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

Ativação das células hepáticas estreladas e fibrose hepática em crianças portadoras de hepatite autoimune tipo 1: estudo imuno-histoquímico de biópsias hepáticas pareadas antes do tratamento e após a remissão clínica

The activation of hepatic stellate cells (HSC) is considered the most important event in hepatic fibrogenesis. The precise mechanism of this process is unknown in autoimmune hepatitis (AIH), and more evidence is needed on the evolution of fibrosis. The aim of this study was to assess these aspects in children with type 1 AIH. We analyzed 16 liver biopsy samples from eight patients, paired before treatment and after clinical remission, performed an immunohistochemical study with anti-actin smooth muscle antibody and graded fibrosisand inflammation on a scale of 0:4 (Batts and Ludwig scoring system). We observedthere was no significant reduction in fibrosis scores after 24± 18 months (2.5 ± 0.93 vs. 2.0± 0.53, P = 0.2012). There was an important decrease in inflammation: portal (2.6 ±0.74 vs. 1.3± 0.89, P = 0.0277), periportal/periseptal (3.0 ±0.76 vs. 1.4 ± 1.06, P = 0.0277), and lobular (2.8 ± 1.04 vs. 0.9± 0.99, P =0.0179). Anti-actin smooth muscle antibodies were expressed in the HSC of the initial biopsies (3491.93 ±2051.48 lm2), showing a significant reduction after remission (377.91 ±439.47 lm2) (P = 0.0117). HSC activation was demonstrated in the AIH of children. The reduction of this activation after clinical remission, which may precede a decrease in fibrosis, opens important perspectives in the follow-up of AIH.