Product Description
This is an OCR edition without illustrations or index. It may have numerous typos or missing text. However, purchasers can download a free scanned copy of the original rare book from GeneralBooksClub.com. You can also preview excerpts from the book there. Purchasers are also entitled to a free trial membership in the General Books Club where they can select from more than a million books without charge. Original Published by: Macmillan in 1919 in 531 pages; Subjects: Mathematics; Education; Mathematics / General; Mathematics / History & Philosophy; ... Read more

Customer Reviews (2)

A History of Mathematics

The Encyclopædia Britannica entry on Florian Cajori says, in part:

Cajori, Florian
born Feb. 28, 1859, St. Aignan, Switz.
died Aug. 14, 1930, Berkeley, Calif., U.S

"Cajori emigrated to the United States in 1875 and taught at Tulane University in New Orleans (1885-88) and at Colorado College (1889-1918), where he also served as dean of the department of engineering (1903-18). In 1918 he became professor of the history of mathematics at the University of California, Berkeley."

Reference: "Cajori, Florian." Encyclopædia Britannica from Encyclopædia Britannica 2006 Ultimate Reference Suite DVD.

The first edition of this History of Mathematics occurred in 1893. Subsequent editions were in 1919, 1980, 1985, and this 5th edition in 1991. Apart from stylistic changes and minor corrections, the main revisions in the 3rd and later editions are said by the editor to be concerned with the eight page chapter on Babylonian mathematics, which was rewritten for the 4th edition and partly rewritten for the 5th. A book of similar title by Cajori, A History of Elementary Mathematics, although based upon this book under review, is a different work.

This is a densely written book, and especially in its later chapters assumes a mathematical background consonant with the period in which it was originally published. This is not a textbook. It is a scholarly discussion on the history of its subject. The historical limit of Cajori's discussion, mentioned in the preface to the 3rd edition and operative still in the current edition, is "the close of World War I." His viewpoint is not that of a modern reader. He is writing during the first blooms of 20th century abstract mathematics. The 1st edition of Whitehead and Russell's Principia Mathematica, which he mentions, occurred over the course of the years 1910 to 1913. The 1st edition of Felix Hausdorff's Grundzüge der Mengenlehre, which he doesn't mention, occurred in 1914. Emmy Noether (1882-1935) didn't arrive at Göttingen until 1915. Bourbaki's first book wasn't published until 1939.

Although Cajori lived until 1930, in revising his book for the 2nd edition (later editions occurred after his death), he could not know where mathematics was going in its methods of abstraction or how educational reforms and popular culture would affect the intellectual background and concerns of future readers, and so he could not write his history to speak to the viewpoint and expectations of a reader in the 21st century. But if you enjoy this sort of thing, this book is worth your attention, regardless of how well you understand everything in it.

The editor of the 3rd edition remarks: "With so vast and complex a subject to set forth within so small a compass, the author of a one-volume history of Mathematics must decide what facts to select, what interrelations to point out, whom to name, how much to explain, and the like. There is no optimum, no one way of doing this. Every one-volume history of mathematics is therefore necessarily very different. Indeed, each of the few that exist are, as the reader can easily verify, strikingly different." (p. iii)

Content may be good but Kindle version is useless
This isn't a proper e-book. It obviously is just a raw OCR scan without any human 'cleaning up'. I feel I should be given a proper copy for my Kindle when it's appropriately formatted. Riddled with misspellings and 'junk'. Especially when dealing with mathematics formatting is key, otherwise the book is unreadable.

Because of this, I am unable to review the actual content of the book. ... Read more

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Customer Reviews (1)

A Reference Book on Ancient Mathematics
The mathematics presented in this book spans the period from ancient times up to 1550. The author discusses the progress in mathematical development, chronologically throughout this period, starting in Egypt and Babylon, then on through Greece, India and Islamic countries, to finally end in Europe. The main focus is the development of trigonometry for use principally in astronomy but also in earth-related matters. Since this is mainly a reference book, the author pulls no mathematical punches. Throughout the book, mathematical discussions are interspersed with very brief historical snippets; these give the subject an always-popular human flavor. In order to provide the reader with a better appreciation for the mathematics of the distant past, the author has retained the techniques used at the time, including the use of the sexagesimal (base 60) number system where applicable. As a result, a reader who has been trained in modern mathematical techniques will need to get used to these ancient ways, especially if his/her objective is to follow, in detail, the mathematical arguments presented. Fortunately, the author provides modern "Explanations" after each ancient digression that has been translated from the ancient texts. The writing style is both authoritative and clear - certainly what one would hope for in any reference/textbook. Those taking university courses on ancient mathematics, as well as the most serious of math buffs, are likely to be the ones who will appreciate this book the most. ... Read more

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Problems that beset Archimedes, Newton, Euler, Cauchy, Gauss, Monge and other greats, ready to challenge today’s would-be problem solvers. Among them: How is a sundial constructed? How can you calculate the logarithm of a given number without the use of logarithm table? No advanced math is required. 100 problems with proofs.
... Read more

Customer Reviews (7)

100 problems
squeezing 100 problems into less than 400 pages is not an easy task. There seems to be consistent tradeoff for brevity at the expense of readability. But if you are fairly experienced in elementary mathematics, most of the stuff can be followed. The degree of difficulty in following problems covered in this book varies greatly. Some can be appreciated by people with middle/high school math backgrounds. Others are very challenging to me (4th undergrad in engineering with good math background relative to undergrad math students). The big advantage of this book is that it deals with widely applicable, and historically significant, and applied math problems. This is in contrast to the bulk of math problem books out there that just deal with recreational problems (puzzle for example. but math olympic problems are recreational too, since the best high school students have to be able to solve them in around an hour). Look at the table of content and you will see. Highly stimulating.

Wonderful book, and a clarification on the term "Elementary"
I love this book, and recommend it very highly, if you're the type who would like to understand, say, why the Fundamental Theorem of Algebra (every polynomial equation as a (possibly complex) root, is true. Yes, it takes intellectual effort to follow the proofs, but that can be incredibly rewarding, once you finally understand.

But this review is mostly to clarify the term "Elementary" in the title. This is used in a technical sense. Many (most?) of the theorems have multiple proofs. And sometimes the clearest proofs involve calculus, and often the calculus of complex variables. But if a proof doesn't involve calculus, then mathematicians refer to these as "elementary". It is in this sense that the title uses the term.

100 Mathematical Triumphs of Genius
100 Great Problems of Elementary Mathematics is such a goldmine of ingenuity that it is hard to comprehend how it could be sold for so low a price. Ten dollars is practically a steal.

This publication, which was translated into English back in 1965, is a concise summary of some of the greatest works of mathematics throughout mankind's history. The problems contained are quite challenging. Many are such that if you understood any one of them, then you would probably know something that even the best math professor nearest you would not. This may sound like an overstatement, but in a day and age where some PhD's in math have either forgotten or never really learned how to determine so little as the square root of a number by just pencil and paper, it is probably not.

It is from analyzing the book's passages of Bernoulli's Power Sum Problem that I was able to achieve a great mathematical triumph after discovering the following challenge found in William Dunham's The Mathematical Universe: determining a precise mathematical formula to figure out how Jakob Bernoulli could take all the positive integers from 1 to 1000, raise each of them to the tenth power, and then add them up to where the sum came up to over 30 digits! I tried to develop algorithms that would work but failed each time, until I, once again, read this volume.

The situations presented are quite difficult to grasp, but once you get to where you know how to apply any one of them in solving mathematical puzzles, you feel elated. I know I did.

For the individual who enjoys looking at mathematics in a historical context and who wants to approach problems that are perhaps not entirely solvable with the use of the calculator and/or the computer, I recommend this book.

best summary of classic problem solutions by masters
Elementary algebra and ingenius ideas are combined to solve some of the most difficult problems in the history of math.This book helped me solve several difficult technical problems .
The concise treatment and cross reference to other solutions is outstanding . This is the finest treatment of advanced mathematical treatments I have ever seen. First published in 1932, it represents the best from the masters and can be used to discover tricks which were helpful to me in algorithm development . The treatment of astronomical problems alone is worth the price .

The best book about elementary problems I have read till now
Perhaps the stress given to geometry gives evidence to the age of the book, but it still represents an example of how a collection of problems should be written. It is too entangled with mathematics to be defined an issue about mathematical games, but also fans of games can find out some enjoying items. Because, if much room has been given to proofs and resolutions, the boundary of elementary curiosity never goes out of sight, even if it can sometimes look like a far horizon. It is surely the best book about elementary problems, mathematical games and jokes I have ever read till now, and I have found its language as clear and straight as a non-English reader (like me) usually finds a non-English writer. ... Read more

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From the blackboard to the graphing calculator, the tools developed to teach mathematics in America have a rich history shaped by educational reform, technological innovation, and spirited entrepreneurship.

In Tools of American Mathematics Teaching, 1800--2000, Peggy Aldrich Kidwell, Amy Ackerberg-Hastings, and David Lindsay Roberts present the first systematic historical study of the objects used in the American mathematics classroom. They discuss broad tools of presentation and pedagogy (not only blackboards and textbooks, but early twentieth-century standardized tests, teaching machines, and the overhead projector), tools for calculation, and tools for representation and measurement. Engaging and accessible, this volume tells the stories of how specific objects such as protractors, geometric models, slide rules, electronic calculators, and computers came to be used in classrooms, and how some disappeared.

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This book is made up of two parts, the first devoted to general, historical and cultural background, and the second to the development of each subdiscipline that together comprise Chinese mathematics. The book is uniquely accessible, both as a topical reference work, and also as an overview that can be read and reread at many levels of sophistication by both sinologists and mathematicians alike.

Customer Reviews (1)

This is the book to read..
for anyone interested in the subject. The author treats both the
native tradition and the reception of Western mathematics with
the highest scholarship. One of the most significant themes is
the conflict between the combinatoric spirit of Chinese mathematics,
akin to modern group theory and algebra, and the synthetic
Euclidean system introduced by European missionaries.
The initial Chinese puzzlement at what seemed an artificially
contrived structure will amuse those who have endured traditional
geometry and those struggling with Chinese itself. ... Read more

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Customer Reviews (2)

Arbitrary collection of some interesting material
Many articles deal with the same issues as Bos's book on geometrical exactness. This includes a nice survey of Descartes's La Geometrie, a curve tracing device designed by Huygens with bearing on the foundations of geometry ("a little cart or boat will serve to square the hyperbola", he said (p. 8), referring to the tractrix), and a discussion of why the Bernoullis, analogously to the Greeks' use of methods beyond ruler and compass, found it appropriate to reduce the construction of transcendental curves to rectification of the lemniscate. Several other articles deal with aspects of calculus in general. These are less satisfactory; Bos has written more interesting texts on these topics elsewhere. A survey of Leibnizian calculus spends most its time making a big deal of fairly technical differences with the modern theory. A discussion of the role of applications in 18th century calculus lines up proof that the calculus was virtually defined through its applications, but warns that these were often applications only in a romantic sense, of little practical value.

Snapshots in the history of mathematics and some philosophy
The history of mathematics is like all other histories of humans. They are characterized by years of steady movement in a certain direction punctuated by occasional bursts of activity that lead to a fundamental change. In politics, the movement can be in several directions, but in mathematics there is the equivalent of thermodynamic entropy. Sir Arthur Eddington called entropy "time's arrow" because you can always use it to determine the movement of time in a system. Mathematics is a system where new discoveries and clarifications are built on the backs of the previous ones. Therefore, if one were to take virtual snapshots of mathematics at any time, it would be fairly easy to determine when the snapshot was taken.
Bos is a professor in the history of mathematics and this book is a collection of some of his most memorable lectures. All of the historical events examined took place before 1900. They are:

*) Huygens, Tractional Motion, and Some Thoughts on the History of Mathematics.
*) The Concept of Construction and the Representation of Curves in Seventeenth-Century Mathematics.
*) The Structure of Descartes's Geometrie.
*) Christiaan Huygens.
*) The Fundamental Concepts of Leibnizian Calculus.
*) The Lemniscate of Bernoulli.
*) Calculus in the Eighteenth Century: The Role of Applications.
*) The Closure Theorem of Poncelet.

The significant points in mathematical history are described using diagrams, formulas and explanations. I learned a great deal about the people and how they approached their mathematics that I didn't already know.
The last three chapters are more philosophical in nature. They are:

*) Elements of Mathematice: They Are No Longer What They Used to Be.
*) "Queen and Servant": The Role Of Mathematics in the Development of the Sciences.
*) Mathematics and Its Social Context: A Dialogue in the Staff Room, with Historical Episodes.

"Elements of Mathematics" is an attempt to create an analogy between the fundamental principles of mathematics and the elements used in the science of chemistry. Like the known elements of chemistry the "elements" of mathematics have grown over time. However, the laws of physics define the limits of the number of elements, whereas the number of mathematical elements is probably infinite.
"Queen and Servant" is derived from the famous book by Eric Temple Bell in which he argues that mathematics is the queen of science. Bell concluded that it provides the foundation upon which all sciences operate. Bos puts forward three ideas:

*) The idea that behind the phenomena of nature there is a deeper reality which is mathematical.
*) The idea that science, in the structure of its theories and the derivation of its results, should take mathematics as an example because the mathematical way of reasoning is the only one that leads to certain knowledge and does so efficiently.
*) The idea that mathematics is the proper language for formulating scientific results, that is, science should search for laws of nature expressed in mathematical form.

"Mathematics and Its Social Context" describes how mathematical truth is independent of the social situation. For example, seven is a prime number in a fascist, communist and democratic society.
These lectures delve into mathematics, in terms of how some of the results were arrived at over time and what the consequences were. Some of them also deal with the role mathematics has in society, the role it plays in social and scientific advancement and how it is an ideal, unsullied by the more variable areas of "truth." I enjoyed the book, the snapshots gave me several new insights into problems that I had encountered before but had not thought of in this way. ... Read more

Product Description
To experience the joy of mathematics is to realize that mathematics is not some isolated subject that has little relationship to the things around us other than to frustrate us with unbalanced checkbooks and complicated computations. Many of the phenomena around us can be described by mathematics. Mathematical concepts are even inherent in the structure of living cells. ... Read more

Customer Reviews (9)

Notice more around you than just what you see
My appreciation for Theoni Pappas is enormous as for an observer and admirer of the world around her and mathematician. These factors cannot be separated, as at first you have to do more than just look around, but you have to have a beautiful mind of a child and be an intellectualist at the same time, not just to take things for granted, but as a child be curious and ask questions and finally as an intellectualistand mathematician find answers to them.
Yet, there is more to it. It is so, as the author popularizes mathematics. She answers the basic questions about role of mathematics in our lives. Most people associate mathematics as calculating especially money, yet in mathematics the theory models or formulaare created, and it occurs that they find application in our material world sometimes evencenturies afterwards. Let us look at some examples in the book "The joy of mathematics": - earthquakes and logarithms- connection lies in the method to calculate earthquakes' magnitudes by means of Richter scale, which is logarithmic, - the catenary & the parabolic curves- who takes as an obvious phenomena- the Golden Gate Bridge in San Francisco- it looks gorgeous, but what it looks like is connected with construction equations, which contribute to the fundamental thing, that it really is invulnerable and cannot be destroyed by the mass itself, as well as additional natural forces. Even Galileo noticed the curve to be parabola, - Thales & the Great Pyramid- Egiptians' calculations of the height of a pyramid were based on shadows and similar triangles, -the Dome of Milan -Gothic plans incorporating the application ofgeometry and symmetry in architecture, and lots of stuff like that. If you like to notice more around you, astound your friends, you should read such books, as there is more beauty around you than what you just see.

engaging
if the discoverable arithmetic of the everyday natural world interests you, try this; and then you may want to explore her other work along this line.

These are vignettes, designed to inspire further exploration
The widely divergent reviews reflect a lack of understanding of the purpose of this book. It is meant to touch on many mathematical ideas, not to go into depth on any one idea. My son read this at age 8, then at 10, and again at 12 - getting something more out of it every time. Many of the ideas intrigued and inspired him to seek out more information on his own, to research and understand more deeply. For that purpose, it deserves the highest rating.

I did not give 5 stars because there are some instances where I did find errors, these do not detract from the purpose of the book, but they are annoying to those of us who try to delve deeper. What I consistently found myself doing is researching from the internet and other print resources. But the idea originated from the overview in the book.

Many recreational mathematics books are inaccessible to beginners or math phobes. This book allows you to sample many, many ideas without feeling overwhelmed by details you may not understand. If you want details, you go explore the world opened up by the book.

Too cursory for much use, very often misleading.
Sorry to say but this book is a dud.While the concept of presenting interesting mathematical facts is great the presentation is so brief, so wrought with errors, and so incomplete that the work is not worth perusing.

Some of the "chapters" have answers at the back of the book and some do not.It appears that the author could not make up her mind wether this was to be a "math tricks" book or a "popular mathematics" presentation substantiated by theory.

There are many other excellent books that are more fulfilling.Journey Through Genius comes to mind.

All in all a disappointing work.

A pathetic little book that could have been good
This book could have been good if the author had done a careful job of writing the text, and perhaps if the illustrations were original, and above all if the author had understood the material she was writing about.Sadly these are often not the case with this book.

Rather, this book gives every sign of being essentially copied from bits of many dozens of other books.All the illustrations appear to be low-quality xerographic copies from other books (clearly used without any permissions).

But worst of all, the book is chock full of misstatements,misconceptions, and sentences that don't convey any meaning.

This book gives the non-expert reader the impression that he or she is learning something, but a great deal of the time this is just the illusion of learning.

I will list a few of the errors and illusory learning that I can readily find:
________
p. 6:The illustration of the cycloid curve should show it to be in a vertical direction where one arch meets another; instead it is at 45 degrees to the vertical.
________
p. 7:It is stated that when marbles are released in a cycloid-shaped container, they will reach the bottom at the same time.This phenomenon occurs for a bowl whose cross-section is an *inverted* cycloid, but that is omitted.
________
p. 13:Both the "impossible tribar" and "Hyzer's optical illusion" are NOT mathematically impossible, contrary to what is written.(They can be constructed in 3 dimensions.)Twistors are mentioned but not defined, even in a rough, metaphoric way -- just not at all.
________
p. 18:It is mentioned that pi cannot be the solution of an algebraic equation with integral coefficients, but there is no discussion in the book of what such an equation is.
__________
p. 19:Also, it is stated that the probability of two randomly chosen integers' being relatively prime is 6/pi.Not only should the correct number be 6/(pi * pi), but the idea of randomly choosing an integer is left completely undiscussed, although there is no known way to do this.
________
p. 38:The Platonic solids (aka regular polyhedra) are discussed here, but although they are defined twice, neither definition is correct.(The author neglects to mention that the faces of such a solid must be *regular* polygons.)
________
p. 45:The Klein bottle is discussed and illustrated here, but there is no mention that a genuine Klein bottle cannot be constructed in ordinary 3-dimensional space.(The familiar model of a Klein bottle depicted here is a self-intersecting version of the real Klein bottle, which does not intersect itself.This is much like the fact that a picture of a knot drawn in the plane must appear as if the knot intersects itself, though it does not do so in space.)
________
p. 46:The illustration at bottom purports to show what the model of the Klein bottle would look like if it were sliced in half.The halves are erroneously shown as identical, but they should be mirror images of each other.
________
p. 78:The title of this page is "Fractals -- real or imaginary?"
This is an entirely misguided question that will only confuse the reader.All mathematical concepts are real within mathematics, and do not exist (except as approximations) in the real world.

It's a worthwhile topic in the philosophy of mathematics, and could well have been introduced in this book, but it has nothing whatsoever to do with fractals per se.
________
p. 91:Here the author attempts to describe a model ofhyperbolic geometry (in a circular disk) devised by Henri Poincaré.However, she gets it exactly backwards, saying that objects get smaller as they approach the boundary of the disk.
(She may have been well-aware of how this model works, but her prose is at best completely ambiguous.)
________
p. 96:Here it is stated that it has been proved that knots cannot exist in more than 3 dimensions.Apparently the author is unfamiliar with an extensive and thriving field of higher-dimensional knots.(For example, a sphere can be knotted in 4-dimensional space.)
________
There are many, many more such gaffes, but I fear I have gone on too long.I just wanted to make it crystal-clear that this book is riddled with erroneous and vacuous statements. ... Read more

Product Description

Here is the classic, much-read introduction to the craft and history of mathematics by E.T. Bell, a leading figure in mathematics in America for half a century. Men of Mathematics accessibly explains the major mathematics, from the geometry of the Greeks through Newton's calculus and on to the laws of probability, symbolic logic, and the fourth dimension. In addition, the book goes beyond pure mathematics to present a series of engrossing biographies of the great mathematicians -- an extraordinary number of whom lived bizarre or unusual lives. Finally, Men of Mathematics is also a history of ideas, tracing the majestic development of mathematical thought from ancient times to the twentieth century. This enduring work's clear, often humorous way of dealing with complex ideas makes it an ideal book for the non-mathematician. ... Read more

Customer Reviews (33)

Neither history nor mathematics
This book contains no mathematics, which is OK since it doesn't pretend to.Unfortunately this leads people to classify it as a history of mathematics, but it is not by any means a history book of any kind---the author makes up facts and distorts them to match his preconceptions and prejudices.Given the worth and content of this book, it's amazing that it's lasted this long.The best way to describe it is as pure kitsch.

This book should never be used as a textbook of any sort, and should never, ever be given to children to `inspire' them.

Men of Mathematics
A great classic for all those who want to find out about the history of mathematics. Highly recommended. Book arrived in the time frame and condition described by the seller.

A Good Biographical Account of Some of the Greatest Mathematicians of History
E.T. Bell's book, "Men of Mathematics" is an excellent account of some of the greatest mathematicians of all time, as well as discussions of some of the events surrounding those men's lives and accomplishments.

For mathematical buffs, a book like this can be rather an enthralling discussion, since numerous anecdotes surrounding the mathematicians' mathematical feats, their conflicts with other mathematicians, etc. are discussed.

I would highly recommend this book for anyone who's a lover of mathematics, or likes to learn about the giant intellects who largely shaped the directions human mathematical endeavors have taken.

Classic on history of math and mathematical physics
I bought this book because it is said that it served as inspiration for a young John Nash to start his career as a mathematician. And I was not disappointed.

As a sign of the times it was written in the style is stodgy, but content, prose, and organization are excellent. It tells stories of some 4 dozen or so influential men (and a few women) of mathematics.. It is both an inspiration and a reference of interesting problems (many old ones and some current). Highly recommended for the mathematically inclined. This book can be read as biographies (ignoring many mathematical details) or with more interest the math itself.

the best single volume history of mathematics?
This is the book that turned John Forbes Nash (A BEAUTIFUL MIND) on to mathematics. I'm not enough of a scholar to know if it's the best single volume history of mathematics, but I can't imagine many better.This is a sturdy modern edition -- handsomely, although not, perhaps, beautifully, printed. ... Read more

Product Description
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

Customer Reviews (2)

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

Product Description
Renowned linguist George Lakoff pairs with psychologist Rafael Nuñez in the first book to provide a serious study of the cognitive science of mathematical ideas.

This book is about mathematical ideas, about what mathematics means-and why. Abstract ideas, for the most part, arise via conceptual metaphor-metaphorical ideas projecting from the way we function in the everyday physical world. Where Mathematics Comes From argues that conceptual metaphor plays a central role in mathematical ideas within the cognitive unconscious-from arithmetic and algebra to sets and logic to infinity in all of its forms.Amazon.com Review
If Barbie thinks math class is tough, what could she possibly think about math as a class of metaphorical thought? Cognitive scientists George Lakoff and Rafael Nuñez explore that theme in great depth in Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being.This book is not for the faint of heart or those with an aversion to heavy abstraction--Lakoff and Nuñez pull no punches in their analysis of mathematical thinking. Their basic premise, that all of mathematics is derived from the metaphors we use to maneuver in the world around us, is easy enough to grasp, but following the reasoning requires a willingness to approach complex mathematical and linguistic concepts--a combination that is sure to alienate a fair number of readers.

Those willing to brave its rigors will find Where Mathematics Comes From rewarding and profoundly thought-provoking. The heart of the book wrestles with the important concept of infinity and tries to explain how our limited experience in a seemingly finite world can lead to such a crazy idea. The authors know their math and their cognitive theory. While those who want their abstractions to reflect the real world rather than merely the insides of their skulls will have trouble reading while rolling their eyes, most readers will take to the new conception of mathematical thinking as a satisfying, if challenging, solution. --Rob Lightner ... Read more

Customer Reviews (27)

Ambiguity and Circumlocution
I just finished David Hume's Enquiry Concerning Human Understanding.Hume accused John Locke (his intellectual "forefather") of publishing writings full of "ambiguity and circumlocution", especially as regards the definition of "innate".

What is an "innate" ability?Is it an ACTUAL ability that is cotemporal with birth?One that appears in some stage of fetal development?Is it the POTENTIAL or CAPACITY for the development of a particular ability?Or is it an IDEA or IMPRESSION?

Well, who knows?But perhaps this tidbit from the authors of "Where Mathematics Comes From" can help settle the debate... in some experiment, babies as young as 3-4 days could tell when there had been a counting "mistake" (i.e. when their "expectations" of quantity were not met), therefore mathematics is "innate".No mention about WHY the experiment was not performed with YOUNGER children (does the "innateness" disappear if the children are too young?) or whether it would be plausible to reproduce this experiment with fetuses using pulses of sound to create expectations, etc.

So, by "innate" these authors mean "appears at 3-4 days-old", apparently after being subjected to the "environment" and learning about cause and effect (i.e. here is one item... if somebody brings another of the same item, there should be two items) for 3-4 days.This is an extremely poor precedent for defining one's terms."By innate, we don't really mean innate but something else.However, we like the sound of innate and it is helpful to our argument to call the ability "innate", so "innate" it is!" (Hume's Ambiguity).

Next, they make arguments about mathematics being a human psychological and cognitive phenomenon.I don't remember the name of the fallacy employed here, but it is something along the lines of the Weak Anthropic Principle... "our minds can only understand the mathematics that is understandable by our minds" (Hume's Circumlocution).So, the authors continue, there is no Platonic Mathematics, but only Human Mathematics and, oh, by the way, here is a list of monkeys and apes who can also do "Human Mathematics."I've never before seen anybody propose a hypothesis, happily and unwittingly disprove their own hypothesis, fail to realize it, and continue for 500 pages on a logical fallacy.

Anyway, all of the above (ambiguity and circumlocution) had appeared by Page 32.

The authors then propose "schemas" (like the Container Schema) which are at least as complex as the ACTUAL IDEAS they describe.

All in all, a very disappointing business.

Highly inspiring
Some readers have dismissed "Where Mathematics Come From" as being "postmodern", i.e. advocating arbitrariness of mathematical reasoning. These reviewers don't seem to have made it to the end. The authors are actually very clear about their viewpoint: They diapprove of the romance of mathematics (that basically claimes, mathematical truths have an objective existence independent of human cognition, and by the same token logic accounts for the (only) correct way of reasoning). But at the same time they acknowledge the universal nature of mathematical reasoning and its effectiveness in dealing with real world phenomena. There is a simple reason for the correspondence between human-made math and reality and it is given in chapter 15: Humans share a common brain structure, they live in fairly similar surroundings dealing with the same basic issues of everday life. Mathematics as well as language is modeled according to real life needs and conditions. Therefore mathematical reasoning is not arbitrary, albeit culturally shaped (c.f. the idea of "essence" leading to the need for axiomatization or the notion that all human reason is some kind of calculation).
It is relatively easy to corroborate the author's thesis, that the development of mathematics can be accurately described in terms of application of metaphorical structures and conceptual blending mechanisms on mathematical concepts and thereby creating new concepts and so forth. Just take a contemporary mathematical andvanced textbook on calculus or algebra and compare it to the writings of mathematicians before the invention of differential calculus (in Lakoffs/Nunez terms: the construction of infinitesimals and the mapping of numbers on the points on a line)or even Euler. The diference is striking: The idea that mathematical insights should rely on some essential axioms whence all mathematical truth can be derivedmust have seemed outlandish to mathematicians before the 19th century (although proved to be incorrect for quite some time now the notion of mathematics as being independent and self-sustaining seems to be quite widespread still).
Of course, by exploiting the possibilies of metaphorical cross-mapping within mathematics itself mathematics has liberated itself from reality to a great extent and turned into an art. Why else would mathematicians claim that beauty, simplicty and truth are closely interrelated?
The authors (like myself) obviously love mathematics and hold mathematicians in high esteem. And even more so by the fact that mathematics is "only" human.

A great reader for anyone who loves mathematics and wonders how it connects to common sense!

Recommended by a Friend - Very Unique
Not a simple book to read, but has soime unique thoughts.Need to have a rather detaile interest in math to enjoy or follow.

Good - but didn't destroy Platonism
There's a lot for professional mathematicians, teachers, and students to chew on in this book.The authors are bold yet constrained in their theses, clearly enunciating their arguments point by point, and seem to be willing and encouraging a dubious reader to enter a debate about the origins of math, and from their the implications.Fortunately they reject the excesses of postmodernism, but for this reviewer they did not succeed in eliminating what they've called "The Romance of Mathematics."Even if granted that all of math starts out from metaphors, and even if it's accepted metaphors are only an anthropic construct, that does not make it so that x^n + y^n = z^n for any n>2.Recommended for the challenges it brings.

wearing blinkers
This book may be seen as valuable if you are a devoted adherent of cognitive science, but as is true of all works having that orientation, it is highly scientistic, predicated on a view of person, mathematics, and world that is tremendously impoverished, impoverishing, partial, severely restricted and constricted. For a fascinating exploration of the same topic by an expert (but highly unusual) mathematician, see Brian Rotman's trilogy (1987, 1993, 2000). (It is not even acknowledged in this book.)

Deep general critiques of scientism are plentiful; a good start is "The Way Things Are", a book of interviews with Huston Smith. You also might have a look at my most recent book: The Unboundaried Self: Putting the Person Back Into the View from Nowhere ... Read more

Product Description
More than 14 percent of the PhD's awarded in the United States during the first four decades of the twentieth century went to women, a proportion not achieved again until the 1980s. This book is the result of a study in which the authors identified all of the American women who earned PhD's in mathematics before 1940, and collected extensive biographical and bibliographical information about each of them. By reconstructing as complete a picture as possible of this group of women, Green and LaDuke reveal insights into the larger scientific and cultural communities in which they lived and worked. The book contains an extended introductory essay, as well as biographical entries for each of the 228 women in the study. The authors examine family backgrounds, education, careers, and other professional activities. They show that there were many more women earning PhD's in mathematics before 1940 than is commonly thought. The material will be of interest to researchers, teachers, and students in mathematics, history of mathematics, history of science, women's studies, and sociology. The data presented about each of the 228 individual members of the group will support additional study and analysis by scholars in a large number of disciplines. Co-published with the London Mathematical Society beginning with Volume 4. Members of the LMS may order directly from the AMS at the AMS member price. The LMS is registered with the Charity Commissioners. ... Read more

Customer Reviews (1)

Fascinating History of Women Mathematicians
Meticulously researched, this book offers an in-depth look at a particular group of interesting women:those who obtained the Ph.D. in mathematics before 1940. The brief biographical entry for each of the 228 women is more a well-written and appreciative "portrait" than a mere listing of historical facts.As a whole, the entries provide the rich background from which the authors draw historical overviews and conclusions about the accomplishments (both great and mundane) of these women.Some of the women became famous; many did not.The surprisingly readable and accessible essay by LaDuke and Green puts these women into historical perspective. It is great fun to refer to the individual entries that augment and reinforce the authors' conclusions.The work (and the accompanying free website that gives even more biographical and bibliographical detail) is a brilliant piece of research and scholarship. ... Read more

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Product Description

General textbooks, attempting to cover three thousand or so years of mathematical history, must necessarily oversimplify just about everything, the practice of which can scarcely promote a critical approach to the subject. To counter this, History of Mathematics offers deeper coverage of key select topics, providing students with material that could encourage more critical thinking. It also includes the proofs of important results which are typically neglected in the modern history of mathematics curriculum.

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Product Description
This book gives a remarkably fine account of the influences mathematics has exerted on the development of philosophy, the physical sciences, religion, and the arts in Western life. ... Read more

Customer Reviews (4)

Best book I have read
Really an excellent book, although it is philosophy of math, it is easy to understand. We could know then, how important the math in promoting the society!

Excellent math overview
Kline's work is a breath of fresh air and an electric shock all in one.His handling of both the history of math and the math itself intertwine so well that you finish the work actually "knowing" for sure what you thought you knew in high school or college math classes.Having read this book one could argue that Kepler was the greatest scienctific mind ever, and actually be able to defend the thesis.A must read.

Enlightening, Interesting and Accessible to All
I agree with the above review and would simply like to add my own thoughts.The book illustrates the fascinating way in which mathematics, society, religion, politics and of course physics have affected each other(it goes both ways!) through out the ages.Furthermore, the author nicelyillustrates the processes by which people think and how those processeshave also changed through the ages (i.e., The Age of Reason versus TheRenisance).This book left me with real insights as to the nature andlimitations of the current state of mathematics and physics.Things arenot as they seem, my friend!Lastly, the author displays an appreciationfor the humor and irony of the history which makes this book hard to putdown at times.I never thought a math/history book could be a "pageturner"... Read it.

A de-mystification of mathematics.
In most mathematics classes, students are presented with a completededifice, and given a floor plan to help them navigate the halls.Whilethis approach works for many people, others need a little more basicinformation.In this book, Morris Kline builds the building, starting withthe mud and straw of the bricks.

"Mathematics in WesternCulture" shows that the history of mathematics is one of hundreds ofyears of people sitting in the sand, drawing shapes and lines, scratchingtheir heads, and trying to figure things out.This is not necessarily Dr.Kline's intention for the book, but this is certainly one of the manymessages to be derived from it.

A fascinating, exciting book which makesmathematics more understandable and accessible. ... Read more

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Customer Reviews (1)

abbreviated
This is a reprint of barely remembered book printed in 1913. It was probably the first to explain to Westerners the history of maths in Japan. Somewhat abbreviated perhaps. You can see how algebra and geometry developed in Japan. With speculation on how calculus from Europe might have influenced 17th century maths manuscripts. While later years brought advances in infinite series and sums.

The book may cause some readers to wonder if there have been other, more comprehensive discussions of Japanese maths. ... Read more

Product Description
Stimulating account of development of basic mathematics from arithmetic, algebra, geometry and trigonometry, to calculus, differential equations and non-Euclidean geometries. Also describes how math is used in optics, astronomy, motion under the law of gravitation, acoustics, electromagnetism, other phenomena.
... Read more

Customer Reviews (10)

Simply amazing
If you are interested at all in any scientific ideas or knowledge than read this book. If you're not, than read this book and you will be.

proofreading???
How about the major error in the example on page 2?Check it out ... he's comparing a job which offers $1800 to start and a raise of $200 a year with a job which offers $1800 to start and a raise of $50 every 6 months ... but when he does a table comparing the two jobs, showing what each would receive during each 6 month period, he incorrectly attributes too much to job two.He writes:

"The first job will pay

"900900100010001100110012001200

"The second job ....

"900950100010501100115012001250"

And concludes the second job, with the $50 raise every 6 months is better.The error is, one would not receive the entire $50 in the six month period ... the raise is per year, so the correct tabulation for job two would be:

900925 950 9751000102510501100

Clearly job one is better, just as intuition and common sense would tell you - if you get $200 more per year, you're going to be doing better than getting $100 more per year.

How can you trust a math book which starts with such a big logical error, that's never been proofread out after all these years?

A guided tour of Rome in one afternoon! (Caveat Emptor)
To the extent that this book collects many historical stories next to simple descriptions of mathematical ideas, reading _some_ parts of this book can be rewarding to _some_ people. But, the book over-reaches and over-promises. I also found the writing style annoying because the actual "per-page" ratio of _useful_ and _relevant_ information is low.

In retrospect, it is obvious that grand claims are always "red flags" - e.g. imagine a tourist on her first trip to Rome signing up for a guided tour that claims to show all of Rome in one afternoon.

Now I understand Physics and its beautiful!
Read this book and grow!This guy is a genius at communicating what is normally seen as difficult ideas.The pace is gentle and there are no mountains to scale - only hills where you can see fantastic physical vistas from.This stuff is beautiful.
Cheers

Almost a Humanities Course on Mathmatics
This book is excellent! Have you ever wondered, where did math come from?What caused/ how has math to developed? If so this book will hit the nail on the head for you. Dr. Kline is fabulous in this book, he explains things very clearly and gives the reader an overview of some of the more practical uses of math. After reading this book you will look at the world with a much better understanding of how math is used in the real world.

Kline also explains why math is so abstract (think of the way American schools teach math). Along with this he explains why math is so precise (due to it's being limited to using inductive reasoning only).

In fact, this book is a humanities course mixed in with the practical usage of mathematics, which all add up to a brilliant text. But don't be mislead, the book is not absent of the actual equations to help you understand some of the math. It's just simplified so as to be short of a textbook on how to do mathematics.


If this review is not helpful to you, or you think it could be improved please email your thoughts to:

HappyReaderTrueReview@yahoo.com

I want my reviews to be helpful to my fellow bookworms. ... Read more

Product Description
Beginning with the Babylonian and Egyptian mathematicians of antiquity, Ivor Grattan-Guinness "succeeds masterfully in viewing the history of mathematics from a new perspective." (Professor Karen Hunger Parshall, editor of Historia Mathematica). He charts the growth of mathematics through its refinement by ancient Greeks and then medieval Arabs, to its systematic development by Europeans from the Middle Ages to the early twentieth century. This book describes the evolution of arithmetic and geometry, trigonometry and algebra; the interplay between mathematics, physics, and mathematical astronomy; and "new" branches such as probability and statistics. Authoritative and comprehensive, The Rainbow of Mathematics is a unique account of the development of the science that is at the heart of so many other sciences. Originally published under the title The Norton History of the Mathematical Sciences. ... Read more

Customer Reviews (5)

Great general reference for history of math (physical science emphasis)
I bought the book to improve my knowledge of the history of mathematics. I'm a High School math teacher, although I never took a history of math course (lots of history which I generally enjoy).Although a significant portion of the math is beyond the High School curriculum, with the density of over 750 pages there are plenty of anecdotes that can be incorporated and interesting to share.

A dense book which covers numerous topics.I've successfully used parts of the book as a personal reference when studying other topics in detail in other books. Referencing this history's summaries on occasion has helped me see the forest through the trees, so to speak. Sometimes this book's summaries of a topic help place a field of mathematics (or a problem within mathematics) in the right context. That said, despite the smallish print and length of the text, the coverage is so wide it's not really going to teach topics in depth as some sort of replacement textbook.

Like any good history, there are many fascinating personalities or shall I say "characters" present in the history of math. Pettiness can be found in any field and some of the mathematician's stories and personal conflicts can be pretty silly in retrospect.

Final observation - while of course one does not need to be a physicist or mathematician to enjoy large parts of the book, one should at least have some calculus. For those looking for a more basic history of the development of algebra and geometry only, look elsewhere. As others noted there's a strong emphasis with physical science mathematics here which can be either fascinating or overwhelming.

A Must-Read for Mathematicians, Scientists, and Engineers
This magnificent work covers mathematics from its recorded beginnings to the end of World War I. It provides remarkable insight into the development of the various branches of mathematics, and into the connections between mathematical and scientific ideas. Readers will find a wealth of interesting and useful information, including a superb
bibliography. Highly recommended!

A Brilliant Rainbow
This much-needed book provides valuable insights into the history of the mathematical sciences. Readers will find a wealth of interesting and useful information, including an excellent bibliography. Highly recommended!

A Magnificent Account of the History of Mathematics
This milestone in the history of mathematics-history covers mathematics from its recorded beginnings to the end of World War I. It is a synthesis of remarkable historical and mathematical scope. Professor Grattan-Guinness has established a new paradigm of excellence in the field of mathematics-history.

An up-to-date, bring-your-own-math history of mathematics
In his introduction, Grattan-Guinness makes several good points about what a history of mathematics should contain. In particular he emphasises "the importance of applications to the physical world", setting his book apart from many other histories that follow the "unfortunate attitude" of "snobbery with regard to pure as opposed to applied mathematics". Partly but not entirely as a result of this approach, Grattan-Guinness covers an enormous amount of material. You are likely to find one or two pages of nontechnical, to-the-point discussion on almost any topic you can think of. This is of course valuable, but it also means that nothing is explained in detail and the book is too fragmented to be very enjoyable as a narrative. As a reference, it complements a history such as Kline's Mathematical Thought in that Grattan-Guinness doesn't bother giving detailed references to very many original works, but instead is strong on modern scholarly interpretations of history and provides lots of references to historical research. ... Read more

Customer Reviews (1)

Excellent Teacher Resource!
This book has everything a teacher could want in order to teach his/her students about the people behind the development of mathematics: sketches of the mathematicians, anecdotes, related activities for each person, andthe solutions. Teaching the history of math to kids really gives them adeeper appreciation of the subject.I have even developed projects fromthis book. This is a great tool to begin with. ... Read more