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A Concise History of Mathematics

This is little more than a sketch of the history of mathematics. In broad strokes, it outlines the relationships among mathematicians and some of their texts, but says almost nothing of substance about the content of their work. Each chapter has a short bibliography, but since the last edition was in 1987 (1st ed. 1948), they are of limited value.

Mathematics Hystorical Development

When I was student ( from elementary school to university level ), I was never taught of history of mathematics ( at least in an organized and formalized way,sometimes there were references to singular and surprising anecdote as Gauss child summing up the first 100 natural numbers in few seconds). I think this is a lack ( probably it is not possible to teach everything in the already rich school program ) of the educational system. For this reason , enjoying mathematics ( especially the simplest one ), I was looking for something to explain and describe how the mathematical thinking developped with time. At the same time, I did not have much time, so I thought from the title that this book was the right one for my needs. Although all the history of mathematics from its dawn to about 1950and written in a short book is something very difficult to realize ( if not impossible ), I very reccomend this beautiful book, where the history of mathematics is well explained in terms of main thinking lines. Furthermore this book is rich of interesting anecdotes and details on personal and accademic relations among the greatest mathematicians of all time.

A Brief Outline of the History of Mathematics
In "A Concise History of Mathematics," Dirk J. Struik succinctly surveys the progression of mathematics: its discoveries, breakthroughs, and distinct personalities. If you enjoy the history of mathematics or if you dislike math (it will help infuse a sense of delight in math) you will find this book very useful. This little volume doesn't cover the subject or any prominent feature of mathematics in exhaustive detail, but it is a precise book with research touching on the foremost pioneers, originators, and timelines caught up in the development of mathematics from their earliest genesis until the start of the last century.

Chapters include:

- The beginnings
- The ancient orient
- Greece
- The beginnings of Western Europe
- 17th century
- 18th century
- 19th century
- First half of 20th century
- And more.

Simple to read and it will help elicit more affection for mathematics as it enlightens the reader in a historical outline regarding this essential subject.

Within this outstanding book is:

- A fine bibliography
- Numerous references (English and various languages).
- A few nice illustrations and pictures
- Many of the most significant details of the history of the mathematics.

Frege, Russell and other important figures are only mentioned in passing. Thus I prefer James Nickel's exhaustive and compelling volume on mathematics. I delight in mathematical truths because they reflect theism as the epistemic foundation. Without the infinite ontic ground of theism and without affirming a wellspring who is the foundation for infinite numbers, the non-theist cannot solve the paradox of infinite immaterial numbers within a finite material world. This is not a problem for the person who ascribes a theistic epistemic source whereas he believes in an infinite, immaterial, and eternal being as the infinite source of mathematics. Non-theists use infinite numbers, yet these numbers have no end, hence they do not comport with their worldview. One must presuppose theism to account for infinite numbers.

"A Concise History of Mathematics" is a marvelous fact-filled book, written in a lucid and crisp style.
The Necessary Existence of God: The Proof of Christianity Through Presuppositional Apologetics

Good infromation - Bad english
I bought the book for a class on history of math, and it covered that topic well.The issue I had was the usage of words only a PHD in English would use. The book lead me to the Internet to find more information about the period that was easier to understand.

Brief History of Mathematics,perhaps too concise.
I rather enjoyed reading this Dover classic on mathematics.Yet,there were some problems with the book.It's a vast topic,that's condensed into a rather small book.The author,Dirk J. Struik,did a wonderful job explaining the great giants of global historical mathematics.Now,if you're looking for the math geniuses of the twentith century,however,you'll miss them here.There is an excellent bibliograghy and many important basic formulas presented.The sketch portraits and rare photograghs add an insightful picture of the character of these esteemed numerologists.After reading this text,I found myself wanting to research more about their contributions to mathematics.Eventhough,i may be more advanced at understanding higher math concepts.This book presents a clear philosophical foundation,often skipped by math instructors today,because of time considerations. So, this book is the ideal text for any fellow math neophyte,who may cringe at the broad scope of classical mathematics. ... Read more

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Excellent brief introduction presents fundamental theory of curves and surfaces and applies them to a number of examples. Topics include curves, theory of surfaces, fundamental equations, geometry on a surface, envelopes, conformal mapping, minimal surfaces, more. Well-illustrated, with abundant problems and solutions. Bibliography.
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Customer Reviews (5)

The most consistent reliableand readable so far
With this book, I hope I have finally broken the code and reached a critical mass in advanced mathematical understanding. These Dover Series books allow "it all to hang out." It is "old school" in the best sense of that phrase: that is, in the sense that they do no "sugar coat" their explanations. They do not "dumb it down" or "fancy it up to" ease the pain. One knows what one is up against when one picks up a book from the "Dover Series." They are always clean and sparse in their explanations.

In this regard, this book is no exception. Professor Struik begins at the beginning and goes straight through to the end without skipping any steps and without passing go to collect his $200. He gives the fundamental conceptions of the theory of curves and surfaces, introducing all of the machinery necessary to understand them in a graduated fashion suitable only to the requirements of the topic itself. Elementary calculus will serve the reader well, especially with a smattering of Linear Algebra thrown in. The author wastes no time with sexy side issues or superfluous explanations: Just the basic facts of the fundamental elements here. Those looking for more advanced topics, should consult those books that use this one as their background.

Explanations are sparse, but never deficient; the same is true of the equations. Notation is straightforward and always clear and economical. It is easy to see that(and why) other books on the same topic have used this one as background, but oddly, those other books have been unable to improve upon this one. Other than the fact that the graphics need updating, and more modern topics are missing, this is a splendid effort. Just what I needed.

Five Stars.

Very Readable Work on Classical Differential Geometry
While it is quite true Dirk Struik's work is on classical differential geometry, the older methods and treatment do not necesarily imply obsolescence or mediocrity as some readers or reviewers suggest in their evaluations.Classical Analysis is still an important branch of Mathematical Analysis.So classical approaches and topics should not be dismissed as a waste of time, useless, outdated or even invalid.Remember Andrew Wiles' recent attack on Fermat's Last Theorem and his ultimate proof of its validity, an event that made headline news.That is a quintessential classical problem in mathematics (i.e., in number theory), only recently resolved.So remember: CLASSICAL Differential Geometry is part of the title.

First of all, this book is very readable, being that it requires no more than 2 years of calculus (with analytic geometry and vector analysis) and linear algebra as prerequisites.Exposure to elementary ordinary and partial differential equations and calculus of variations are highly desirable, but not absolutely necessary.There are numerous carefully drawn diagrams of geometric figures incorporated throughout the book for illustration and, of course, better understanding.Topological methods are not used in the book, and the concept of manifolds not mentioned, much less treated.So this is an older work that bridges the very foundational and applied aspects of differential geometry with vector analysis, a field and body of knowledge widely used nowadays in the sciences and engineering and exploited in applications such as geodesy.For those insisting on modern approaches and want to omit studying foundations and historical development, please read up on other books such as O'Neill and Spivak.(Also, there are tons of other newer works, i.e., on "modern differential geometry", I am unfamiliar with.They are probably availble for browsing in college bookstores.)

The author begins by leading the reader from analytic geometry in 3-dimensions into theory of surfaces, done the old fashion or classical way, i.e., utilizing vector calculus and not much more.Along the way, he takes the reader through subjects such as Euler's theorem, Dupin's indicatrix and various methods for surfaces.Then he continues with developing important fundamental equations underlying surfaces, e.g., Gauss-Weingarten equations, looks at Gauss and Codazzi equations, and proceeds to geodesics and variational methods.He includes a somewhat detailed treatment of the Gauss-Bonnet theorem as he progresses.He ends up with introducing concepts in conformal mapping, which plays an important role in differential geometry, minimal surfaces and various applications, one of which is geodesic mapping useful in geodesy, surveys and map-making.He does all of it with clarity and focus, including problems or "exercises" as he calls it, in under 240 pages - brevity that is rare in many mathematical books and works these days.

For those with a mind for or bent on applications, e.g., applied physics (geophysics), applied mathematics, astronomy, geodesy and aerospace engineering, this book would be an excellent introduction to differential geometry and the classical theories of surfaces - being that one need not worry about abstract analysis and topological aspects of mathematics.Perhaps the title should be "Topics in Classical Differential Geometry" or "Introduction to the Theory of Surfaces in Classical Differential Geometry".But one must keep in mind that Dirk Struik is an old MIT hand and contemporary of Norbert Wiener, also at MIT, and Richard Courant (and many great German-educated mathematicians) who lived and worked in the early to mid-20th century, a long time ago and before computers became commonplace, an era in which total abstraction in mathematics and physics was not quite widely emphasized, but clear concrete thinking was important.A good friend of mine and co-worker who studied at the University of California, Berkeley, told me he had great respect for the classical geometers such as Struik and Eisenhart, understanding that they built ideas from a scatch and wrote in such a way that readers can discern the physical origins of geometry, in particular of differential geometry, a subject that supposedly started with Gauss during the early or mid-19th century when he performed survey work for his government in Germany.(The term "torsion" introduced and sed by Struik in the first few chapters of the book comes from classical mechanics, and is commonly employed in mechanical structures/structural engineering nowadays.)

I for one am an aerospace engineer.There were one or more occasions where I consulted the book for formulas and expressions of curved surfaces and spheroids in my work of flight navigation (flying over the ellipsoidal Earth, as one example).I am sure that are other areas, e.g., space engineering, where classical methods of differential geometry embodied in Struik's book can come in handy.

The only problem I have with the book is that the "exercises" do not come with solutions, but I do not think that is a major drawback unless one uses it as a textbook for a course that requires assignments and drill exercises.

Judge for yourself by borrowing this book to read, i.e., if you are interested, can tell whether you like or dislike it on the first pass, and for what reasons one way or another.Find out for yourself.

classical
This is a survey of classical i.e. early 20th century differential geometry and not a more "modern" abstract treatment.

Good treatment of classical differential geometry
Struik's book provides solid coverage of curve and surface theory from the classical point of view, i.e. the kind of stuff Monge, Serret, Frenet and Gauss did. I agree that the book should be on the shelves of mathematicians. A number of classical topics are simply not in vogue these days, and one can find them discussed at length in Struik, or in the exercises. In this sense the book certainly has a more geometric flavor than a number of contemporary texts.

However, Struik can't be used to understand what is happening today. For these purposes,books by O'Neill and do Carmo would be more appropriate. The discussion of manifolds and coordinate charts, the discussion of connection forms, differential forms, covariant derivatives, exterior derivatives, pullbacks and pushforwards can be found in these texts. This is the language of modern geometry.It leads on naturally to tensors, fibre bundles, de Rham cohomology and so on and so forth.The emphasis in modern geometry is on global phenomena, the interaction between local and global (e.g. Morse theory or De Rham cohomology), and the attempt to do everything in an algebraic setting (projective modules, spectral sequences, categories etc.) For this purpose, Struik is useless, though he does have some coverage of forms (he calls them by their earlier name of 'pfaffians').

The price of the book makes it an attractive purchase.

Struik's book - a classic on classical differential geometry
I simply cannot believe I am the first reviewer of this book!This book should be on the shelf of every mathematician interested in geometry, every computer graphics specialist, everyone interested in solid modelling.Forten bucks, you get a great summary of a wide range of topics in"classical differential geometry" -- the stuff geometers wereinterested in one hundred years ago.Today it's gauge and string theory --but the topics discussed in this book are timeless, and many have seenremarkable renaissances in recent years. It is a wonderful little book ...I am using it to teach a basic differential geometry course next year. ... Read more

Customer Reviews (1)

Valuable
This is an excellent source book. One of the many benefits of reading original sources is that it frees us from the often dogmatic single-mindedness of modern textbooks. I would like to illustrate this by looking at some excerpts from this book on the topic of power series. The modern approach to power series representations of functions is of course to find the coefficients by repeated differentiation. However, history teaches us that this approach is backwards and obscures some important insights.

The first published derivation of the so-called general Maclaurin series (by Taylor in 1715, excerpted here on pp. 329-333) was based on entirely different ideas than that of repeated differentiation, namely Newton's forward-difference formula. It may be summarised as follows. An infinite polynomial A+Bx+Cx^2+Dx^3+... has infinite degrees of freedom. Therefore we expect to be able to construct an infinite polynomial passing through an infinite number of given points, just as a parabola of the form y=Ax^2+Bx+C can be constructed going through essentially any three points, but not any four, owing to its having three coefficients. Newton's forward-difference formula constructs such a polynomial, namely a polynomial which takes the same values as a given function at the x-values 0,b,2b,3b,.... Taylor's derivation of his series consists in letting b go to zero is this formula. The nowadays more popular method of finding the series by repeated differentiation was not published until decades later by Maclaurin in 1742 (pp. 338-340). Thus history alerts us to the fact that the blind-computation approach favoured today robs us of an opportunity to "see" the infinitely many degrees of freedom of a power series in an illuminating way that is based on open-minded reasoning.

A related lesson from history concerns the binomial series. Today it is popular to derive the general binomial series by finding its Maclaurin series through repeated differentiation. Particular binomial expansions are then obtained by plugging numbers into this general series. Again this textbook approach based on blind computations has a much more vivid and interesting counterpart in history. When we read Newton we realise that the binomial series is not a "theorem" to be "proved." Although we often say that we "use a binomial expansion" to find some integral or other, this is really just a time-saving device, not a fundamental and substantial reliance on some profound theorem in all its generality. The classical applications of binomial series expansions do not by any means require that we "prove" the binomial series in general. When Newton wants to find the power series for sqrt(1+x^2) he does not say "I apply the binomial theorem with the exponent 1/2." Rather he simply says "I extract the root." Indeed, it is very straightforward to "extract the root" without knowing anything about the binomial series: what we want is a power series such that
sqrt(1+x^2)=A+Bx+Cx^2+Dx^3+...
or in other words,
1+x^2=(A+Bx+Cx^2+Dx^3+...)(A+Bx+Cx^2+Dx^3+...).
Very well, we simply multiply out the right hand side and identify coefficients with the left hand side, and there's our root. The benefit of calling this a binomial expansion is that we can shortcut out algebraic labours by using the quick and easy way of thinking about the binomial series by analogy with the integer-exponent case. We can do so with good conscience knowing that we could always fall back on the direct algebraic way to "extract the root" if pressed for justification. Here are Newton's own words:

"Fractions are reduced to infinite series by division; and radical quantities by extraction of the roots, by carrying out those operations in the symbols just as they are commonly carried out in decimal numbers. These are the foundations of these reductions: but extractions of roots are shortened by this theorem [the binomial theorem]." (pp. 285-286) ... Read more

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Thoughtful, readable survey explores "flowering" of science in New England from Colonial times to the Civil War. Describes contributions of scientists Benjamin Franklin and Eli Whitney, engineers George Washington Whistler and Cyrus Field, naturalists Gray, Agassiz and Dana; medical accomplishments of Holmes, Morton and Jarvis, beginnings of Darwinism, much more.
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Technological history at its best
In this excellent book, Dirk Struick strikes an admirable balance among technological history, brief biographies, and social history.He clearly establishes the context for each period and type of invention, reminding us that there is nothing automatic about scientific and technological advance. Economic interests, social hierarchies, and religion all influence the way such developments occur. Unlike many histories of technology, this one is a pleasure to read. ... Read more

Product Description
This digital document, covering the life and work of Dirk Jan Struik, is an entry from Contemporary Authors, a reference volume published by Thomson Gale. The length of the entry is 683 words. The page length listed above is based on a typical 300-word page. Although the exact content of each entry from this volume can vary, typical entries include the following information:

• Place and date of birth and death (if deceased)
• Family members
• Education
• Professional associations and honors
• Employment
• Writings, including books and periodicals
• A description of the author's work
• References to further readings about the author

Product Description
Both volumes are well illustrated. Measure 11 X 17 cm. ... Read more