Analyse av værforhold ved utvalgte skredsituasjoner i Ørsta

The meteorological phenomena of avalanches in Nivane, ørsta, Norway, for the period from 1960 to 1980 and for the years 1941 and 1942 have been analysed. We have analysed the meteorological phenomena of some situations which could possibly lead to an avalanche. The main reason for avalanches in Nivane, ørsta, seems to be strong winds from the NW combined with intense snow fa11, corresponding to a precipitation of 75 - 100 mm during the last 3 days before the avalanche. When cold periods precede the avalanche condition only a minor quantity of new snowfall is necessary to release an avalanche. In such situations the wind direction is not so decisive for the relase of an avalanche.NTN

Une interprétation byzantine de Diophante

In this paper, we develop a new interpretation, based on the notion of proportion, of Diophantus's procedure, which is applicable to all cases of equations with two unknowns that Diophantus (ca.A.D. 250) solves. This interpretation was proposed for the first time by Maximus Planudes (ca.A.D. 1255 - 1305) in his commentary on problem II 8 of theArithmetica.Copyright 1998 Academic Press. Dans cet article on propose une nouvelle interprétation, basée sur la notion de la proportion, de la méthode utilisée par Diophante à la résolution des équations à deux inconnues. L'interprétation par proportion avait été proposée pour la première fois par Maxime Planude dans son commentaire au problème II 8 desArithmétiques.Copyright 1998 Academic Press. Στην εργασια πρτεινεται μια νεα ερμηνεια, με χρηση αναλγιων, της μεθδυ πυχρησιμπιει Διφαντς για να επιλυσει τις εξισωσεις με δυ αγνωστυς. H ερμηνεια αυτη ειχε πρταθει για πρωτη φρα απ τν Mαξιμ Πλανυδη στ σχλι τυ στ πρβλημα B 8 τωνAριθμητικων. Copyright 1998 Academic Press. AMS 1991 subject classification: 01 A 20 © 1998 Academic Press

The Meaning of Hypostasis in Diophantus’ Arithmetica

Historians of ancient philosophy and theological writers often come up against the puzzling issue of understanding the meaning of the term hypostasis used by different ancient authors. One could hardly expect that the same issue would be of interest for historians of ancient mathematics. Indeed, altogether absent from the works of Euclid, Archimedes, and Apollonius, and scarcely appearing in a nonmathematical context in the works of Heron and Nicomachus, the term hypostasis and its cognates appear 127 times in the six books of Diophantus’ Arithmetica preserved in Greek. This chapter examines Diophantus’ use of the term hypostasis and argues in favour of interpreting it as a term for numbers qua specific, individual entities. It is composed of three parts. The first part discusses the different statuses of numbers in a worked-out problem according to Diophantus’ general method, and the relevant issue of the Diophantine conception of an arithmetical problem; the second part investigates all instances of the term within Diophantus’ text; and the third part surveys briefly the testimonies of the Byzantine commentators of the Arithmetica, which provide further evidence supporting the interpretation proposed in this paper. © 2015, Springer International Publishing Switzerland

Analytical Reasoning and Problem-Solving in Diophantus’s Arithmetica : Two Different Styles of Reasoning in Greek Mathematics

Over the past few decades, the question regarding the proper understanding of Diophantus’s method has attracted much scholarly attention. “Modern (i.e., post-Vietan) algebra”, “algebraic geometry”, “arithmetic”, “analysis and synthesis”, have been suggested by historians as suitable contexts for describing Diophantus’s resolutory procedures, while the category of “premodern algebra” has recently been proposed by other historians to this end. The aim of this paper is to provide arguments against the idea of contextualizing Diophantus’s modus operandi within the conceptual framework of the ancient analysis and to examine the few instances, in the preserved books of the Arithmetica, which might be regarded as linked to practices belonging to the field of analysis

Diophantus and Premodern Algebra: New Light on an Old Image

As a theme of historical research, Diophantus' Arithmetica (ca 300 AD) raises two main issues that have been most intensively debated: the first concerns the proper understanding of Diophantus' practice, while the other relates to the identification of the mathematical tradition to which this practice belongs. Since the time of medieval Islam, through the Renaissance and the early modern period, the thesis that the work of Diophantus belongs to the history of algebra has enjoyed broad consensus among mathematicians, despite the fact that the term `algebra' was introduced in the language of mathematics five centuries after Diophantus. Thus, as early as the Middle Ages, the work of Diophantus was recognized by mathematicians as a work on algebra, avant la lettre. The consensus was maintained during the nineteenth and the most part of the twentieth century-this time among historians of mathematics. It is essential to stress, however, that, when associating Diophantus' work with algebra, premodern mathematicians on the one hand and modern historians of mathematics on the other did not start from the same understanding of algebra. Those mathematicians understood algebra with its premodern meaning and, accordingly, characterized the Arithmetica as `algebraic' in the premodern meaning of the term. In contrast, modern historians of mathematics approach the Arithmetica mostly through the viewpoint of a loose understanding of modern algebra and, precisely for this reason, their accounts are often exposed to anachronism. This explains why some contemporary historians of ancient mathematics are reluctant to accept the conclusions of the traditional historiography, while others deny without reservation any relation of Diophantus' practice to algebra. However, criticizing the methodology by which the traditional historiography has reached its conclusion does not necessarily mean that the conclusion itself was wrong. This paper discusses some crucial issues related to Diophantus' problem solving practice, thus, giving support to the traditional view of the algebraic character of his work, but put in a totally new framework of ideas

Greek geometrical analysis: A new interpretation through the "Givens"-terminology

This paper consists of two parts. In the first part the linguistic, philosophical, and mathematical aspects of the terms 'dothen' and 'dedomenon', which appear in Greek mathematical texts related to the method of analysis, are discussed. It is argued that these terms, usually translated by the same word (' given'), have a slightly different meaning: The term 'dothen' means the casually possible, and is used, as a rule, in the intermediate steps of a syllogism, while the term 'dedomenon' means the necessarily possible, which arises as a result of a syllogism. In the light of this discrimination, a new interpretation of Greek geometrical analysis is proposed in the second part. It is argued that the analytical part of the solution of a geometrical problem comprises two separate parts, named here 'hypothetical' and 'confirmatory'. The 'hypothetical' part is an upward movement of searching for preconditions, while the 'confirmatory' part, which is developed with use of the terms 'dothen' and 'dedomenon', is a course of deductions, and is directed downwards. Finally, the new interpretation of analysis is illustrated through an example from Pappus's mathematical practice

Theory of ratios in Nicomachus'

The voluminous Treatise of the four mathematical sciences of Georgios Pachymeres is the most renowned quadrivium produced in Byzantium. Among its specific features, historians of mathematics have pointed out, is the inclusion of Diophantus, besides Nicomachus and Euclid, in the sources for the arithmetical section and, accordingly, the incorporation of series of problems and problem-solving in its contents. The present paper investigates the “Diophantine portion” of Pachymeres' treatise and it shows that it is structured according to two criteria intrinsically characterized by seriality: on one hand, the arrangement in which the problems are presented in book I of Diophantus' Arithmetica; on the other hand, for those problems of which the enunciation involves ratio, the order in which Nicomachus discusses the kinds of ratios in his Arithmetical introduction. Furthermore, it analyses the solutions that Pachymeres offers and argues that Nicomachus' Arithmetical introduction provides the necessary tools for pursuing them

The way of Diophantus: Some clarifications on Diophantus' method of solution

In the introduction of the Arithmetica Diophantus says that in order to solve arithmetical problems one has to "follow the way he (Diophantus) will show." The present paper has a threefold objective. Firstly, the meaning of this sentence is discussed, the conclusion being that Diophantus had elaborated a program for handling various arithmetical problems. Secondly, it is claimed that what is analyzed in the introduction is definitions of several terms, the exhibition of their symbolism, the way one may operate with them, but, most significantly, the main stages of the program itself. And thirdly, it is argued that Diophantus' intention in the Arithmetica is to show the way the stages of his program should be practically applied in various arithmetical problems. © 2006 Elsevier Inc. All rights reserved