出版社: Springer
副标题: 3th Edition
出版年: 201082
页数: 662
定价: USD 69.95
装帧: Hardcover
丛书: Undergraduate Texts in Mathematics
ISBN: 9781441960528
内容简介 · · · · · ·
From the reviews of the third edition: "The author’s goal for Mathematics and its History is to provide a “bird’seye view of undergraduate mathematics.” (p. vii) In that regard it succeeds admirably. ... Mathematics and its History is a joy to read. The writing is clear, concise and inviting. The style is very different from a traditional text. ... The author has done a wonder...
From the reviews of the third edition: "The author’s goal for Mathematics and its History is to provide a “bird’seye view of undergraduate mathematics.” (p. vii) In that regard it succeeds admirably. ... Mathematics and its History is a joy to read. The writing is clear, concise and inviting. The style is very different from a traditional text. ... The author has done a wonderful job of tying together the dominant themes of undergraduate mathematics. ... While Stillwell does a wonderful job of tying together seemingly unrelated areas of mathematics, it is possible to read each chapter independently. I would recommend this fine book for anyone who has an interest in the history of mathematics. For those who teach mathematics, it provides lots of information which could easily be used to enrich an opening lecture in most any undergraduate course. It would be an ideal gift for a department’s outstanding major or for the math club president. Pick it up at your peril — it is hard to put down!" (Richard Wilders, MAA Reviews) From the reviews of the second edition: "This book covers many interesting topics not usually covered in a present day undergraduate course, as well as certain basic topics such as the development of the calculus and the solution of polynomial equations. The fact that the topics are introduced in their historical contexts will enable students to better appreciate and understand the mathematical ideas involved...If one constructs a list of topics central to a history course, then they would closely resemble those chosen here." (David Parrott, Australian Mathematical Society) "The book...is presented in a lively style without unnecessary detail. It is very stimulating and will be appreciated not only by students. Much attention is paid to problems and to the development of mathematics before the end of the nineteenth century... This book brings to the nonspecialist interested in mathematics many interesting results. It can be recommended for seminars and will be enjoyed by the broad mathematical community." (European Mathematical Society) "Since Stillwell treats many topics, most mathematicians will learn a lot from this book as well as they will find pleasant and rather clear expositions of custom materials. The book is accessible to students that have already experienced calculus, algebra and geometry and will give them a good account of how the different branches of mathematics interact." (Denis Bonheure, Bulletin of the Belgian Society)
作者简介 · · · · · ·
John Stillwell is a professor of mathematics at the University of San Francisco. He is also an accomplished author, having published several books with Springer, including The Four Pillars of Geometry; Elements of Algebra; Numbers and Geometry; and many more.
目录 · · · · · ·
Preface to the Second Edition
Preface to the First Edition
1 The Theorem of Pythagoras
2 Greek Geometry
3 Greek Number Theory
· · · · · · (更多)
Preface to the Second Edition
Preface to the First Edition
1 The Theorem of Pythagoras
2 Greek Geometry
3 Greek Number Theory
4 Infinity in Greek Mathematics
5 Number Theory in Asia
6 Polynomial Equations
7 Analytic Geometry
8 Projective Geometry
9 Calculus
10 Infinity Series
11 The Number Theory Revival
12 Elliptic Functions
13 Mechanics
14 Complex Numbers in Algebra
15 Complex Numbers and Curves
16 Complex Number and Functions
17 Differential Geometry
18 NonEuclidean Geometry
19 Group Theory
20 Hypercomplex Numbers
21 Algebraic Number Theory
22 Topology
23 Simple Groups
24 Sets, Logic, and Computation
25 Combinatorics
Bibliography
Index
· · · · · · (收起)
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時代抗疫totoro (Veritas vincit)
Common Notions 1. Things which are equal to the same thing are also equal to one another. 2. If equals be added to equals, the wholes are equal. 3. If equals be subtracted from equals, the remainders are equal. 4. Things which coincide with one another are equal to one another. 5. The whole is greater than the part. It appears that Euclid’s intention was to deduce geometric propositions from v...20110815 17:50
Common Notions 1. Things which are equal to the same thing are also equal to one another. 2. If equals be added to equals, the wholes are equal. 3. If equals be subtracted from equals, the remainders are equal. 4. Things which coincide with one another are equal to one another. 5. The whole is greater than the part. It appears that Euclid’s intention was to deduce geometric propositions from visually evident statements (the postulates) using evident principles of logic (the common notions). Actually, he often made unconscious use of visually plausible assumptions that are not among his postulates. His very first proposition used the unstated assumption that two circles meet if the center of each is on the circumference of the other (Heath (1925), p. 242).
回应 20110815 17:50 
時代抗疫totoro (Veritas vincit)
The impossibility of squaring the circle was proved by Lindemann (1882), in a very strong way. Not only is π undefinable by rational operations and square roots; it is also transcendental, that is, not the root of any polynomial equation with rational coefficients. Like Wantzel’s work, this was a rare example of a major result being proved by a minor mathematician. In Lindemann’s case the ex...20110815 17:40
The impossibility of squaring the circle was proved by Lindemann (1882), in a very strong way. Not only is π undefinable by rational operations and square roots; it is also transcendental, that is, not the root of any polynomial equation with rational coefficients. Like Wantzel’s work, this was a rare example of a major result being proved by a minor mathematician. In Lindemann’s case the explanation is perhaps that a major step had already been taken when Hermite (1873) proved the transcendence of e. Accessible proofs of both these results can be found in Klein (1924). Lindemann’s subsequent career was mathematically undistinguished, even embarrassing. In response to skeptics who thought his success with π had been a fluke, he took aim at the most famous unsolved problem in mathematics, “Fermat’s last theorem” (see Chapter 11 for the origin of this problem). His efforts fizzled out in a series of inconclusive papers, each one correcting an error in the one before. Fritsch (1984) has written an interesting biographical article on Lindemann. One ruler and compass problem is still open: which regular ngons are constructible? Gauss discovered in 1796 that the 17gon is constructible and then showed that a regular ngon is constructible if and only if n = 2m p1 p2 · · · pk, where the pi are distinct primes of the form 22h + 1. (This problem is also known as circle division, because it is equivalent to dividing the circumference of a circle, or the angle 2π, into n equal parts.) The proof of necessity was actually completed by Wantzel (1837). However, it is still not explicitly known what these primes are, or even whether there are infinitely many of them. The only ones known are for h = 0, 1, 2, 3, 4.
回应 20110815 17:40 
時代抗疫totoro (Veritas vincit)
The regular polyhedra will make another important appearance in connection with yet another 19thcentury development, the theory of finite groups and Galois theory. Before the regular polyhedra made this triumphant comeback, they also took part in a famous fiasco: the Kepler (1596) theory of planetary distances. Kepler’s theory is summarized by his famous diagram (Figure 2.3) of the five polyh...20110815 17:40
The regular polyhedra will make another important appearance in connection with yet another 19thcentury development, the theory of finite groups and Galois theory. Before the regular polyhedra made this triumphant comeback, they also took part in a famous fiasco: the Kepler (1596) theory of planetary distances. Kepler’s theory is summarized by his famous diagram (Figure 2.3) of the five polyhedra, nested in such a way as to produce six spheres of radii proportional to the distances of the six planets then known. Unfortunately, although mathematics could not permit any more regular polyhedra, nature could permit more planets, and Kepler’s theory was ruined when Uranus was discovered in 1781.
回应 20110815 17:40 
時代抗疫totoro (Veritas vincit)
The Elements was in fact too subtle for most mathematicians, let alone their students, so that in time Euclid’s geometry was boiled down to the simplest and driest propositions about straight lines, triangles, and circles. This part of the Elements is based on the following axioms (in the translation of Heath (1925), p. 154), which Euclid called postulates and common notions. Postulates Let th...20110815 17:40
The Elements was in fact too subtle for most mathematicians, let alone their students, so that in time Euclid’s geometry was boiled down to the simplest and driest propositions about straight lines, triangles, and circles. This part of the Elements is based on the following axioms (in the translation of Heath (1925), p. 154), which Euclid called postulates and common notions. Postulates Let the following be postulated: 1. To draw a straight line from any point to any point. 2. To produce a finite straight line continuously in a straight line. 3. To describe a circle with any center and distance. 4. That all right angles are equal to one another. 5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
回应 20110815 17:40

時代抗疫totoro (Veritas vincit)
The Pythagorean theorem was the first hint of a hidden, deeper relationship between arithmetic and geometry, and it has continued to hold a key position between these two realms throughout the history of mathematics. This has sometimes been a position of cooperation and sometimes one of conflict, as followed the discovery that √2 is irrational (see Section1.5). It is often the case that new id...20110814 21:59
The Pythagorean theorem was the first hint of a hidden, deeper relationship between arithmetic and geometry, and it has continued to hold a key position between these two realms throughout the history of mathematics. This has sometimes been a position of cooperation and sometimes one of conflict, as followed the discovery that √2 is irrational (see Section1.5). It is often the case that new ideas emerge from such areas of tension, resolving the conflict and allowing previously irreconcilable ideas to interact fruitfully. The tension between arithmetic and geometry is, without doubt, the most profound in mathematics, and it has led to the most profound theorems.
回应 20110814 21:59 
時代抗疫totoro (Veritas vincit)
Most of the problems solved by Diophantus involve quadratic or cubic equations, usually with one obvious trivial solution. Diophantus used the obvious solution as a stepping stone to the nonobvious, but no account of his method survived. It was ultimately reconstructed by Fermat and Newton in the 17th century, and this socalled chord–tangent construction will be considered later. Here, we nee...20110814 22:01
Most of the problems solved by Diophantus involve quadratic or cubic equations, usually with one obvious trivial solution. Diophantus used the obvious solution as a stepping stone to the nonobvious, but no account of his method survived. It was ultimately reconstructed by Fermat and Newton in the 17th century, and this socalled chord–tangent construction will be considered later. Here, we need it only for the equation x2 +y2 = 1, which is an ideal showcase for the method in its simplest form.
A trivial solution of this equation is x = −1, y = 0, which is the point Q on the unit circle (Figure 1.5). After a moment’s thought, one realizes will meet the circle at a second rational point R. This is because substitutionof y = t(x + 1) in x2 + y2 = 1 gives a quadratic equation with rational coefficients and one rational solution (x = −1); hence the second solution must also be a rational value of x. But then the y value of this point will also be rational, since t and x will be rational in (3). Conversely, the chord joining Q to any other rational point R on the circle will have a rational slope. Thus by letting t run through all rational values, we find all rational points R Q on the unit circle.
回应 20110814 22:01 
時代抗疫totoro (Veritas vincit)
We have already said that the perpendicular distances from P to the axes are the numbers x, y. The distance between points on the same perpendicular to an axis should therefore be defined as the difference between the appropriate coordinates. In Figure 1.11 this is x2−x1 for RQ and y2−y1 for PQ. But then the Pythagorean theorem tells us that the distance PR is given by That is, Since this...20110814 22:05
We have already said that the perpendicular distances from P to the axes are the numbers x, y. The distance between points on the same perpendicular to an axis should therefore be defined as the difference between the appropriate coordinates. In Figure 1.11 this is x2−x1 for RQ and y2−y1 for PQ. But then the Pythagorean theorem tells us that the distance PR is
given by
That is,
Since this construction applies to arbitrary points P, R in the plane, we now have a general formula for the distance between two points.
回应 20110814 22:05 
時代抗疫totoro (Veritas vincit)
There he founded a school whose members later became known as the Pythagoreans. The school’s motto was “All is number,” and the Pythagoreans tried to bring the realms of science, religion, and philosophy all under the rule of number. The very word mathematics (“that which is learned”) is said to be a Pythagorean invention. The school imposed a strict code of conduct on its members, which i...20110814 22:09
There he founded a school whose members later became known as the Pythagoreans. The school’s motto was “All is number,” and the Pythagoreans tried to bring the realms of science, religion, and philosophy all under the rule of number. The very word mathematics (“that which is learned”) is said to be a Pythagorean invention. The school imposed a strict code of conduct on its members, which included secrecy, vegetarianism, and a curious taboo on the eating of beans. The code of secrecy meant that mathematical results were considered to be the property of the school, and their individual discoverers were not identified to outsiders. Because of this, we do not know who discovered the Pythagorean theorem, the irrationality of √2, or other arithmetical results that will be mentioned in Chapter 3.
回应 20110814 22:09

時代抗疫totoro (Veritas vincit)
The impossibility of squaring the circle was proved by Lindemann (1882), in a very strong way. Not only is π undefinable by rational operations and square roots; it is also transcendental, that is, not the root of any polynomial equation with rational coefficients. Like Wantzel’s work, this was a rare example of a major result being proved by a minor mathematician. In Lindemann’s case the ex...20110815 17:40
The impossibility of squaring the circle was proved by Lindemann (1882), in a very strong way. Not only is π undefinable by rational operations and square roots; it is also transcendental, that is, not the root of any polynomial equation with rational coefficients. Like Wantzel’s work, this was a rare example of a major result being proved by a minor mathematician. In Lindemann’s case the explanation is perhaps that a major step had already been taken when Hermite (1873) proved the transcendence of e. Accessible proofs of both these results can be found in Klein (1924). Lindemann’s subsequent career was mathematically undistinguished, even embarrassing. In response to skeptics who thought his success with π had been a fluke, he took aim at the most famous unsolved problem in mathematics, “Fermat’s last theorem” (see Chapter 11 for the origin of this problem). His efforts fizzled out in a series of inconclusive papers, each one correcting an error in the one before. Fritsch (1984) has written an interesting biographical article on Lindemann. One ruler and compass problem is still open: which regular ngons are constructible? Gauss discovered in 1796 that the 17gon is constructible and then showed that a regular ngon is constructible if and only if n = 2m p1 p2 · · · pk, where the pi are distinct primes of the form 22h + 1. (This problem is also known as circle division, because it is equivalent to dividing the circumference of a circle, or the angle 2π, into n equal parts.) The proof of necessity was actually completed by Wantzel (1837). However, it is still not explicitly known what these primes are, or even whether there are infinitely many of them. The only ones known are for h = 0, 1, 2, 3, 4.
回应 20110815 17:40 
時代抗疫totoro (Veritas vincit)
The regular polyhedra will make another important appearance in connection with yet another 19thcentury development, the theory of finite groups and Galois theory. Before the regular polyhedra made this triumphant comeback, they also took part in a famous fiasco: the Kepler (1596) theory of planetary distances. Kepler’s theory is summarized by his famous diagram (Figure 2.3) of the five polyh...20110815 17:40
The regular polyhedra will make another important appearance in connection with yet another 19thcentury development, the theory of finite groups and Galois theory. Before the regular polyhedra made this triumphant comeback, they also took part in a famous fiasco: the Kepler (1596) theory of planetary distances. Kepler’s theory is summarized by his famous diagram (Figure 2.3) of the five polyhedra, nested in such a way as to produce six spheres of radii proportional to the distances of the six planets then known. Unfortunately, although mathematics could not permit any more regular polyhedra, nature could permit more planets, and Kepler’s theory was ruined when Uranus was discovered in 1781.
回应 20110815 17:40 
時代抗疫totoro (Veritas vincit)
Common Notions 1. Things which are equal to the same thing are also equal to one another. 2. If equals be added to equals, the wholes are equal. 3. If equals be subtracted from equals, the remainders are equal. 4. Things which coincide with one another are equal to one another. 5. The whole is greater than the part. It appears that Euclid’s intention was to deduce geometric propositions from v...20110815 17:50
Common Notions 1. Things which are equal to the same thing are also equal to one another. 2. If equals be added to equals, the wholes are equal. 3. If equals be subtracted from equals, the remainders are equal. 4. Things which coincide with one another are equal to one another. 5. The whole is greater than the part. It appears that Euclid’s intention was to deduce geometric propositions from visually evident statements (the postulates) using evident principles of logic (the common notions). Actually, he often made unconscious use of visually plausible assumptions that are not among his postulates. His very first proposition used the unstated assumption that two circles meet if the center of each is on the circumference of the other (Heath (1925), p. 242).
回应 20110815 17:50 
時代抗疫totoro (Veritas vincit)
The Elements was in fact too subtle for most mathematicians, let alone their students, so that in time Euclid’s geometry was boiled down to the simplest and driest propositions about straight lines, triangles, and circles. This part of the Elements is based on the following axioms (in the translation of Heath (1925), p. 154), which Euclid called postulates and common notions. Postulates Let th...20110815 17:40
The Elements was in fact too subtle for most mathematicians, let alone their students, so that in time Euclid’s geometry was boiled down to the simplest and driest propositions about straight lines, triangles, and circles. This part of the Elements is based on the following axioms (in the translation of Heath (1925), p. 154), which Euclid called postulates and common notions. Postulates Let the following be postulated: 1. To draw a straight line from any point to any point. 2. To produce a finite straight line continuously in a straight line. 3. To describe a circle with any center and distance. 4. That all right angles are equal to one another. 5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
回应 20110815 17:40
这本书的其他版本 · · · · · · ( 全部7 )

高等教育出版社 （2011）9.1分 109人读过

Springer （2001）暂无评分 8人读过

Springer （2012）暂无评分 1人读过

Springer （2010）暂无评分 1人读过
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4 有用 [已注销] 20130814
就个人而言，将数学和历史结合起来学习，比单纯学习数学有趣很多倍。
3 有用 [已注销] 20111210
最后一章关于组合数学的在其他历史书中很难见到，目前在纯数学中也不太受重视。
1 有用 宿命论 20150524
真佩服這些老數學家，憑空吧啦噠搞了個東西，還證了自己搞的東西…請收下我一學期的膝蓋！中國古代數學很輝煌，排個top 1也沒關係~可惜燒了坑了全沒了，誰都不曉得~
1 有用 任平生 20120102
相当有特色
1 有用 风乍起 20201010
复习下曾经学过的数学知识
1 有用 风乍起 20201010
复习下曾经学过的数学知识
0 有用 Rigel 20200713
可以在学完数分或者高数之后读。
1 有用 宿命论 20150524
真佩服這些老數學家，憑空吧啦噠搞了個東西，還證了自己搞的東西…請收下我一學期的膝蓋！中國古代數學很輝煌，排個top 1也沒關係~可惜燒了坑了全沒了，誰都不曉得~
4 有用 [已注销] 20130814
就个人而言，将数学和历史结合起来学习，比单纯学习数学有趣很多倍。
1 有用 任平生 20120102
相当有特色