| The Greek mathematician Diophantus of Alexandria's Arithmetica, which dealing with problems in indeterminate analysis, is systematically studied in this doctoral dissertation. Based on the research of other historians of mathematics, by further expansion and exploration of historical materials, a series of new results are reached. The paper contains five parts:1. The process of the historical transmission on Diophantus' Arithmetica is checked in this paper. It points out the placement of four Arabic books and six Greek books within the original Arithmetica, and analyzes historically the formal differences in the two kinds of extant texts. (1) The form in which we know the Greek Arithmetica(with the fragment of the De polygonis numeris appended to it)can be definitively traced back to the time of Plaximus Planudes, who ever gathered and collated it in thirteenth century. (2) The remaining books were believed to be lost, until the discovery of a medieval Arabic translation of four of the remaining books in a manuscript in the Shrine Library in Meshed in Iran. The manuscript was discovered in 1968 by F.Sezgin. The four Arabic books have been the subject of previous publications by Dr.Rashed and Dr.Sesiano. This paper viewed the four Arabic books not as an entity in itself but as part of the Arithmetica as a whole. (3) For the formal division of the solution of each problem, Arabic text contains six parts: (i) protasis, (ii) diorismos, (iii) ekthesis, (iv) analysis, (v) synthesis, and (vi) symperasma, whereas the Greek contains only parts (i)-(iv), the Greek text of Books I-III and "IV"-"VI" is, aside from unsystematic interpolations, the same as the original written by Diophantus; The situation with regard to the Arabic text is more complicated: part of the Arithmetica (at least Books I-VII) plus early additions were rewritten by a commentator (Hypatia?), who added the synthesis; the final statements were added by a Greek scholiast, and the resulting version was translated into Arabic.2. The origin and influence of Arithmetica are analyzed in the paper. (1) Did Diophantus learn his methods from Euclid's Data or did he have access to the Babylonian material? These questions cannot be answered, there are only some historical clues which we have to pay attention to. For example, I, 27, 28, 29, 30 are also 4 types solved by Babylonians. Interestingly, the answers to these problems are all same, reminding us of the common Babylonian practice of having the same answers to a series of related problems. However, Diophantus' solution is strictly algebraic, unlike the quasigeometric Babylonian solution. (2) Diophantus' work, the only example of a genuinely algebraic work surviving from ancient Greece, was highly influential. Not only was it commented on in late antiquity, but it was also carefully studied by many famous mathematicians such as R.Descartes, F.Viete and P.Fermat and led them to numerous general results which Diophantus himself only hinted. Perhaps more important, however, is the fact that this work, as a work of algebra, was in effect a treatise on the analysis of problems. The synthesis was never given by Diophantusbecause it only amounted to an arithmetic computation. Thus Diophantus' work is at the opposite end of the spectrum from the purely synthetic work of Euclid.3. The methods of Diophantus are discussed in the paper. Most of Diophantus' problem can be written as a set of k equations in more than k unknowns. Often there are infinitely many solutions. For these problems, Diophantus generally gives only one solution explicitly, but one can easily extend the method to give other solutions. Thus, one of central work is to explicate the generality of Diophantus' solution.4. The translation is essentially a literal one, and may thus be awkward in some passages; we chose to translate in this way, in order to give an idea of the form and expression of the text. When desirable, the sense of passages has been made clearer by additions which appear in parentheses. The footnotes consist... |
PDF Full Text Request