The Mathematics of Collision and the Collision of Mathematics in the 17th Century

Thesis (Ph.D.) - Indiana University, History and Philosophy of Science, 2015This dissertation charts the development of the quantitative rules of collision in the 17th century. These were central to the mathematization of nature, offering natural philosophy a framework to explain all the changes of nature in terms of the size and speed of bodies in motion. The mathematization of nature is a classic thesis in the history of early modern science. However, the significance of the dynamism within mathematics should not be neglected. One important change was the emergence of a new language of nature, an algebraic physico-mathematics, whose development was intertwined with the rules of collision. The symbolic equations provided a unified system to express previously diverse kinds of collision with a new representation of speed with direction, while at the same time collision provided a practical justification of the otherwise "impossible" negative numbers. In private manuscripts, Huygens criticized Descartes's rules of collision with heuristic use of Cartesian symbolic algebra. After he successfully predicted the outcomes of experiments using algebraic calculations at an early meeting of the Royal Society, Wallis and Wren extended the algebraic investigations in their published works. In addition to the impact of the changes in mathematics itself, the rules of collision were shaped by the inventive use of principles formulated by 'thinking with objects,' such as the balance and the pendulum. The former provided an initial framework to relate the speeds and sizes of bodies, and the latter was key both in the development of novel conservation principles and made possible experimental investigations of collision. This dissertation documents the formation of concepts central to modern physical science, and re-evaluates the mathematics of collision, with implications for our understanding of major figures in early modern science, such as Descartes and Huygens, and repercussions for the mathematization of nature

A large discourse concerning algebra: John Wallis's 1685 <i>Treatise of algebra</i>

A treatise of algebra historical and practical (London 1685) by John Wallis (1616-1703) was the first full length history of algebra. In four hundred pages Wallis explored the development of algebra from its appearances in Classical, Islamic and medieval cultures to the modern forms that had evolved by the end of the seventeenth century. Wallis dwelt especially on the work of his countrymen and contemporaries, Oughtred, Harriot, Pell, Brouncker and Newton, and on his own contribution to the emergence of algebra as the common language of mathematics. This thesis explores why and how A treatise of algebra was written, and the sources Wallis used. It begins by analysing Wallis's account of mathematical learning in medieval England, never previously investigated. In his researches on the origins and spread of the numeral system Wallis was at his best as a historian, and initiated many modern historiographical techniques. His summary of algebra in Renaissance Europe was less detailed, but for Wallis this part of the story set the scene for the English flowering that was to be his main theme. The influence of Oughtred's Clavis on Wallis and his contemporaries, and Wallis's efforts to promote the book, are explored in detail. Wallis's controversial account of Harriot's algebra is also examined and it is argued that it was better founded than has sometimes been supposed and that Wallis had direct access to Harriot's algebra through Pell. Many other chapters of A treatise of algebra contain mathematics that can be linked or traced to Pell, a hitherto unsuspected secret of the book. The later chapters of the thesis, like the final part of A treatise of algebra, explore Wallis's Arithmetica infinitorum and the work which arose from it up to Newton's foundation of modern analysis, and include a discussion of Brouncker's treatment of the number challenges set by Fermat. The thesis ends with a summary of contemporary and later reactions to A treatise of algebra and an assessment of Wallis's view of algebra and its history

Complex numbers from 1600 to 1840

This thesis uses primary and secondary sources to study advances in complex number theory during the 17th and 18th Centuries. Some space is also given to the early 19th Century. Six questions concerning their rules of operation, usage, symbolism, nature, representation and attitudes to them are posed in the Introduction. The main part of the thesis quotes from the works of Descartes, Newton, Wallis, Saunderson, Maclaurin, d'Alembert, Euler, Waring, Frend, Hutton, Arbogast, de Missery, Argand, Cauchy, Hamilton, de Morgan, Sylvester and others, mainly in chronological order, with comment and discussion. More attention has been given tp algebraists, the originators of most advances in complex numbers, than to writers in trigonometry, calculus and analysis, who tended to be users of them. The last chapter summarises the most important points and considers the extent to which the six questions have been resolved. The most important developments during the period are identified as follows: (i) the advance in status of complex numbers from 'useless' to 'useful'. (ii) their interpretation by Wallis, Argand and Gauss in arithmetic, geometric and algebraic ways. (iii) the discovery that they are essential for understanding polynomials and logarithmic, exponential and trigonometric functions. (iv) the extension of trigonometry, calculus and analysis into the complex number field. (v) the discovery that complex numbers are closed under exponentiation, and so under all algebraic operations. (vi) partial reform of nomenclature and symbolism. (vii) the eventual extension of complex number theory to n dimensions

The teaching of science in English dissenting academies 1662-1800

This thesis attempts a new assessment of the place of science teaching in the English Dissenting Academies. It examines the approach to science teaching, seeking to account for the presence of science in the curricula of the Academies and its relationship to the wider educational aims. The content of science courses is examined together with the texts used and the relationships between the science taught and contemporary trends in scientific thought. Some attention is also given to the forms of teaching used in Dissenting Academies, and comparisons with the Universities. Solutions to these problems have been sought in a variety of sources: lecture notes, correspondence, published texts, formal minutes, prospectuses. Chapter 1 introduces the subject, and Chapters 2 to 8 cover the Academies. As there are over 90 known Academies it is not possible to examine each one in equal depth, thus four (Northampton, Moorfields/Stepney/Hoxton, Warrington and Hackney) have been chosen as case studies. These Academies were selected because sufficient material survives, and their dates collectively allow continuous and overlapping coverage from 1701 to 1796. The remaining Academies are discussed rather more briefly. Chapter 9 draws together the findings of Chapters 2 to 8, and attempts an assessment of the Academies by their approach to and teaching of scientific subjects. The most significant point about the science teaching in the Academies is the varied quality, which ranged from the exceptional to the merely perfunctory. In almost one half, there is no evidence to suggest that science was taught at all. Generalisations cannot be made on the strength of the excellence of individual Academies or tutors, for example Priestley, Forster or Dalton, all of whom taught in these institutions. The reasons for including the subject on the curriculum also vary including specialist courses for students intending to follow medical or commercial careers, but in most Academies, the central reason is related to Christian belief, and the use of scientific knowledge to support the theological argument from design. An antithesis between the Academies' aim to train ministers and at the same time, to offer a broad general education, can be perceived in attitudes towards science, and it is possible to suggest a more convincing reason for the decline of the Academies than those hitherto advanced

Reconceptualizing Mathematics Education

This dissertation is to explore theoretically mathematics education in the United States and the need for reconcepualizing mathematics education. Mathematics education needs reconceptualizing because students know very little mathematics by the time they graduate from high school. Mathematics has become a subject to be feared and dreaded for centuries. High school teachers blame middle school teachers, middle school teachers blame elementary teachers, and elementary teachers blame parents for their students\u27 lack of preparedness in mathematics. Elementary teachers express frustration in teaching mathematics because of their own lack of content knowledge and lack of preparation for the mathematics component of their profession. Regardless of who is to blame, most students entering high school are not prepared to problem solve nor are they interested in mathematics except as the dreaded requirement needed to graduate. Because I have been involved in mathematics education for more than three decades, I have seen many programs come and go. I have seen different types of pedagogy be the in way to teach mathematics. Naturally, technology has influenced mathematics education tremendously in the last decade. Unfortunately, many mathematics educators use technology as a crutch instead of using it to enhance mathematics education. Mathematics education in the United States has been debated for over three centuries. The debate is ongoing. Standardized testing has become a way of life in schools today. Teachers are expected to tell the students exactly what they are supposed to know in mathematics. Standardized tests do not allow students to be creative or struggle in their quest for knowledge because teachers must make sure they have covered the material for the test. The No Child Left Behind Act of 2001 (NCLBA) adds to the problem of mathematics education. The shortage of mathematics teachers throughout the nation is acute. Compliance with the NCLBA requires more mathematics teachers than can possibly be found. My purpose in writing this dissertation is to convey my thoughts and ideas about how the study of mathematics developed, how mathematics education progressed throughout history how mathematics education is progressing today, and how mathematics education will progress in the future. In my opinion, teacher preparation of elementary and middle school teachers will be a very strong component in the reconceptualization of mathematics education. Mathematics teachers at all levels should be grounded in a history of mathematics and be cognizant of the development of mathematics education throughout the relatively short history of America. Furthermore, a dialogue must be implemented and maintained between mathematics educators at all levels. With the implementation of this dialogue, mathematics education will become a subject of intrigue and beauty and will no longer remain the subject to be feared and dreaded

EMPIRICAL CHARACTERIZATION OF SOFTWARE QUALITY

The research topic focuses on the characterization of software quality considering the main software elements such as people, process and product. Many attributes (size, language, testing techniques etc.) probably could have an effect on the quality of software. In this thesis we aim to understand the impact of attributes of three P’s (people, product, process) on the quality of software by empirical means. Software quality can be interpreted in many ways, such as customer satisfaction, stability and defects etc. In this thesis we adopt ‘defect density’ as a quality measure. Therefore the research focus on the empirical evidences of the impact of attributes of the three P’s on the software defect density. For this reason empirical research methods (systematic literature reviews, case studies, and interviews) are utilized to collect empirical evidence. Each of this research method helps to extract the empirical evidences of the object under study and for data analysis statistical methods are used. Considering the product attributes, we have studied the size, language, development mode, age, complexity, module structure, module dependency, and module quality and their impact on project quality. Considering the process attributes, we have studied the process maturity and structure, and their impact on the project quality. Considering the people attributes, we have studied the experience and capability, and their impact on the project quality. Moreover, in the process category, we have studied the impact of one testing approach called ‘exploratory testing’ and its impact on the quality of software. Exploratory testing is a widely used software-testing practice and means simultaneous learning, test design, and test execution. We have analyzed the exploratory testing weaknesses, and proposed a hybrid testing approach in an attempt to improve the quality. Concerning the product attributes, we found that there exist a significant difference of quality between open and close source projects, java and C projects, and large and small projects. Very small and defect free modules have impact on the software quality. Different complexity metrics have different impact on the software quality considering the size. Product complexity as defined in Table 53 has partial impact on the software quality. However software age and module dependencies are not factor to characterize the software quality. Concerning the people attributes, we found that platform experience, application experience and language and tool experience have significant impact on the software quality. Regarding the capability we found that programmer capability has partial impact on the software quality where as analyst capability has no impact on the software quality. Concerning process attributes we found that there is no difference of quality between the project developed under CMMI and those that are not developed under CMMI. Regarding the CMMI levels there is difference of software quality particularly between CMMI level 1 and CMMI level 3. Comparing different process types we found that hybrid projects are of better quality than waterfall projects. Process maturity defined by (SEI-CMM) has partial impact on the software quality. Concerning exploratory testing, we found that exploratory testing weaknesses induce the testing technical debt therefore a process is defined in conjunction with the scripted testing in an attempt to reduce the associated technical debt of exploratory testing. The findings are useful for both researchers and practitioners to evaluate their project

Attempts to measure annual stellar parallax : Hooke to Bessel

Imperial Users onl

Re-developing knowledge creation capability: Innovating in Indian pharmaceutical industry under the TRIPS regime

The transition to a new technology, market or regulatory regime can be difficult for any organisation to manage. Technological and institutional change has proven to be a big cause for the failure of established firms and many examples exist of such failures. The Trade Related intellectual property rights agreement (TRIPs), as part of The World Trade Organization (WTO) agreement, represents such an institutional change for knowledge based industries from developing countries. As a result of the TRIPs agreement all of the WTO member countries will move from no or partial patent protection to fully fledged patent protection. This represents a radical break with the past in which developing countries typically had only weak levels of patent protection. Against this backdrop, the research examines the learning processes involved in the development of innovative R&D capabilities within the context of the Indian pharmaceutical industry, in response to the strengthening of patent law. In the last decade much research has addressed the process of dynamic learning within firms, however this has predominantly focused on firms from advanced countries. Previous research on developing countries mainly focused on building the minimum knowledge base essential for production and innovation activity. In recent years limited research has begun to explore dynamic learning in firms from developing countries. However, there still remains a scarcity of research which examines firm-level learning processes central to the development of advanced level capabilities. This research addresses this deficiency by applying the conceptual understanding developed within advanced countries to a developing countries context. This is operationalised through a set of research activities which investigate firm-level learning, knowledge creation and innovative capability within the context of the Indian pharmaceutical industry. The substantive conclusions are that the development of new capabilities involves the removal of redundant capabilities, coupled with the acquisition of new knowledge. The findings also indicate that Indian firms are hiring Indian scientists educated or working overseas in multinational pharmaceutical R&D and collaborating with Indian and overseas research institutes and universities to acquire capabilities in innovative R&D. Furthermore, inter-firm differences in learning processes suggest that at a firm level, learning is neither linear nor automatic and requires a deliberate strategy. The thesis also provides important insights into knowledge creation capabilities that have significant implications with respect to innovative activity for firms from other developing countries