Book Description

This volume includes a selection of 19 classic papers on the history of Greek mathematics that were published during the 20th century and affected significantly the state of the art of this field. It is divided into six thematic sections and covers all the major issues of the Greek mathematical production. First, the inclusion in one volume of a considerable number of papers that had been published for the first time in old, and in certain cases hard to find, scientific journals representing turning-points in the history of the field, constitutes a particularly useful aid for all those working on the history of mathematics. Second, by means of the selected papers and the introductory texts of six well-known modern historians of ancient mathematics that accompany them, the reader can follow the ways the historiography of Greek mathematics developed. Finally, the introductory texts that precede each chapter help the reader to approach critically the selected papers and at the same time to get an idea of the issues being further clarified by the new historiographical approaches.

Book Description

Book Description
Ancient Greece was the birthplace of science, which developed in the Hellenized culture of ancient Rome. This volume locates science within ancient Greek society and culture, investigates its impact upon that society, and identifies it as a cultural phenomenon deserving no less attention than literary or artistic creativity.Chapters by seventeen international experts examine the role and achievement of science and mathematics in Greek antiquity through discussion of the linguistic, literary, political, religious, sociological, and technological factors which influenced scientific thought and practice. Greek science was both motivated and constrained by wholly 'unscientific' cultural interests, and by ideas and biases arising from the language and the paradigms of the day. For example, it is here argued that the prediction of eclipses was not a concern of ancient astronomers until after 'non-scientific' authors such as the historian Livy, elaborating on a good story with a moral, suggested that it should be.Familiar classical authors, such as Homer, Polybius, Cicero, and Pliny are here seen in a new light. Less-studied classical authors, such as Euclid, Hero, Galen, and Ptolemy, are also considered, and attention is drawn to areas where there is potential for new research and where editions and translations are still needed. ... Read more

Book Description
iA History of Mathematics: From Mesopotamia to Modernity/i covers the evolution of mathematics through time and across the major Eastern and Western civilizations. It begins in Babylon, then describes the trials and tribulations of the Greek mathematicians. The important, and often neglected, influence of both Chinese and Islamic mathematics is covered in detail, placing the description of early Western mathematics in a global context. The book concludes with modern mathematics, covering recent developments such as the advent of the computer, chaos theory, topology, mathematical physics, and the solution of Fermat's Last Theorem.Containing more than 100 illustrations and figures, this text, aimed at advanced undergraduates and postgraduates, addresses the methods and challenges associated with studying the history of mathematics. The reader is introduced to the leading figures in the history of mathematics (including Archimedes, Ptolemy, Qin Jiushao, al-Kashi, al-Khwarizmi, Galileo, Newton, Leibniz, Helmholtz, Hilbert, Alan Turing, and Andrew Wiles) and their fields. An extensive bibliography with cross-references to key texts will provide invaluable resource to students and exercises (with solutions) will stretch the more advanced reader. ... Read more

Customer Reviews (2)

A historiography-geek history
Hodgkin is a historiography geek with no interest in writing a history of mathematics other than to nitpick about details. Basically, each chapter summarises the conventional story---usually rather scornfully, and too briefly for anyone to gain from it---and then dwells on a myriad of minuscule objections to this version raised by highly specialised historians and published (for a reason, I would say) in highly specialised journals. This piling up of obscure historiographical hypotheses rarely makes a coherent point, let alone does it contribute to any substantial understanding of the history of mathematics.

Refreshing math history
Mr. Hodgkin gives a great overview of the history of mathematics, the current state of historical arguments, and all the references (including websites) for further study.At 262 pages it is very readable - I was not looking for a ponderous work with every possible fact catalogued.His approach is refreshingly irreverent and even funny:
"10th century Damascus must surely have been unique as a place where copying the text of Euclid could earn you a living." and
"Perhaps rather than decrying the 'low level' of geometry present in Vitruvius's architecture, we should think about the fact that it was a Roman, rather than a Greek, who bothered to write such a treatise....We have different cultures (cohabiting in the same empire) with different ideas of what a book is for."
I have slogged my way through many math histories without learning half as much, and to be entertained as well is more than one hopes for in such a book. ... Read more

Book Description
This is the first edition of the original text of the advanced part of the most important work on conic sections written in antiquity and one of the most influential works in mathematics. It is also the first literal English translation of it ever to be published. The purpose of the work is to make available, to those interested in the history of science and to mathematicians, a version of the work as close to the original as possible. This part of Apollonius' Conics is lost in the original Greek, and only an Arabic translation made in the 9th century survives. This text has never been published previously, and all "editions" of this part of Apollonius' work are based on the Latin translation from the Arabic published by Edmund Halley in 1710, which suffers from Halley's insufficient knowledge of Arabic and his use of a single manuscript. The present edition is based on all known manuscripts. Its other improvements over Halley's edition are: 1) the Arabic text with a full critical apparatus; 2) an accurate English translation (until now only a loose paraphrase, based on Halley's translation, has been available in English); 3) a commentary to elucidate both mathematical and historical difficulties. This book will replace Halley's edition and all its derivatives as the standard edition of this part of Apollonius' work. ... Read more

Book Description
The philosopher Immanuel Kant writes in the popular introduction to his philosophy: "There is no single book about metaphysics like we have in mathematics.If you want to know what mathematics is, just look at Euclid's Elements." (Prolegomena Paragraph 4) Even if the material covered by Euclid may be considered elementary for the most part, the way in which he presents essential features of mathematics in a much more general sense, has set the standards for more than 2000 years. He displays the axiomatic foundation of a mathematical theory and its conscious development towards the solution of a specific problem.We see how abstraction works and how it enforces the strictly deductive presentation of a theory.We learn what creative definitions are and how the conceptual grasp leads to the classification of the relevant objects. For each of Euclid's thirteen Books, the author has given a general description of the contents and structure of the Book, plus one or two sample proofs.In an appendix, the reader will find items of general interest for mathematics, such as the question of parallels, squaring the circle, problem and theory, what rigour is, the history of the platonic polyhedra, irrationals, the process of generalization, and more. This is a book for all lovers of mathematics with a solid background in high school geometry, from teachers and students to university professors.It is an attempt to understand the nature of mathematics from its most important early source. ... Read more

Customer Reviews (3)

Interesting survey of the Elements
The material in Euclid's Elements may be divided into four categories of very different degrees of interest for modern readers. (a) Elementary material. To keep us interested when covering tedious proofs of obvious things Artmann discusses foundational issues (as seen by Euclid and contrasted with the modern view), the principles that guide the overall structure of the books, historical topics, etc. (b) Well-known material. This category includes some basic geometry (Pythagoras's theorem, etc.), but primarily it includes all of Euclid's number theory. This is very interesting stuff but less exotic than other parts of the Elements since these pearls have been kept polished and accessible (see, for example, the historically enlightened books by Stillwell, esp. "Elements of Number Theory" and "Numbers and Geometry"). (c) Incomprehensible material. Some parts of the Elements appear mysterious to the modern reader, especially some aspects of "geometric algebra" and of course the theory of incommensurability. A truly faithful guide to the Elements would make it its mission to clarify these things, but Artmann is not that committed, often preferring instead the easy way of looking for agreement with modern mathematical-aestetical principles and commenting on those things instead (e.g. discussions of the role of generalisations and the relation between problems and theories). (d) Constructions. This is the most rewarding part. First there is the remarkable construction of the regular pentagon in book IV. Euclid's construction draws on all previous books, in accordance with his aim to hide his masterplan and unveil it in a flash of brilliance just as we though he was getting lost in a mass of technicalities. Artmann adds helpful commentary on how the principles of construction may be understood through possible earlier constructions that used marked rulers and similar triangles (not developed by Euclid until book VI). The similar triangles proof uses a neat property of the pentagon: a side and a diagonal are in "extreme and mean ratio" (i.e. "golden ratio"), so constructing this ratio is one way to construct the pentagon. Euclid brings this up in connection with the marvellous constructions of the regular polyhedra in book XIII --- the culmination of the entire Elements. "For the construction of the dodecahedron, Euclid starts with a cube and constructs what can be called a 'roof of a house' over each of its faces". The pentagonal faces of the dodecahedron are made up of a quadrilateral piece of one roof and a triangular gable from the roof on the adjacent side. To make this work we must choose the right length for our beams, i.e. we must divide the side of the cube in extreme and mean ratio. The construction topics are not only the most rewarding in themselves but also the starting points of Artmann's most enthusiastic excursions, including the modern algebraic view of constructions as developed by Gauss, the group theoretical view of symmetries and polyhedra, appearances of these figures in art and architecture, etc.

Roots of mathematics in our Western Culture
This is a Renaissance book by a Renaissance man.Artmann gives a full summary of the "Elements", using considerable modern notation.It is accurate and detailed, and the various themes he traces (such as Symmetry, or Incommensurables) let him include a wide range of topics: architecture, design, sculpture, myth, history -- even philology and poetry.Some may think he limits himself too narrowly to the classical Greeks, does too little digging in the Babylonian or Egyptian parts of the story.

To Artmann's credit, his book disregards the smallscale disputes amongst superspecialists ("all modern translations of Elements are satisfactory").He overturns the fashionable idea that the "Two Cultures" cannot communicate.So, Rilke has something to say -- perhaps not to Hilbert, but to the widely cultured mathematician, or to the general reader -- about Contradiction, or Widerspruch.

About the pre-Euclidean origins of mathematics in Greece, he overmodestly disclaims specialist knowledge.An example:he traces the earliest technical work on the dodecahedron and the icosahedron via pre-Euclideans such as Theaetetus (Plato's friend), and up to the highly abstract Group Theory work on isomorphisms of the 1990s A.D. -- and does this well and surefootedly. Too bad his modesty barred him ("I leave that to the specialists") from analyzing the pre-history of Euclid's Book XII, the classical ancestor of our integral calculus.The fact is that he knows a great deal about Eudoxus (another friend of Plato's).Perhaps more detail in a Second Edition?

His work on the so-called Euclidean Algorithm (finding a greatest common factor) is another valuable contribution.Its autobiographical flavor is reminiscent of Archimedes in "Sand Reckoner".It allows him to stake out a clear and non-partisan position on the "where is the algebra?" question, on which scholarly debates often produce more heat than light.

So multi-faceted a book, one could wish an Index fuller than a mere 2 pages.Typos are too frequent for a good house like Springer, including two I found in names of authors or book titles.But the book's cultural sweep is admirable throughout, its bibliography good.

TL Heath's 1933 report about the Cambridge undergraduate, so struck by Euclid ("a book to be read in bed or on a holiday") may have been exaggerated, making him over into a Young Werther.But Artmann's charming and learned book really is hard to put down, on or off holiday.

[note: this is a lightly revised version of a review I submitted a few days ago.-Malcolm Brown]

Roots of mathematics in our Western Culture
This is a Renaissance book by a Renaissance man.Artmann gives a full summary of the "Elements", using considerable modern notation.It is accurate and detailed, and the various themes he traces (such as Symmetry, or Incommensurables) let him include a wide range of topics: architecture, design, sculpture, myth, history -- even philology and poetry.

He largely disregards smallscale battles amongst the superspecialists ("all modern translations of Elements are satisfactory").He overturns the fashionable idea that the "Two Cultures" cannot communicate.(Rilke has things to say, perhaps not to Hilbert, but to the widely cultured mathematician, about Widerspruch!)

About the pre-Euclidean origins of mathematics, he overmodestly disclaims specialist knowledge.An example:his tracing of the earliest technical work on dodecahedrons and icosahedrons via pre-Euclideans such as Theaetetus (Plato's friend), and on up to the Group Theory work on isomorphisms of the 1990s A.D. is done well and surefootedly. Too bad his modesty barred him ("I leave that to the specialists") from analyzing the pre-history of Euclid's Book XII, the classical ancestor of our integral calculus.The fact is that he knows a great deal about Eudoxus (another friend of Plato's).Perhaps more detail in a Second Edition?

His work on the so-called Euclidean Algorithm (finding a greatest common factor) also contributes importantly.Its autobiographical flavor is reminiscent of that of Archimedes' in "Sand Reckoner".It allows him to stake out a clear and non-partisan position on the question "where is the algebra?" question, on which scholarly debates often produce more heat than light.

So multi-faceted a book, one could wish a fuller Index.But the cultural sweep is admirable throughout.TL Heath's 1933 report about the Cambridge undergraduate, so struck by Euclid ("a book to be read in bed or on a holiday") may have exaggerated, making him over into a Young Werther.But Artmann's charming and learned book really is hard to put down, even at vacationtime. ... Read more

Book Description
This is an updated edition of an original and controversial book. As well as revising parts of the text and substantially updating the bibliography, in a new Appendix the author takes a more polemical stance and enters into a discussion of the nature and range of different interpretations. The book is divided into three parts; Interpretation, Evidence, and Later developments. The first part presents several new interpretations of the idea of ratio in early Greek mathematics and illustrates them in detailed discussions of several texts. Part Two focuses on the sources themselves, and questions the depth of modern knowledge of Plato's Academy during his lifetime, the source of our text of Euclid's Elements, and modern understanding of early Greek mathematics. The final part contrasts some of the evidence from early and late antiquity and then gives a historical account, since the seventeeth century, of the theory of continued fractions, our version today of the mathematics underlying the reconstruction.From reviews of the first edition:'...a real treat.' Greece and Rome'...cites an impressive array of evidence...The result should be widely read by classicists and mathematicians as well as historians of mathematics.' ISIS'...he enters into classical scholarship here with a really 'new reconstruction' of early Greek mathematics.' Nature '...this fascinating book...will arouse the interest and command the admiration of any historically minded lover of mathematics with a taste for the unorthodox.' Institute of Mathematics and its Applications'This book, speculative in the best sense, engages the ancient material on its own terms in setting forth what the Greeks might have thought and done...While the book represents an important departure in historical research in its reaching beyond the spare formalism of surviving materials to an understanding of motivation and perception, its careful documentations and technical descriptions make it valuable in a more traditional way.' Zentralblatt fur Mathematik ... Read more

Customer Reviews (2)

A new landmark
"THE MATHEMATICS OF PLATO'S ACADEMY"
Second Edition
Fowler.

The first impression on receiving this book in your hands is the heavy weight.But this is not only true physically, due to the high quality of the cartridge paper, it is also true intellectually.Thus the second impression reinforces the first.The caliber of the scholarship exhibited in this tome is of the highest order, doing full justice to an investment in so expensive a paper.

Nothing less than the most complete exposition possible of ancient Greek mathematics as taught at the Platonic Academy in Athens, is presented, based on all currently available sources.

The author labors to guide the reader with diagrams, definitions, explanations, cross-references, commentaries and modern mathematical symbols to provide a clear, detailed and thorough account.He even starts from the photographic plates of Greek papyri.This is a major work of scholarship that itself deserves to become a classic; a model of its kind.

Just in case amazon readers accuse me of obsequious flattery, abject servility and distasteful onesidedness, allow me one criticism.The influence of the Ionian philosopher-mathematicians, Thales, Anaxagoras, Anaximander and Anaximenes on Plato's Academy is not covered.

A magnificent twenty-one page bibliography testifies to the author's detailed background research, and whets the reader's appetite for further reading.

Finally, three separate indexes show that the author is making every effort to help his reader as much as he can.Could one ask for more ?

A brilliant, sprawling book
Two things are certain if you really want to know what mathematics was done in Plato's Academy, and before Euclid: Your heart will break at the lack of evidence, and you will have to read this book.

Fowler details how thin the surviving evidence is, even for such basics as when Euclid's ELEMENTS were written. Drawing on other careful classicists he demolishes now traditional stories about the Pythagoreans and the irrational, Plato's Academy, even Euclid's own style in the Elements. He shows them coming from heavy interpretations of extremely vague (and often late) sources. Plates in the book show how desperately scanty are the physical remains of any mathematical writing within centuries of Plato's death. Even the first and second century AD leave us only a few scraps of Euclid.

On the positive side, Fowler gives a persuasive account of a method of reciprocal subtraction which he calls "anthyphairesis". It lay within the grasp of Athenian geometers, and suits some remarks Plato makes on mathematics, and suits traditions on geometers Plato knew, and goes far to unify and explain much of Euclid. It was apparently cited by Aristotle (under the name "antanairesis"). Probably, it really was used in the period. It also makes some very pretty geometry. Regular pentagons make a lot of sense anthyphairetically. Anyone trying to read the later books of Euclid, especially books X and XIII, will get tremendous help from this book. Conversely, you can hardly read much of this book without reading Euclid.

The book is not well organized. It spends many pages at a time on mathematical reconstructions that could not possibly have been used by the Greeks, so as to show beyond question that they could not have been. And it probably pushes its point too far. That is what classicists do. They push a point for all it is worth and perhaps more. These flaws are inevitable when you work on such important questions on so little evidence. Fowler assembles enormous amounts of classical textual evidence and later scholarship. He gives some nice mathematics including an appendix on the later arithmetized incarnation of anthyphairetic methods as continued fractions.

If you are determined to ask what math Plato knew and promoted, and what existed before Euclid--and so you are determined to break your heart--then you must read this book. ... Read more

Book Description
This book is at once an analytical study of one of the most important mathematical texts of antiquity, the Mathematical Collection of the fourth-century AD mathematician Pappus of Alexandria, and also an examination of the work's wider cultural setting. This is one of very few books to deal extensively with the mathematics of Late Antiquity. It sees Pappus' text as part of a wider context and relates it to other contemporary cultural practices and opens new avenues to research into the public understanding of mathematics and mathematical disciplines in antiquity. ... Read more

Customer Reviews (1)

Interesting project
Cuomo starts this book by suggesting we don't understand the late-classical era, that great confusing muddle which starts around 300 AD when Constantine transferred the capital of the Roman Empire to Constantinople and legalized Christianity. It ends with the Muslim conquest of Alexandria. In crude terms, it is what Gibbon called the 'fall of the Roman Empire.' Cuomo uses the 'Arch of Constantine' as a metaphoric reference. For most contemporary art historians, the arch is a pastiche of scavenged sculptures from earlier and finer artistic efforts. Scavenged is the key word here. The late-antiquity (according to Gibbon) was the moral equivalent of Sodom and Gomorrah.

I don't know how many people still take "Rome's Fall' as a moral litmus test, but I suspect the story still holds a lot of weight. It's this icon that Cuomo targets.

In general terms, I couldn't be more pleased with the project. Unfortunately, it doesn't really get off the ground. Cuomo isn't very forth coming on what she makes of the era. It seems she simply likes pastiche.

She starts her iconoclastic journey well, suggesting the subject of her book might never have existed. It is hard to argue the point. We know almost nothing about Pappus, the man. Unfortunately, the fictional Pappus concept seems to have been mentioned for shock value, and not pursued seriously. I would have been interested in hearing details on the process of putting mathematic lectures on scrolls for academic, social or bureaucratic purposes. Maybe ghost writing was a common practice. This emphasis on the 'media' itself seems critical to Cuomo's case (a role the Arch of Constantine served), but it is entirely ignored.

Cuomo then takes us down an entertaining bunny hole involving legal torture and highly paid astrologers. By taking this route, she hopes to convince us that mathematics was about as important to our late-classical delinquents as, well, ourselves. The legal discussion shows mathematical knowledge put one socially above those who could expect torture during any legal cross-examination. The astrological references show desperate young parents prayed for their off-spring to become mathematicians.

So far, so good, but Cuomo then launches into a book by book deconstruction of the works ascribed to Pappus (whoever he was), and in this the reader starts to wonder just what she wants to say. The less than stunning conclusion is that Pappus had careerist interests and said different things to target groups in hopes of enhancing his authority.

I was less than impressed.

One might surmise Cuomo has a bigger goal, but if it exists, it is very subtle. Of these subtle arguments, the chief seems to be that the standard historiography associates the development of Greek mathematics exclusively with Plato's philosophy (the Proclus (411-485) perspective). Cuomo points out contradictions in this line of reasoning made by Pappus (? 320 ?) and Iamblichus (250?-330?). In this, Cuomo hints at disputing the role of the Neo-Platonic synthesis. Proclus, as the heir to Plato's academy, plays a pivotal role in this. Cuomo seeks to uncover the real mathematician hidden by Proclus and later Neo-Platonic Christians.

If this is really what she hints at, I would be surprised. I am just grasping at straws... The unfortunate fate of the interested reader. ... Read more

Book Description
Reprint of 1910 Cambridge University Press, Second edition. Cloth, [vi] 387 pp. With a Supplement Containing an Account of Fermat's Theorems and Problems Connected with Diophantine Analysis and Some Solutions of Diophantine. ... Read more

Book Description
1889. Contents: Thales; Pythagoras and His School; The Geometers of the Fifth Century B.C.-Hippocrates of Chios: Democritus; Archytas; Eudoxus; The Successors of Eudoxus-I. Menaechmus; The Successors of Eudoxus-II. Deinostratus; The Successors of Eudoxus-III. Aristaeus; and Theaetetus. ... Read more