Pappus of Alexandria: Book 4 of the Collection: Edited With Translation and Commentary by Heike Sefrin-Weis (Sources and Studies in the History of Mathematics and Physical Sciences) 2010th Edition by Heike Sefrin-Weis (PDF)

Description

Although not so well known today, Book 4 of Pappus’ Collection is one of the most important and influential mathematical texts from antiquity. The mathematical vignettes form a portrait of mathematics during the Hellenistic “Golden Age”, illustrating central problems – for example, squaring the circle; doubling the cube; and trisecting an angle – varying solution strategies, and the different mathematical styles within ancient geometry.This volume provides an English translation of Collection 4, in full, for the first time, including: a new edition of the Greek text, based on a fresh transcription from the main manuscript and offering an alternative to Hultsch’s standard edition, notes to facilitate understanding of the steps in the mathematical argument, a commentary highlighting aspects of the work that have so far been neglected, and supporting the reconstruction of a coherent plan and vision within the work, bibliographical references for further study.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Pappus’s Book 4 is most famous for its “meta-theoretical passage,” “a locus classicus for methodology in ancient mathematics” (p. 271), where problems are divided into a hierarchy of three classes based on their complexity:”We say that there are three kinds of problems in geometry, and that some [of the problems] are called ‘plane’, others ‘solid’, and yet others ‘linear’. Now, those that can be solved by means of straight line and circle, one might fittingly call ‘plane’. For the lines by means of which problems of this sort are found have their genesis in the plane as well. All those problems, however, that are solved when one employs for their invention either a single one or even several of the conic sections, have been called ‘solid’. For it is necessary to use the surfaces of solid figures—I mean, however, (surfaces) of cones—in their construction. Finally, as a certain third kind of problems the so-called ‘linear’ kind is left over. For different lines, besides the ones mentioned, are taken for their construction, which have a more varied and forced genesis, because they are generated out of less structured surfaces, and out of twisted motions.” (p. 144) “An error of the following sort seems to be not a small one for geometers, [namely] when a plane problem is found by means of conics or of linear devices …, and summarily, whenever it is solved from a nonkindred kind.” (p. 145)The mathematical content of Book 4 is largely concerned with various of these higher curves that can be used to solve the three classical construction problems. For example, “For the duplication of the cube a certain line is introduced by Nicomedes” (p. 126), namely the conchoid, which “Nicomedes himself has proved … can be described with an instrument” (p. 127). Similarly, “For the squaring of the circle a certain line has been taken up by Dinostratus and Nicomedes and some other more recent (mathematicians)” (p. 131), namely the quadratrix, which is defined as the intersection of a rotating and a horizontally moving line coordinated in a particular way.The quadratrix is a very interesting case, because it involves a difficulty that cuts to the very core of what it means to solve a mathematical problem. For the coordination of the motions is highly problematic, as Pappus reports:”Sporus, however, is with good reason displeased with [the quadratrix] … [since it] takes into the assumption the very thing for which it seems to be useful. For how is it possible when two points start from B, that they move, the one along the straight line to A, the other along the arc to D, and come to a halt [at their respective end points] at the same time, unless the ratio of the straight line AB to the arc BED is known beforehand? For the velocities of the motions must be in this ratio, also.” (p. 132)Pappus tackles the matter thus: “Now, this genesis of the [quadratrix] is, as has been said, rather mechanical; it can however be made the subject of a geometrical analysis by means of loci on surfaces” (p. 137), by assuming a cylindrical helix as given.As Sefrin-Weis notes, this analysis is apparently “intended to meet, or rather perhaps to circumvent, the objections raised by Sporus, so as to ‘geometrize’ the curve” (p. 256). However, it does so only indirectly and partially. Again as Sefrin-Weis notes: “Because we are using analysis, the result is not a constructive solution, or a constructive genesis of the curve. This is also not intended. The content really is a mathematical analysis of the genesis, establishing unique determinateness for the ‘target curve'” (p. 262). Thus “Pappus does not achieve, and does not believe he has achieved, the quadrature of the circle. He has not ‘saved’ the genesis of the quadratrix in the sense that the curve can now be constructed geometrically, and he does not claim to have ‘solved’ the problem. … Even so, the analytical determination has achieved something. Its effect is that the quadratrix, although not constructible, can be investigated geometrically, without conceptual inconsistencies, qua locus curve for a certain symptoma. It is well-defined, uniquely determined.” (p. 265)Sefrin-Weis offers a fascinating speculation on the basis of this case:”Apollonius for one argued that the helix should be placed alongside circle and straight line as a basic, unanalyzed principle in geometrical argumentation. Does this imply an anti-essentialist thrust, a turn towards making locus-properties, i.e., relations, the final objects of mathematics? Are the basic items all loci, as it were, characterized as such via ‘defining’ relations? That would make Apollonius a forerunner of the paradigm shift toward algebra that occurred in the seventeenth century. It cannot be ruled out.” (p. 266)”If such was the case, and there was an Apollonian programme to implement a new paradigm for mathematics, one in which operationalism, and the manipulation of relations are key ideas, we would have to say that the programme did not carry the day in antiquity, and the ancient research project of symptoma-mathematics might have died out precisely for that reason: re-channeling into the mainstream essentialist approach. What we see in Coll. IV, and what the seventeenth century readers saw as well, would then be like the remnants of a large-scale re-orientation project for mathematics which was abandoned, with the remnants still bearing the traces of the revolutionary ideas behind them, of this push toward operationalizing geometry into a proto-algebraic discipline. Such an ideological clash, an unsuccessful frontal attack against the ruling paradigm, which in turn was backed by non-negotiable essentialist convictions and preconceptions, and which in the end prevailed, would explain why Apollonius’ minor analytical works were lost, why his Konika were stripped of their analysis-parts, which in the original must have been dominant (Pappus groups the Konika with the analytical works), and recast by Eutocius in purely synthetic form, and also why no works of the authors who worked on the analysis of loci on surfaces are preserved. The essentialists in the field of epistemology/theory of science would have won the day, and forced the continuation of the old paradigm. Such a speculation is tempting. But it is equally possible that the mathematicians, including Apollonius, went along with the essentialist views on the nature of science and explanation, or—and that is perhaps the most likely option—that they did not reflect on such questions at all and just went ahead doing their mathematics of symptoma-curves.” (pp. 266-267)

⭐

Keywords

Free Download Pappus of Alexandria: Book 4 of the Collection: Edited With Translation and Commentary by Heike Sefrin-Weis (Sources and Studies in the History of Mathematics and Physical Sciences) 2010th Edition in PDF format
Pappus of Alexandria: Book 4 of the Collection: Edited With Translation and Commentary by Heike Sefrin-Weis (Sources and Studies in the History of Mathematics and Physical Sciences) 2010th Edition PDF Free Download
Download Pappus of Alexandria: Book 4 of the Collection: Edited With Translation and Commentary by Heike Sefrin-Weis (Sources and Studies in the History of Mathematics and Physical Sciences) 2010th Edition 2010 PDF Free
Pappus of Alexandria: Book 4 of the Collection: Edited With Translation and Commentary by Heike Sefrin-Weis (Sources and Studies in the History of Mathematics and Physical Sciences) 2010th Edition 2010 PDF Free Download
Download Pappus of Alexandria: Book 4 of the Collection: Edited With Translation and Commentary by Heike Sefrin-Weis (Sources and Studies in the History of Mathematics and Physical Sciences) 2010th Edition PDF
Free Download Ebook Pappus of Alexandria: Book 4 of the Collection: Edited With Translation and Commentary by Heike Sefrin-Weis (Sources and Studies in the History of Mathematics and Physical Sciences) 2010th Edition