A obra de triangulis omnimodis libri quinque de Johann Müller Regiomontanus (1436 1476): uma contribuição para o desenvolvimento da trigonometria

Trigonometry, branch of mathematics related to the study of triangles, developed from practical needs, especially relating to astronomy, Surveying and Navigation. Johann Müller, the Regiomontanus (1436-1476) mathematician and astronomer of the fifteenth century played an important role in the development of this science. His work titled De Triangulis Omnimodis Libri Quinque written around 1464, and published posthumously in 1533, presents the first systematic exposure of European plane and spherical trigonometry, a treatment independent of astronomy. In this study we present a description, translation and analysis of some aspects of this important work in the history of trigonometry. Therefore, the translation was performed using a version of the book Regiomontanus on Triangles of Barnabas Hughes, 1967. In it you will find the original work in Latin and an English translation. For this study, we use for most of our translation in Portuguese, the English version, but some doubt utterance, statement and figures were made by the original Latin. In this work, we can see that trigonometry is considered as a branch of mathematics which is subordinated to geometry, that is, toward the study of triangles. Regiomontanus provides a large number of theorems as the original trigonometric formula for the area of a triangle. Use algebra to solve geometric problems and mainly shows the first practical theorem for the law of cosines in spherical trigonometry. Thus, this study shows some of the development of the trigonometry in the fifteenth century, especially with regard to concepts such as sine and cosine (sine reverse), the work discussed above, is of paramount importance for the research in the history of mathematics more specifically in the area of historical analysis and critique of literary sources or studying the work of a particular mathematician

A difícil aceitação dos números negativos: um estudo da teoria dos números de Peter Barlow (1776-1862)

The present study seeks to present a historico-epistemological analysis of the development of the mathematical concept of negative number. In order to do so, we analyzed the different forms and conditions of the construction of mathematical knowledge in different mathematical communities and, thus, identified the characteristics in the establishment of this concept. By understanding the historically constructed barriers, especially, the ones having ontologicas significant, that made the concept of negative number incompatible with that of natural number, thereby hindering the development of the concept of negative, we were able to sketch the reasons for the rejection of negative numbers by the English author Peter Barlow (1776 -1862) in his An Elementary Investigation of the Theory of Numbers, published in 1811. We also show the continuity of his difficulties with the treatment of negative numbers in the middle of the nineteenth century

Fundamentos da filosofia moral de Peter Singer

This research aims to reconstruct and explain the argument proposed by Peter Singer to justify the principle of equal consideration of interests (PECI). The PECI is the basic normative principle according to people should consider the interests of all sentient beings affected when somebody taking a moral decision. It is the join that Singer proposes between universalizability and the principle of equal consideration of interests that constitutes a compelling reason to justify it. The universalizability requires to disregard the numerical differences, putting yourself in other people s shoes, and to consider preferences, interests, desires and ideals of those affected. Singer joins universalizability to normative principle and molds the form and content of his theory. The first chapter introduces the discussion will be developed in this essay. The second chapter deals the historical and philosophical viewpoint from which Singer starts his studies. The third chapter is about the Singer s critiques of naturalism, intuitionism, relativism, simple subjectivism and emotivism. The fourth chapter exposes the design of universal prescriptivism proposed by R. M. Hare. The universal prescriptivism indicates, in the Singer s viewpoint, a consistent way to create the join between the universalizability and PECI. It highlights also the criticism designed by J. L. Mackie and Singer himself to universal prescriptivism. The second part of this chapter shows briefly some of the main points of the classical conception of utilitarianism and its possible relationship with the theory of Singer. The fifth chapter introduces the Singer s thesis about the origin of ethics and the universalizability as a feature necessary to the point of view of ethic, and the way which this argument is developed to form the PECI. The sixth chapter exposes the main distinctions that characterize the PECI. Finally the seventh chapter provides a discussion about the reasons highlighted by Singer for one who wants orient his life according to the standpoint of ethics. This structure allows explaining the main ideas of the author concerning the theoretical foundations of his moral philosophy