Thesis (Ph. D.)--University of Pennsylvania, 1911. / "The Introductionis arithmeticae libri duo ... is the basis of the present essay"--P. 1. Includes bibliographical references (p. 1).

Euclid’s Method of finding Pythagorean triples is a commonly accepted and applied technique. This study focuses on a myriad of other methods behind finding such Pythagorean triples. Specifically, we discover whether or not other ways of finding triples are special cases of Euclid’s Method.

This study concerns a teaching design with the purpose to facilitate the students’ objectification of the Pythagorean Theorem. Twelve 14-year old students (N=12) participated in the study before the theorem was introduced to them at school. The design incorporated ideas from the ‘embodied mind’ framework, history and realistic mathematics, linking ‘embodied verticality’ with ‘perpendicularity’. The qualitative analyses suggested that the participants were led to the conquest of the ‘first level of objectification’ (through numbers) of the Pythagorean Theorem, showing also evidence of appropriate ‘fore-conceptions’ of the ‘second level of objectification’ (through proof) of the theorem. The triangle the sides of which are associated with the Basic Triple (3,4,5) served as a primary instrument for the students’ objectification, mainly, by facilitating their ‘generic abstraction’ of the Pythagorean Triples.

Submitted by MARCOS EDER CUPAIOLI (marcoscupaioli@hotmail.com) on 2016-09-13T14:53:52Z No. of bitstreams: 1 Dissertação-MARCOS-EDER-CUPAIOLI-Matemática-Final Repositório.pdf: 3030917 bytes, checksum: 5fee5216541ccf70c5a9acb075b9976f (MD5) / Rejected by Felipe Augusto Arakaki (arakaki@reitoria.unesp.br), reason: Solicitamos que realize uma nova submissão seguindo as orientações abaixo: No campo “Versão a ser disponibilizada online imediatamente” foi informado que seria disponibilizado o texto completo porém no campo “Data para a disponibilização do texto completo” foi informado que o texto completo deverá ser disponibilizado apenas 6 meses após a defesa. Caso opte pela disponibilização do texto completo apenas 6 meses após a defesa selecione no campo “Versão a ser disponibilizada online imediatamente” a opção “Texto parcial”. Esta opção é utilizada caso você tenha planos de publicar seu trabalho em periódicos científicos ou em formato de livro, por exemplo e fará com que apenas as páginas pré-textuais, introdução, considerações e referências sejam disponibilizadas. Se optar por disponibilizar o texto completo de seu trabalho imediatamente selecione no campo “Data para a disponibilização do texto completo” a opção “Não se aplica (texto completo)”. Isso fará com que seu trabalho seja disponibilizado na íntegra no Repositório Institucional UNESP. Por favor, corrija esta informação realizando uma nova submissão. Agradecemos a compreensão. on 2016-09-14T21:55:41Z (GMT) / Submitted by MARCOS EDER CUPAIOLI (marcoscupaioli@hotmail.com) on 2016-09-15T01:47:57Z No. of bitstreams: 1 Dissertação-MARCOS-EDER-CUPAIOLI-Matemática-Final Repositório.pdf: 3030917 bytes, checksum: 5fee5216541ccf70c5a9acb075b9976f (MD5) / Approved for entry into archive by Juliano Benedito Ferreira (julianoferreira@reitoria.unesp.br) on 2016-09-15T18:14:38Z (GMT) No. of bitstreams: 1 cupaioli_me_me_sjrp.pdf: 3030917 bytes, checksum: 5fee5216541ccf70c5a9acb075b9976f (MD5) / Made available in DSpace on 2016-09-15T18:14:38Z (GMT). No. of bitstreams: 1 cupaioli_me_me_sjrp.pdf: 3030917 bytes, checksum: 5fee5216541ccf70c5a9acb075b9976f (MD5) Previous issue date: 2016-08-19 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Este trabalho aborda um conjunto de atividades experimentais com a finalidade de demonstrar um dos mais belos e importantes teoremas da Matemática: o Teorema de Pitágoras. São conhecidas mais de 400 demonstrações, aqui optamos por utilizar uma demonstração devido a Rudolf Wolf, por possibilitar uma abordagem geométrica lúdica através da dissecção de figuras planas. Inicialmente apresentamos o conceito geral de semelhança e áreas das figuras planas que utilizam propriedades e áreas de polígonos equidecomponíveis. Posteriormente, realizamos um breve resgate histórico sobre diversas demonstrações do Teorema e da vida de Pitágoras. Destacamos, também, uma maneira de achar algumas ternas pitagóricas, utilizando a sequência de Fibonacci. Por fim, foram propostas e desenvolvidas atividades experimentais em sala de aula com a utilização de moldes em EVA, explorando o Teorema de Pitágoras e algumas de suas aplicações. / This work contains a set of experimental activities in order to prove one of the most beautiful and important theorems in Mathematics: the Pythagorean Theorem. There are known more than 400 proofs, here we chose to use a proof due to Rudolf Wolf, by allowing a playful geometric approach by dissection of plane figures. Initially we present the general concept of similarity and areas of plane figures using properties and areas of equidecomposable polygons. Later, we do a brief historical review of some proofs of Theorem and Pythagoras's life. We also highlight a way to find some Pythagorean triples using the Fibonacci sequence. Finally, it was proposed and developed experimental activities in the classroom with the use of molds EVA, exploring the Pythagorean theorem and some of its applications.

It has been shown that alkaloids from various sources are vital as lead compounds in medicinal research and thus also the efficient synthesis of these. With the goal of developing a general synthetic route that can potentially access pyrrolizidine, indolizidine, quinolizidine and possibly lehmizidine alkaloid skeletons, a modified route that has been shown to produce pyrrolizidines was employed towards the synthesis of indolizidine alkaloid 223AB. Within this synthesis, a 6-endo-dig hydroamination-cyclization step was attempted for construction of the bicyclic system. For this purpose, a selection of titanium-based catalysts were synthesized in order to determine their regiochemical outcome. For the purpose of investigating ab initio the mechanism of regioselective hydroamination, the skills and methods involved in computational chemistry were acquired through a study into amide rotational barriers: A range of novel 2-oxo-2H-chromen-7-yl dimethylcarbamates were synthesised containing either an oxygen or sulphur in the α-position to the carbonyl or thiocarbonyl group of the amide moiety. Microwave synthesis was essential for the successful synthesis of some of the sulphur containing carbamates. The barriers to internal rotation of each of these compounds were investigated as follows. Variable Temperature and Exchange Spectroscopy NMR was performed on these compounds and the barrier to free amide rotation was calculated. Each of these compounds were also modeled ab initio and the gas phase barrier to rotation calculated. These three sets of data were compared and the influence of the α-heteroatom on rotation for amides and thioamides evaluated. / Thesis (M.Sc.) - University of KwaZulu-Natal, Pietermaritzburg, 2008.

Thesis (Ph. D.)--University of Chicago, 1941. / Lithoprinted. "Private edition, distributed by the University of Chicago libraries."

SILVA, Danniel Emanuel Bruno. O teorema de Pitágoras: abordagem no cotidiano da educação matemática e suas diversas demonstrações. 2014. 55 f. Dissertação (Mestrado Profissional em Matemática em Rede Nacional) - Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2014. / Submitted by Jessyca Silva (jessyca@mat.ufc.br) on 2017-08-21T19:55:33Z No. of bitstreams: 1 2014_dis_debsilva.pdf: 1316565 bytes, checksum: f9f30b38921af75056aba52ae1360c98 (MD5) / Rejected by Rocilda Sales (rocilda@ufc.br), reason: Bom dia, Na folha de aprovação não deve constar as assinaturas dos membros da banca. Rocilda on 2017-08-22T13:51:25Z (GMT) / Submitted by Jessyca Silva (jessyca@mat.ufc.br) on 2017-08-23T14:35:03Z No. of bitstreams: 1 2014_dis_debsilva.pdf: 1008247 bytes, checksum: e15ea5795b214cc268197d35bb86e018 (MD5) / Rejected by Rocilda Sales (rocilda@ufc.br), reason: Boa tarde, encontrei alguns pequenos erros na Dissertação de DANNIEL EMANUEL, e já envie uma cópia do email para ele avisando quais eram os erros. 1 – FOLHA DE ROSTO (centralize o nome Fortaleza e o ano da Dissertação) 2- FICHA CATALOGRÁFICA (o título da Dissertação que aparece na ficha catalográfica deve estar em letra minúscula, só a primeira letra deve ser maiúscula) 3- FOLHA DE APROVAÇÃO (está faltando seu nome na parte superior da folha de aprovação, acima do título da Dissertação) 4- EPÍGRAFE (retire o sublinha que aparece no nome EINSTEIN) 5- NUMERAÇÃO DAS PÁGINAS (coloque a numeração das páginas no CANTO SUPERIOR DIREITO) Atenciosamente, on 2017-08-23T15:50:40Z (GMT) / Submitted by Jessyca Silva (jessyca@mat.ufc.br) on 2017-08-23T19:33:51Z No. of bitstreams: 1 2014_dis_debsilva.pdf: 1008605 bytes, checksum: 92cfe1499970b1c1f1a10d77725f68b3 (MD5) / Approved for entry into archive by Rocilda Sales (rocilda@ufc.br) on 2017-08-25T11:13:23Z (GMT) No. of bitstreams: 1 2014_dis_debsilva.pdf: 1008605 bytes, checksum: 92cfe1499970b1c1f1a10d77725f68b3 (MD5) / Made available in DSpace on 2017-08-25T11:13:24Z (GMT). No. of bitstreams: 1 2014_dis_debsilva.pdf: 1008605 bytes, checksum: 92cfe1499970b1c1f1a10d77725f68b3 (MD5) Previous issue date: 2014 / This work is about the Pythagorean Theorem and its various statements, addressing its importance and as has been shown in mathematics education. We will present some of the history of Pythagoras, as well as on his theorem, and well see a search on their importance, and how students receive this information. We'll talk a bit about how the textbooks, from elementary school to the middle, deal about it, and finally we will see some statements about this theorem seeking to open a lot more range of options for students and lovers of mathematics teachers. These statements are part of the collection assembled by Loomis, who managed to submit more than three hundred statements of the theorem in a publication of 1940. / Este trabalho trata sobre o Teorema de Pitágoras e suas diversas demonstrações, abordando sua importância e como vem sendo apresentado na educação matemática. Apresentaremos um pouco da história de Pitágoras, bem como, sobre o seu teorema, e veremos uma pesquisa sobre a sua importância, e como os alunos recebem essa informação. Falaremos um pouco, sobre como os livros didáticos, tanto do ensino fundamental como no médio, tratam sobre o assunto, e por fim veremos algumas demonstrações sobre este Teorema buscando abrir muito mais o leque de opções para os alunos e professores amantes da matemática. Essas demonstrações fazem parte do acervo reunido por Loomis, que conseguiu apresentar mais de trezentas demonstrações do teorema em uma publicação de 1940.

The sixth-century-BCE Pythagorean communities included both male and female members. This thesis focuses on the Pythagorean women and aims to explore what reasons lie behind the prominence of women in Pythagoreanism and what roles women played in early Pythagorean societies and thought. In the first chapter, I analyse the social conditions of women in Southern Italy, where the first Pythagorean communities were founded. In the second chapter, I compare Pythagorean societies with ancient Greek political clubs and religious sects. Compared to mainland Greece, South Italian women enjoyed higher legal and socio-political status. Similarly, religious groups included female initiates, assigning them authoritative roles. Consequently, the fact that the Pythagoreans founded their communities in Croton and further afield, and that in some respects these communities resembled ancient sects helps to explain why they opened their doors to the female gender to begin with. The third chapter discusses Pythagoras’ teachings to and about women. Pythagorean doctrines did not exclusively affect the followers’ way of thinking and public activities, but also their private way of living. Thus, they also regulated key aspects of the female everyday life, such as marriage and motherhood. I argue that the Pythagorean women entered the communities as wives, mothers and daughters. Nonetheless, some of them were able to gain authority over their fellow Pythagoreans and engage in intellectual activities, thus overcoming the female traditional domestic roles. The fourth chapter argues that another contributing factor to the status of the Pythagorean women is the doctrine of metempsychosis. This belief led the Pythagoreans to adopt similar behaviours towards other ensouled beings. Therefore, since men and women were believed to have the same souls, they were treated with the same respect and received the same education. Finally, the fifth chapter explores how the Pythagorean views on women are taken up and developed in Plato’s Republic. I argue that, although the Pythagoreans never went as far as to have philosopher-queens and abolish private families, they took the first step towards Plato’s ‘gender equality’ theory. Overall, that of women in Pythagoreanism is the first documented case of female engagement with ancient philosophy: Pythagorean men and women lived together according to the same lifestyle, were educated on the same doctrines and played equally integral roles in the intellectual community.

This study concerns a teaching design with the purpose to facilitate the students’ objectification of the Pythagorean Theorem. Twelve 14-year old students (N=12) participated in the study before the theorem was introduced to them at school. The design incorporated ideas from the ‘embodied mind’ framework, history and realistic mathematics, linking ‘embodied verticality’ with ‘perpendicularity’. The qualitative analyses suggested that the participants were led to the conquest of the ‘first level of objectification’ (through numbers) of the Pythagorean Theorem, showing also evidence of appropriate ‘fore-conceptions’ of the ‘second level of objectification’ (through proof) of the theorem. The triangle the sides of which are associated with the Basic Triple (3,4,5) served as a primary instrument for the students’ objectification, mainly, by facilitating their ‘generic abstraction’ of the Pythagorean Triples.

Few would refute the didactic nature of stained glass windows, paintings, and sculptures used in Spanish cathedrals during the Counter-Reformation. For hundreds of years the artistic renderings of biblical narratives and of Catholic dogmata had aided both the literate and illiterate alike to internalize the teachings of the Church. In contrast, the seemingly complex web of semiotic signs that form part of the aural and visual spectacle of the auto sacramental has understandably led some to question if such productions could have truly held much meaning for commoners with little formal education. However, as a theatrical genre, the auto sacramental does not deviate much from the literal meaning and allegorical symbolism of the more static art forms that adorn cathedral walls and altarpieces. The usage of ships, highwaymen, and courtroom trials represent some of the most prominent symbols utilized by playwrights to create a Counter-Reformatory drama that facilitated the audiences' ability to decode the plays' allegorical meaning. The repeated use of these semiotic signs allowed the culturally literate public in urban centers across Spain to draw upon their intertextual knowledge of such symbols to appreciate and understand these Corpus Christi performances. Modern readers less familiar with these semiotic signs and their meaning experience an additional handicap because of their inability to see the visual spectacle designed, if not as the primary didactic tool of the genre, then at least as an effective complement to the instructive dialogue that takes place between the different characters of the auto. In spite of these additional challenges that the modern reader faces, the auto sacramental offers some insight into sixteenth- and seventeenth-century Spain that cannot be found in the more secular genres of the time. The added effort to investigate and understand the missing links of intertextual knowledge open a window that offers a panorama of a largely unexplored landscape of early-modern, Spanish society.