we examine how the large Traité was adapted to a textbook - the Traité élémentaire, we take a look at the second edition of the large Traité, and conclude the body of the thesis with some final remarks.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Role of an artefact of Dynamic algebra in the conceptualisation of the algebraic equality

In this contribution, we explore the impact of Alnuset, an artefact of dynamic algebra, on the conceptualisation of algebraic equality. Many research works report about obstacles to conceptualise this notion due to interference of the previous arithmetic knowledge. New meanings need to be assigned to the equal sign and to letters used in algebraic expressions. Based on the hypothesis that Alnuset can be effectively used to mediate the conceptual development necessary to master the algebraic equality notion, two experiments have been designed and implemented in Italy and in France. They are reported in the second part of this pape

Rethinking the Concept of Exclusion in Patent Law

Patent law’s broad exclusionary rule is one of its defining features. It is unique within intellectual property as it prohibits acts of independent creation. Even if a second inventor had no connection or aid from an initial inventor, patent law allows the first inventor to stop the second. Even though a number of pressing problems can be traced to this rule, it remains untouchable; it is thought to be essential for incentivizing invention. But is it really our only choice? And why is it so different from our otherwise widespread reliance on free entry and competition in markets? The current rule and its anti-competitive stance are defended as being economically necessary as well as being administratively manageable. This article questions both of these justifications. As an alternative, the article explores a narrower type of exclusion suggested by Learned Hand some fifty years ago. The article finds that his reform ideally could provide for the same set of inventive projects (if not more) as the current rule while it could avoid many of the pitfalls bedeviling the current system. Learned Hand’s suggested rule models itself on copyright where infringement extends only to copyists and thus allows generally free entry and competition. Interestingly, despite the competitive pressures and their reduction in the magnitude of the reward to the initial inventor, this ‘free entry system’ can provide for the same set of inventive projects as the current rule and because of the competitive pressures, it can do so with improved social welfare. Furthermore as to administration, though there are surely difficulties in both monitoring and adjudicating such a copying-based patent rule, there are important unappreciated self-enforcement benefits. Though far from advocating an immediate doctrinal change, these results suggest at least a conceptual reorientation wherein prevention of copying and its resulting economic undercutting and not the per se prevention of competition become the ideal goals of the patent system. Rather than being a necessary economic and administrative feature, patent law’s broad conceptualization of exclusion may be an artifact that we would jettison if only we could

The mathematical works of Bernard Bolzano published between 1804 and 1817

The purpose of the thesis is to assess the mathematical achievements of Bernard Bolzano on the basis of the five early published works. The material is divided into the areas of the foundations of mathematics, geometry and analysis. In making this assessment there have been two principal considerations. Firstly, any judgement of the significance of Bolzano's work should be made in the light of the historical context, so considerable space is devoted to the relevant 18th century sources. Secondly, as a general framework to the thesis there is the question of how Bolzano's general views about mathematical proofs and concepts are related to his achievements. The main claim and conclusion of the thesis is that this relationship was unusually clear and significant in the case of Bolzano's work. There is an Appendix containing the first English translation of all five of Bolzano's works as well as the German texts of their first editions

Vagueness in mathematics talk

The Cockcroft Report claimed that "mathematics provides a means of communication which is powerful, concise and unambiguous". Such precision in language may be a conventional aim of mathematics, particularly when communicated in writing. Nonetheless, as this thesis demonstrates, vagueness is commonplace when people talk about mathematics. In this thesis, I examine the circumstances in which vagueness arises in mathematics talk, and consider the practical purposes which speakers achieve by means of vague utterances in this context. The empirical database, which is considered in Chapters 4 to 7, consists almost entirely of transcripts of mathematical conversations between adult interviewers (including myself) and one or two children. The data were collected from clinical interviews focused on a small number of tasks, and from fragments of teaching. For the most part, the pupils involved in the study were aged between 9 and 12, although the age-range in Chapter 7 extends from 4 to 25. I draw on a number of approaches to discourse associated with 'pragmatics' -a field of linguistics - to analyse the motives and communicative effectiveness of speakers who deploy vagueness in mathematics talk. I claim that, for these speakers, vagueness fulfills a number of purposes, especially 'shielding', i. e. self-protection against accusation of being wrong. Another purpose is to give approximate information; sometimes to achieve shielding, but also to provide the level of detail that is deemed to be appropriate in a given situation. A different purpose, associated with a particular form of vagueness (of reference), is to compensate for lexical gaps in pursuit of effective communication of concepts and ideas. I show, in particular, how speakers use the pronouns 'it' and 'you' in mathematics talk to communicate concepts and generalisations. Some consideration is given to the intentions of 'expert speakers of mathematics when they deploy vague language. Their purposes include some of those identified for novices. Teachers also use vagueness as a means of indirectness in addressing pupils; this strategy is associated with the redress of 'face threatening acts'. My thesis is that vagueness can be viewed and presented, not as a disabling feature of language, but as a subtle and versatile device which speakers can and do deploy to make mathematical assertions with as much precision, accuracy or as much confidence as they judge is warranted by both the content and the circumstances of their utterances. I report on the validation and generalisation of my findings by an Informal Research Group of school teachers, who transcribed and analysed their own classroom interactions using the methods I had developed

The crucial role of proof--a classical defense against mathematical empiricism

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Linguistics and Philosophy, 1993.Includes bibliographical references (leaves 135-137).by Catherine Allen Womack.Ph.D