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This Elibron Classics book is a facsimile reprint of a 1921 edition by the Clarendon Press, Oxford. ... Read more

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more than just history
It should be noted that this is one of a two volume set. This author also compiled and commented upon The Elements ofEuclid in three volumes [also available here].

Theseworkswere first brought to my attention by my Greeklanguage professor nearly 40 years ago as the best English language source on Greek Mathematics.

Just as the Greeks did not view `pure' mathematics or geometry as a lifes-work so to younger readers [through collage] the methods of logic may prove most useful.

For we retired `geezers' not quite ready for Oprah reruns and made for T.V. `romances' it may be the stimulation ofthe brain by the problems [which are documented and solved infull], the history andthe `awe' of how much these did `without computers';

Academically great
This is not a terribly exciting book to read, but it is a superior reference for looking up Greek mathematicians.It is apparent that the author is partial to Euclid, as his section is close to a third of the book, (see the author's version of the Elements)but being a Euclid fan myself I can forgive this easily.Even the most obscure mathematicians are covered in good detail along with what they proved, as well as how they proved it.For those interested in historical mathematics, this book is invaluable. Note:This is a two volume set.I thought it was only one and I only purchased the second.Be sure to get both. ... Read more

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This Elibron Classics book is a facsimile reprint of a 1921 edition by the Clarendon Press, Oxford. ... Read more

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Among other things, Aaboe shows us how the Babylonians did calculations, how Euclid proved that there are infinitely many primes, how Ptolemy constructed a trigonometric table in his Almagest, and how Archimedes trisected the angle. Some of the topics may be familiar to the reader while the others will seem surprising or be new. By treating episodes, Aaboe is able to give the reader the details of representative pieces of ancient mathematics, bringing clarity and dispelling such myths as the assertion that the greeks allowed only ruler and compass in constructions. ... Read more

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A Fascinating Look at the Early History of Mathematics
It is very rare for any intellectual discipline today to be built on the foundation that is thousands of years old. The discipline for which this observation holds most unequivocally is mathematics: the discoveries and tools that have been created well over two thousand years ago are still as valid and relevant today as they were when they first appeared.

This book begins with the Babylonian mathematics and explores their use of the number system which had number sixty as its base. The author uses images of the original cuneiform clay tablet and through a series of intuitive steps shows how we can deduce what their number system looked like and how arithmetic operations were carried out. It is interesting to see how to do arithmetic in the base sixty in its own right, since it is not a number system that is used often. Nonetheless, the Babylonian number system is the source of our own way of dividing time and measuring angles in terms of minutes and seconds, and the book makes a persuasive case that this is actually a very compact way of writing down very small numbers and working with them efficiently. Unfortunately, after some interesting early developments Babylonian mathematics did not progress too far and remained on a relatively rudimentary level.

The bulk of the book deals with Greek mathematics. This is really where the story of mathematics as we understand it today begins, and Greeks already showed a remarkable level of mathematical sophistication. The author presents a few of the most important discoveries of Greek mathematics, primarily in geometry, although Greeks did make many other important contributions. Several important theorems are worked out following the original presentation as much as possible. Nonetheless many concessions were necessary in order to make the text legible for the modern reader.

One of the beast features of this book is that it's not just a description of ancient mathematics - there are numerous exercises throughout the text that aim to engage the reader and draw him or her in into the actual mathematical practice. It is quite remarkable in a way to be having the same thought processes that Euclid or Pythagoras might have been having all those centuries ago. In this limited sense we are able to achieve a sort of union of minds that is hard to imagine in any other sphere of human endeavor.

Early and timeless beauty in mathematics
While mathematics has a long history, in many ways it was not until the publication of Euclid's Elements that it became an abstract science. Babylonian mathematics, the topic of the first chapter, largely dealt with counting and the focus in this book is on the notations the Babylonians used to represent numbers, both integers and fractions. Although their notation had its' limits, we still use it today for time and angle measure.
And then there was Euclid, and all was ordered. There is no reason to believe one way or another that Euclid was the first to prove the theorems in his classic work, but there is no doubt as to his organizational genius. His "rigorous" setting down of the principles of geometric thought was truly a turning point in abstract mathematics, If you are not impressed when reading the material of the second chapter, taken from Euclid, then you have no aesthetic appreciation for what mathematics is. While the mathematics has been cleaned, the beauty has never been topped.
The next chapter is about the greatest genius before Newton, Archimedes. In fact, had he been blessed with better notation, it is possible that he would have invented, or at least pre-invented calculus. If even half of the legends about his mechanical skill are true, they are still amazing. Apparently, entire armies and navies were terrified at the rumor that one of his mechanical devices was about to be used. The crispness of his theorems and the logical progression will be just as instructive thousands of years from now.
The final chapter describes how Ptolemy was able to construct trigonometric tables. Using the chords of circles, he was able to construct tables that can still be used today. Civilization improves and mathematicians continue to expand the mathematical field and refine earlier work. However, the elegance of earlier work still shines through, and in this book you can experience some of the earliest mathematical diamonds, hewn from thought and destined to survive as long as humans do. ... Read more

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Excellent reference
This book is a slightly condensed version of Sir Heath's two volume History of Greek Mathematics (originally published in 1922).In the preface he notes that the two volume set was primarily geared towards classical scholars whereas this set is essentially the same material condensed for the general reader.

I have not seen a copy of History Vol 2 but from looking at the first volume it seems that mostly what was removed is historical recount and interpretation (ie how one mathematician interpreted the work of another).While that commentary is interesting in some cases, all of the graphs and equations contained within the first volume are reproduced here so I would generally recommend this over the two volume set.

And the material that is here is very high quality and comprehensive in scope, covering everything from the pre-Pythagorean era to Plato and Euclid and beyond to Diophantus and algebra.There is also a decent amount of material on Egyptian and Arabic mathematicians in antiquity.

Highly recommended. ... Read more

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A sequel to Unexpected Links Between Egyptian and Babylonian Mathematics (World Scientific, 2005), this book is based on the author s intensive and ground breaking studies of the long history of Mesopotamian mathematics, from the late 4th to the late 1st millennium BC. It is argued in the book that several of the most famous Greek mathematicians appear to have been familiar with various aspects of Babylonianmetric algebra,a convenient name for an elaborate combination of geometry, metrology, and quadratic equations that is known from both Babylonian and pre-Babylonian mathematical clay tablets.The book s use ofmetric algebra diagramsin the Babylonian style, where the side lengths and areas of geometric figures are explicitly indicated, instead of wholly abstractlettered diagramsin the Greek style, is essential for an improved understanding of many interesting propositions and constructions in Greek mathematical works. The author s comparisons with Babylonian mathematics also lead to new answers to some important open questions in the history of Greek mathematics. ... Read more

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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

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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

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Not light weight
The scholar or academic may find this book quite engaging -- I'm just an amateur mathematician with an added mild interest in the history.For me, it's rather pedantic (over my pay grade as it were).

The Inner Workings of the Modern "Abstraction" of Experience from Nature
While Klein's LECTURES & ESSAYS contains accounts of the distinctive features of the modern world (from modern science to the rise of capitalism) that are likely to appear as more "reader-friendly" than the ones contained in this volume on Mathematical Thought, here Klein dives more thoroughly than ever into the inner workings of the "abstraction" of the modern Cartesian-like Life-World of experience (what Husserl calls Lebenswelt) out of the World confronted by pre-modern men.The abstraction in question proceeds through a gradual uprooting of measure (as the Greek, arithmos) from its metaphysical basis, and an "analytical" re-grounding of the uprooted measure in our experience (matter).

What pre-modern thinkers had intuited as the meta/pre/supra-empirical ground of reasoning/measuring, modern thinkers qua modern leave behind in the name of a faculty of reasoning/measuring positing itself (de facto) autonomously of any purely intelligible Being, until the "function" of measuring comes to replace "symbolically" or "in symbolic forms" (i.e., in the imagination and its certainties) its original substance (Being/Nature) as the formal "foundation" of our experience: "logical" certainty is elevated (through a new "constructive" imagination) to the status of truth proper--Truth is sought as the Logic of experience (historicism).

Opening with an extended, precise examination of the pre-modern philosophical understanding of measure, Klein's volume on Mathematical Thought proceeds to explore the inner life of the writings of early modern thinkers with whom the general orientation of pre-modern philosophical reason (distinguished from the practical reasoning of merchants) is rejected in favor or what will come to be known as "instrumental reason".The modern "ordering" of our experience according to the "laws" of a new and exact science of Logic, brings about the eclipse of a pre-modern philosophical ordering of experience according to experience's own natural principles of constitution.Now "abstract" and "exact" laws want to replace Nature as the authentic basis of practical life and judgment.What is called "reason" is no longer maieutic or Socratic, but foundational or dogmatic.

We are here at the dawn of the modern goddess Reason, wherein reason(ing) is posited nominally or formally (and thus also authoritatively), as the foundation of the constitution of our experience.As an authority, the new goddess determines--through the very "logic" of its self-determination--the MORAL (or practical) place of all "events" in our experience; it does not discover the natural or original place of things within a Kosmos transcending the domains of any and all experience.With the new Reason is thus born a new Morality.The modern world is the brave inheritor of both.

Related Readings: Leo Strauss, "Epilogue"; Edmund Husserl, The Crisis of European Sciences; Jerrold J. Katz, Realistic Rationalism.

On 'arithmos' and 'general magnitude'
It's hard to say something about this wonderful book without sounding pompous.Generally, I try to avoid terms like 'classic' and 'essential', but they keep coming to mind.

The original was written in the mid 1930s.As Klein writes in this version's preface, "This study was originally written and published in Germany during rather turbulent times."

The late Jacob Klein spent his post war years teaching Platonic philosophy at St. John's College.There, he was known as something of a lovable elitist.Professors tell a story about Klein being partial to the number 12.He claimed that there were an exclusive 12 philosophers, 7 Greek and 5 German.The word got out and he soon received a letter from 4,000 American philosophers begging to differ with his opinion.

While many might call this book 'philosophy of math,' I doubt Dr. Klein would agree.The book is without much in the way of serious math.It is more concerned with the symbols of math and how they are used.Quoting from the first paragraph of the introduction:

"Creation of a formal mathematical language was of decisive significance for the constitution of modern mathematical physics. If the mathematical presentation is regarded as a mere device, preferred only because the insights of natural science can be expressed by "symbols" in the simplest and most exact manner possible, the meaning of the symbolism as well as of the special methods of the physical disciplines in general will be misunderstood. True, in the seventeenth and eighteenth century it was still possible to' express and communicate discoveries concerning the "natural" relations of objects in non mathematical terms, yet even then -or, rather, particularly then - it was precisely the mathematical form, the mos geometricus, which secured their dependability and trustworthiness. After three centuries of intensive development, it has finally become impossible to separate the content of mathematical physics from its form. The fact that elementary presentations of physical science which are to a certain degree nonmathematical and appear quite free of presuppositions in their derivations of fundamental concepts (having recourse, throughout, to immediate "Intuition") are still in vogue should not deceive us about the fact that it is impossible, and has always been impossible, to grasp the meaning of what we nowadays call physics independently of its mathematical form. Thence arise the insurmountable difficulties in which discussions of modern physical theories become entangled as soon as physicist or nonphysicists attempt to disregard the mathematical apparatus and to present the results of scientific research in popular form. The intimate connection of the formal mathematical language with the content of mathematical physics stems from the special kind of conceptualization which is a concomitant of modern science and which was of fundamental importance in its formation."

While this iconoclastic promise is a bit difficult to extract from the somewhat professional philosophic prose, there is a wonderful essay in "Biographies of Scientific Objects," edited by Lorraine Daston that serves as an excellent commentary.The essay called "Mathematical Entities in Scientific Discourse" credits Klein with a new perspective from which to interpret the transition of ancient and medieval traditions to the new mathematical physics of the seventeenth century. His was the seemingly narrow-but only deceptively so-perspective of the ancient concept of "arithmos", compared to the concept of number in its modern, symbolic sense. In Klein's own words, the underlying thematics of the book never loses sight of the "general transformation, closely connected with the symbolic understanding of number, of the scientific consciousness of later centuries."

Although the Greek conceptualization of mathematical objects was indeed based upon the notion of arithmos, this notion should not be thought of as a concept of "general magnitude." It never means anything other than "a definite number of definite objects," or an "assemblage of things counted". Likewise, geometric figures and curves, commensurable and incommensurable magnitudes, ratios, have their own special ontology which directs mathematical inquiry and its methods.

In contradistinction to Greek parlance, "general magnitude," according to Klein, is clearly a modern concept.Proving this case is the project of both books.

I think you will find reading this material an interesting journey.

Klein's work is a masterpiece of philosophical exegesis.
Klein's work examines the generally unsuspected foundations of modern algebraic mathematics.He charts the development of a new kind of intentionality which lies at the heart of modern mathematical practice, with an explicit affirmation that this mode of intentionality is exemplary for all of modern thought.Beginning from the classical foundations of mathematics, he follows the subject carefully through every turn of ideation until he has completed his thesis. On the basis of this thorough-going evaluation and exegesis of mathematical thought, he identifies Francois Viete as the true founder of this modern symbolic intentionality.But he does not rest with this, proceeding to show how Descartes, Stevin, and Wallis each draw out of this foundation conclusions which are familiar to the modern thinker.This reader knows of no other work of this kind that has so deeply penetrated the foundations of what we call modernity. ... Read more

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The beginnings of Greekmathematics
Thebookwasveryinteresting, It showsmethe new aspect of Greek mathematics.
Itisrecommendedif you like to know the depth of Greekmathematics. ... Read more

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This book provides a way to understand a momentous development in human intellectual history: the phenomenon of deductive argument in classical Greek mathematics. The argument rests on a close description of the practices of Greek mathematics, principally the use of lettered diagrams and the regulated, formulaic use of language. ... Read more

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James Gow's A Short History of Greek Mathematics (1884) provided the first full account of the subject available in English, and it today remains a clear and thorough guide to early arithmetic and geometry. Beginning with the origins of the numerical system and proceeding through the theorems of Pythagoras, Euclid, Archimedes and many others, the Short History offers in-depth analysis and useful translations of individual texts as well as a broad historical overview of the development of mathematics. Parts I and II concern Greek arithmetic, including the origin of alphabetic numerals and the nomenclature for operations; Part III constitutes a complete history of Greek geometry, from its earliest precursors in Egypt and Babylon through to the innovations of the Ionic, Sophistic, and Academic schools and their followers. Particular attention is given to Pythagorus, Euclid, Archimedes, and Ptolemy, but a host of lesser-known thinkers receive deserved attention as well. ... Read more

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A History of Mathematics: From Mesopotamia to Modernity 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-Khwasizmi, 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

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Shockingly remarkable
Although the chapter topics follow the current model of history of mathematics text books (compare the table of contents Victor J. Katz's history of mathematics; notably similar), the text has a strength, depth, and honesty found all too seldom in a text book mathematical history.This is not the typical text-book on technical history that can be dismissed (as Victor J. Katz's should be) as "a pack of lies" with only "slight exageration" (to quote William Berkson's Fields of Force).

Also, the text is bold enough to quote and translate the actual and typical style of presentation used in Bourbaki meetings: "tu es demembere foutu Bourbaki" ("you are dismmembered [..]) [a telegram sent by Bourbaki group to Cartan, informing him that his book was accepted and would be published].Luke Hodgkin's text dispenses with the asterisk (see p.241).

A History of Mathematics
A slightly more descriptive title for this book would be On the History of Mathematics, because the book is not a chronology and detailed narrative of the development of mathematics over the course of human history, but rather a careful, questioning look at selected past moments in mathematics. It does not attempt to tell a comprehensive story of its subject, and in fact ponders at times how such a story should be told. The writing style is polished and reflective. The author often compares the methods, notation, meanings, and possible intentions of earlier mathematicians to those of our own, and contemplates what the differences might imply for our understanding of the texts. The book is a scholarly, thoughtful overview, and would work well as an introductory supplement to more comprehensive general histories of mathematics.

Hodgkin refers often throughout the text to Fauvel and Gray's The History of Mathematics: A Reader.

Brief Contents
Introduction
1.Babylonian mathematics
2.Greeks and 'origins'
3.Greeks, practical and theoretical
4.Chinese mathematics
5.Islam, neglect and discovery
6.Understanding the scientific revolution
7.The calculus
8.Geometry and space
9.Modernity and its anxieties
10.A chaotic end?
Conclusion
Bibliography
Index

"We have not, unfortunately, resisted the temptation to cover too wide a sweep, from Babylon in 2000 BCE to Princeton 10 years ago. We have, however, selected, leaving out (for example) Egypt, the Indian contribution aside from Kerala, and most of the European eighteenth and nineteenth centuries. Sometimes a chapter focuses on a culture, sometimes on a historical period, sometimes (the calculus) on a specific event or turning-point. At each stage our concern will be to raise questions, to consider how the various authorities address them, perhaps to give an opinion of our own, and certainly to prompt you for one.

"Accordingly, the emphasis falls sometimes on history itself, and sometimes on historiography: the study of what historians are doing." (4)

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

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Isaac Barrow is largely responsible for that preservation and promulgation of the Euclidean tradition which invigorated the physical science and mathematics of Newton and others, and allowed for an ongoing engagement with classical Greek mathematics. Barrow's philosophy of mathematics remains relevant to many key issues still at the forefront of modern philosophies of mathematics. The tradition of mathematics as a liberal discipline, which is the main topic of this book, reaches back to the educational ideals of Plato and Aristotle, and was brought to an early state of near perfection in the Euclidean school of Greek geometry. Within this framework mathematics is practiced as the means of training and cultivating our formal powers: the intellect and imagination function together to achieve a high degree of familiarity with the geometrical structures, patterns and relations that we find embedded in the world around us. The ultimate goal is liberation of the spirit from exclusive preoccupation with mundane concerns.However, we remain at the same time anchored to a realistic philosophical conception of the world, in which our mathematical notions arise by abstraction from real, objective properties of things outside ourselves. This practice of mathematics as a liberal discipline is exemplified nowhere more completely than in the work of Isaac Barrow. The interplay between these modern philosophies and liberal educational ideals forms a recurrent theme of this book. Separate chapters are devoted to Barrow's development as a translator of Greek mathematics and as a mathematician in his own right; to his contributions to the birth of the infinitesimal calculus; to his conception of geometry as the most fundamental and realistic form of mathematics; to his treatment of physical space and time; and to the philosophical and theological implications of the constructive ideal which lies at the heart of classical geometry. ... Read more

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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.

The audience of the book includes scholars from history and philosophy of mathematics and mathematical sciences, scholars from history of science, students in the field of history of mathematics and history of sciences.

Product Description
This is an EXACT reproduction of a book published before 1923. This IS NOT an OCR'd book with strange characters, introduced typographical errors, and jumbled words.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more