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Here is the classic, muchread 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 boo...
Here is the classic, muchread 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 nonmathematician.
作者简介 · · · · · ·
埃里克·坦普尔·贝尔(Eric Temple Bell)1883年出生于苏格兰的阿伯丁。早年就学于英格兰。1902年到美国，进斯坦福大学学习，l904年取得文学士学位。1908年在华盛顿大学做研究生，兼事教学，1909年获该校文学硕士学位。1911年进哥伦比亚大学，1912年获该校哲学博士学位。此后回华盛顿大学任数学讲师，1921年成为教授。1924年夏～1928年夏任教于芝加哥大学，1926年上半年任教于哈佛大学，随之受聘为加州理工学院的数学教授。
贝尔是美国国家科学院院士，曾任美国数学协会主席，美国数学学会和美国科学促进会副主席，《美国数学学会会报》、《美国数学学报》和《科学哲学》编委。他曾获美国数学学会的博歇(Bocher)奖。其著作除本书外，还包括《紫色的蓝宝石》(1924)、《代数的算术》(1927)、《揭穿科学之谜》和((科学的...
埃里克·坦普尔·贝尔(Eric Temple Bell)1883年出生于苏格兰的阿伯丁。早年就学于英格兰。1902年到美国，进斯坦福大学学习，l904年取得文学士学位。1908年在华盛顿大学做研究生，兼事教学，1909年获该校文学硕士学位。1911年进哥伦比亚大学，1912年获该校哲学博士学位。此后回华盛顿大学任数学讲师，1921年成为教授。1924年夏～1928年夏任教于芝加哥大学，1926年上半年任教于哈佛大学，随之受聘为加州理工学院的数学教授。
贝尔是美国国家科学院院士，曾任美国数学协会主席，美国数学学会和美国科学促进会副主席，《美国数学学会会报》、《美国数学学报》和《科学哲学》编委。他曾获美国数学学会的博歇(Bocher)奖。其著作除本书外，还包括《紫色的蓝宝石》(1924)、《代数的算术》(1927)、《揭穿科学之谜》和((科学的皇后》(1931)、《命理学》(1933)以及《探索真理》(1934)等。
贝尔在其最后一部著作《最后的问题》出版之前，于1960年12月逝世。
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費馬經常被認為是十七世紀最偉大的數學家，不像笛卡爾那麼全能：During his long vagabondage in Holland Descartes occupied himself with a number of studies in addition to his philosophy and mathematics. Optics, chemistry, physics, anatomy, embryology, medicine, astronomical observations, and meteorology, including a study of the rainbow, all claimed their share of his restless activity. Any man today ...
20111202 11:36
費馬經常被認為是十七世紀最偉大的數學家，不像笛卡爾那麼全能：
人家雖然業餘，但是只熱愛純數學：During his long vagabondage in Holland Descartes occupied himself with a number of studies in addition to his philosophy and mathematics. Optics, chemistry, physics, anatomy, embryology, medicine, astronomical observations, and meteorology, including a study of the rainbow, all claimed their share of his restless activity. Any man today spreading his effort over so diversified a miscellany would write himself down a fiddling dilettante.（引自前一章）
對於一個母語是法語的傢伙來說，懂得拉丁文跟西班牙文也許並沒有十分了不起，但是能用這兩種語言寫詩仍然是值得稱道的：As for Descartes and Fermat, each of them, entirely independently of the other, invented analytical geometry. [...] Fermat seems never to have been tempted, as both Descartes and Pascal were, by the insidious seductiveness of philosophizing about God, man, and the universe as a whole ...
公務員是業餘數學家的合適主業；附註，天朝的就算樂：His knowledge of the chief European languages and literatures of Continental Europe was wide and accurate, and Greek and Latin philology are indebted to him for several important corrections. In the composition of Latin, French, and Spanish verses, one of the gentlemanly accomplishments of his day, he showed great skill and a fine taste.
丟番圖的《算術》一書的裝幀留白看來不小，一共讓費馬寫出了47個定理。並且已經全部被後來者一一證實。在他擁有的全套定理中，既有重要的，也有僅僅是趣味性的。關於甚麼樣的定理是重要的，感謝作者，為我們這些外行讀者提供了一個解釋，並針對費馬給出了很好的範例：Fermat's work as a King's councillor was an aid rather than a detriment to his intellectual activities. Unlike other public servants  in the army for instance  parliamentary councillors were expected to hold themselves aloof from their fellow townsmen and to abstain from unnecessary social activities lest they be corrupted by bribery or otherwise in the discharge of their office. Thus Fermat found plenty of leisure.
在那47個定理中，最有名的還是他的最後定理，畢竟沒幾個定理能困擾住全世界所有的數學家三百年之久的（當然，也有可能是因為並非所有好腦袋都跑去搞數論樂。當今數學界不是有這麼個說法麼：“如果你在一個問題上卡住樂，其中一個辦法是讓陶哲軒對它感興趣。”）。對於那些僅僅因為腦子還不錯就認為自己好了不起的人來說，如果不能分清楚厚積薄發於天賦異稟之間的關係，作者建議他們去試一試費馬最後定理（雖然我認為最終摘得桂冠的那一位也並非天才，後詳）：It is difficult if not impossible to state why some theorems in arithmetic are considered 'important' while others, equally difficult to prove, are dubbed trivial. One criterion, although not necessarily conclusive, is that the theorem shall be of use in other fields of mathematics. Another is that it shall suggest researches in arithmetic or in mathematics generally, and a third that it shall be in some respect universal. Fermat's theorem just stated satisfies all of these somewhat arbitrary demands: it is of indispensable use in many departments of mathematics, including the theory of groups, which in turn is at the root of the theory of algebraic equations; it has suggested many investigations, of which the entire subject of primitive roots may be recalled to mathematical readers as an important instance; and finally it is universal in the sense that it states a property of all prime numbers  such general statements are extremely difficult to find and very few are known.
此書成書年代頗早（1937年），費馬定理1994年被證出來了，可以肯定，作者如果現在重寫這章，那麼長度大約是老版本的兩至三倍，有人用樂一本書的篇幅才把這個故事講完（http://book.douban.com/subject/1322358/）。不過這則八卦在那本書里倒沒有出現過：Something rarer than grubby patience or the greatly overrated 'infinite capacity for taking pains' is needed to find a way through the wilderness. Those who imagine genius is nothing more than the ability to be a good bookkeeper may be recommended to exert their infinite patience on Fermat's Last Theorem.
拿到這市值一分錢獎金的人名為安德魯懷爾斯，專業數學教授。八年苦心耕耘，運用幾百年後最先進的技巧，寫掉100多頁的公式加術語，終於把費馬最後定理給證了出來，然後，這份證明報告還是濃縮版的（《算術》的空白地方確實寫不下）。費馬如果真的如他自己所說，有一個“十分美妙的證明”，唯一可以確定的是那絕不是懷爾斯給出的那一種。哪一種更美妙，答案顯而易見——In 1908 the late Professor Paul Wolfskehl (German) left 100,000 marks to be awarded to the first person giving a complete proof of Fermat's Last Theorem. The inflation after the World War reduced this prize to a fraction of a cent, which is what the mercenary will now get for a proof.
這段高斯發表於1874年的言論，對120年後問世的費馬最後定理的首例冗長證明來說，恰好是一個絕妙的評述。我想我就在這裡結束。：）This seems to be an appropriate place to quote some famous remarks of Gauss concerning the favourite field of Fermat's interests and his own. [...] 'The higher arithmetic presents us with an inexhaustible store of interesting truths  of truths, too, which are not isolated, but stand in a close internal connexion, and between which, as our knowledge increases, we are continually discovering new and sometimes wholly unexpected ties. A great part of its theories derives an additional charm from the peculiarity that important propositions, with the impress of simplicity upon them, are often easily discoverable by induction, and yet are of so profound a character that we cannot find their demonstration till after many vain attempts; and even then, when we do succeed, it is often by some tedious and artificial process, while the simpler methods may long remain concealed.'
回应 20111202 11:36 
中譯本把所有作為形容詞的gay都翻譯成了“放蕩的”。 Descartes gambled with enthusiasm  and some success. Whatever he undertook he did with his whole soul.雖然人家不是搞概率的⋯⋯ November 10th, 1619, then, is the official birthday of analytic geometry and therefore also of modern mathematics. Eighteen years were to pass before the method was published. In the meantime Descartes went on wi...
20111129 16:07
中譯本把所有作為形容詞的gay都翻譯成了“放蕩的”。
雖然人家不是搞概率的⋯⋯Descartes gambled with enthusiasm  and some success. Whatever he undertook he did with his whole soul.
這寫法！November 10th, 1619, then, is the official birthday of analytic geometry and therefore also of modern mathematics. Eighteen years were to pass before the method was published. In the meantime Descartes went on with his soldiering. On his behalf mathematics may thank Mars that no halfspent shot knocked his head off at the battle of Prague. A score or so of promising young mathematicians a few years short of three centuries later were less lucky, owing to the advance of that science which Descartes, dream inspired.
腦中不禁浮現出一坨圓錐曲線方程組成的二次元人形生物舉著劍把一群大塊頭水手逼到甲板盡頭的詭異畫面⋯⋯... Unfortunately for their plans, Descartes understood their language. Whipping out his sword he compelled them to row him back to the shore, and once again analytical geometry escaped the accidents of battle, murder, and sudden death.
... as he dryly remarks, he soon discovered that the number of those who understand man is negligible in comparison with the number of those who think they understand geometry.
網上流傳的那個有關笛卡爾臨終前為瑞典女王創造樂個心形曲線當情書段子實在是有夠胡扯。克莉絲汀是恐怖的女人！！！不過依此書來看，其智力並沒有得到笛卡兒的高度認可，好吧他老人家是拿自己作的參考系：This somewhat masculine young woman was then nineteen, already a capable ruler, reputedly a good classicist (of this, more later), a wiry athlete with the physical endurance of Satan himself, a ruthless huntress, an expert horsewoman who thought nothing of ten hours in the saddle without once getting off, and finally a tough morsel of femininity who was as hardened to cold as a Swedish lumberjack. With all this she combined a certain thick obtuseness toward the frailties of less thickskinned beings. Her own meals were sparing; so were those of her courtiers. Like a hibernating frog she could sit for hours in an unheated library in the middle of a Swedish winter; her hangerson begged her through their chattering teeth to throw all the windows wide open and let the merry snow in. Her cabinet, she noted without a qualm, always agreed with her. She knew everything there was to be known; her ministers and tutors told her so. As she got along on only five hours' sleep she kept her toadies hopping through the hoop nineteen hours a day. ... had not the obtuse Christine got it into her immovable head that five o'clock in the morning was the proper hour for a busy, hardboiled young woman like herself to study philosophy. ... Christine appears to have lacked a normal human skin as well as nerves. ... He (Descartes) tried to make up his rest by lying down in the afternoons. She soon broke him of that.
由於短短幾個月內笛卡兒所受的各種折騰，作者將笛卡兒的死大半歸咎於瑞典女王：... He (Descartes) had chanced to interrupt one of the lessons in Greek. To his amazement Descartes learned that the vaunted classicist Christine was struggling over grammatical puerilities which, he says, he had mastered by himself when he was a little boy. His opinion of her mentality thereafter appears to have been respectful but low.
女王陛下那時候才二十多歲吧，一天睡5個小時就夠的學習狂，要體諒從小愛賴床的笛老人家確實有點困難。Thus he died on 11 February 1650, aged 54, a sacrifice to the overweening vanity of a headstrong girl.
即使翻譯成中文也不減其色的類比。Jacques Hadamard:... we shall quote Jacques Hadamard. He remarks first that the mere invention of coordinates was not Descartes' greatest merit, because that had already been done 'by the ancients'  a statement which is exact only if we read the unexpressed intention into the unaccomplished deed. Hell is paved with the halfbaked ideas of 'the ancients' which they could never quite cook through with their own steam.
It is quite another thing to recognize [as in the use of coordinates] a general method and to follow to the end the idea which it represents. It is exactly this merit, whose importance every real mathematician knows, that was preeminently Descartes' in geometry; it was thus that he was led to what ... is his truly great discovery in the matter; namely, the application of the method of coordinates not only to translate into equations curves already defined geometrically, but, looking at the question from an exactly opposite point of view, to the a priori definition of more and more complicated curves and, hence, more and more general ... Directly, with Descartes himself, later, indirectly, in the return which the following century made in the opposite direction, it is the entire conception of the object of mathematical science that was revolutionized. Descartes indeed understood thoroughly the significance of what he had done, and he was right when he boasted that he had as far surpassed all geometry before him as Cicero's rhetoric surpasses the ABC.
回应 20111129 16:07 
中譯開頭引了Edgar Poe的詩句，“光榮歸於希臘，輝煌歸於羅馬”，英文版中沒有。引得還算確切吧，尤其參照本章節最後一段來看：As Whitehead has observed, 'No Roman lost his life because he was absorbed in the contemplation of a mathematical diagram.'說的當然是阿基米德。 Of all the ancients Archimedes is the only one who habitually thought with the unfettered freedom that the greater mathematicians perm... (1回应)
20111129 13:14
中譯開頭引了Edgar Poe的詩句，“光榮歸於希臘，輝煌歸於羅馬”，英文版中沒有。引得還算確切吧，尤其參照本章節最後一段來看：
說的當然是阿基米德。As Whitehead has observed, 'No Roman lost his life because he was absorbed in the contemplation of a mathematical diagram.'
（阿基米德的穿越性）Of all the ancients Archimedes is the only one who habitually thought with the unfettered freedom that the greater mathematicians permit themselves today with all the hardwon gains of twentyfive centuries to smooth their way, for he alone of all the Greeks had sufficient stature and strength to stride clear over the obstacles thrown in the path of mathematical progress by frightened geometers who had listened to the philosophers. [...] Had the Greek mathematicians and scientists followed Archimedes rather than Euclid, Plato, and Aristotle, they might easily have anticipated the age of modern mathematics, which began with Descartes (1596~1650) and Newton in the seventeenth century, and the age of modern physical science inaugurated by Galileo (1564~1642) in the same century, by 2,000 years.
（畢達哥拉斯無理數發現的各種表述版本）... the common whole numbers 1,2,3 ... are insufficient for the construction of mathematics even in the rudimentary form in which he (Pythagoras) knew it. [...] This was what knocked his theory flat: it is impossible to find two whole numbers such that the square of one of them is equal to twice the square of the other. [...] Actually Pythagoras found his stumblingblock in geometry: the ratio of the side of a square to one of its diagonals cannot be expressed as the ratio of any two whole numbers. This is equivalent to the statement above about squares of whole numbers. In another form we would say that the square root of 2 is irrational, that is, is not equal to any whole number or decimal fraction, or sum of the two, got by dividing one whole number by another.
（在自己塗滿橄欖油的皮膚上畫圖很萌，說實話，很有點香豔的場景，橄欖油甚麼的）Archimedes made his own occasions. Sitting before the fire he would rake out the ashes and draw in them. After stepping from the bath he would anoint himself with olive oil, according to the custom of the time, and then, instead of putting on his clothes, proceed to lose himself in the diagrams which he traced with a fingernail on his own oily skin.
此處首度印證前述某段（於Introduction中），一併引來：In short he (Archimedes) used his mechanics to advance his mathematics. This is one of his titles to a modern mind: he used anything and everything that suggested itself as a weapon to attack his problems.
It must not be imagined that the sole function of mathematics  'the handmaiden of the sciences'  is to serve science. Mathematics has also been called 'the Queen of the Sciences.' If occasionally the Queen has seemed to beg from the sciences she has been a very proud sort of beggar, neither asking nor accepting favours from any of her more affluent sister sciences. What she gets she pays for.
比費馬最後定理還誇張，雖然沒它有名。幾乎忘記尺規作圖中的“尺”是指不帶刻度的直邊了。接前：... 'the three problems of antiquity': to trisect an angle; to construct a cube having double the volume of a given cube; to construct a square equal to a circle. None of these problems is possibly with only straightedge and compass, although it is hard to prove that the third is not, and the impossibility was finally proved only in 1882.
說流毒無窮也不為過，竟然！此處也印證了之前談到歐多克斯時作者的觀點之一，即：All constructions effected with other implements were dubbed 'mechanical' and, as such, for some mystical reason known only to Plato and his geometrizing God, were considered shockingly vulgar and were rigidly taboo in respectable geometry. Not till Descartes, 1,985 years after the death of Plato, published his analytical geometry, did geometry escape from its Platonic straightjacket
緊隨其後：As we shall see, his (Plato's) total influence on the development of mathematics was probably baneful.
柏拉圖，嫉妒⋯⋯哇。But he did recognize what Eudoxus was and became his devoted friend until he began to exhibit something like jealousy towards his brilliant protégé.
1回应 20111129 13:14 
After all, the whole purpose of science is not technology  God knows we have gadgets enough already; science also explores depths of a universe that will never, by any stretch of the imagination, be visited by human beings or affect our material existence. So we shall attend also to some of the things which the great mathematicians have considered worthy of loving understanding for their intrinsi...
20111129 10:38
After all, the whole purpose of science is not technology  God knows we have gadgets enough already; science also explores depths of a universe that will never, by any stretch of the imagination, be visited by human beings or affect our material existence. So we shall attend also to some of the things which the great mathematicians have considered worthy of loving understanding for their intrinsic beauty.
Lagrange believed that a mathematician has not thoroughly understood his own work till he has made it so clear that he can go out and explain it effectively to the first man he meets on the street.
（此處中文版翻譯有誤。）... the first time something new is studied the details seem too numerous and hopelessly confused, and no coherent impression of the whole is left on the mind. Then, on returning after a rest, it is found that everything has fallen into place with its proper emphasis  like the development of a photographic film.
It is probably correct to say that as a class they (mathematicians) have tended slightly to the left in their political opinions.
An impartial account of western mathematics, including the award to each man and to each nation of its just share in the intricate development, could be written only by a Chinese historian. He alone would have the patience and the detached cynicism necessary for disentangling the curiously perverted pattern to discover whatever truth may be concealed in our variegated occidental boasting.
... the nineteenth century, prolonged into the twentieth, was, and is, the greatest age of mathematics the world has ever known. Compared to what glorious Greece did in mathematics the nineteenth century is a bonfire beside a penny candle.
（離散和連續）From the earliest times two opposing tendencies, sometimes helping one another, have governed the whole involved development of mathematics. Roughly these are the discrete and the continuous.
Geometry partakes of both the continuous and the discrete. A major task of mathematics today is to harmonize the continuous and the discrete, to include them in one comprehensive mathematics, and to eliminate obscurity from both.
Bertrand Russell:If we care to inspect the symbols in nature's great book through the critical eyes of modern science we soon perceive that we ourselves did the writing, and that we used the particular script we did because we invented it to fit our own understanding. Some day we may find a more expressive shorthand than mathematics for correlating our experiences of the physical universe  unless we accept the creed of the scientific mystics that everything is mathematics and is not merely described for our convenience in mathematical language. If 'Number rules the universe' as Pythagoras asserted, Number is merely our delegate to the throne, for we rule Number.
（以上，雕塑般冷峻而嚴厲的美）'Mathematics, rightly viewed, possesses not only truth but supreme beauty  a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.'
The nineteenth century, on this scale, contributed to mathematical knowledge about five times as much as was done in the whole of preceding history.
（以上，巴比倫人首度發現了“證明”之於數學的重要性，而非希臘人。）More important than the technical algebra of these ancient Babylonians is their recognition  as shown by their work  of the necessity for proof in mathematics. Until recently it had been supposed that the Greeks were the first to recognize that proof is demanded for mathematical propositions.
Mathematics then has had four great ages: the Babylonian, the Greek, the Newtonian (to give the period around 1700 a name), and the recent, beginning about 1800 and continuing to the present day. Competent judges have called the last the Golden Age of Mathematics.
回应 20111129 10:38

費馬經常被認為是十七世紀最偉大的數學家，不像笛卡爾那麼全能：During his long vagabondage in Holland Descartes occupied himself with a number of studies in addition to his philosophy and mathematics. Optics, chemistry, physics, anatomy, embryology, medicine, astronomical observations, and meteorology, including a study of the rainbow, all claimed their share of his restless activity. Any man today ...
20111202 11:36
費馬經常被認為是十七世紀最偉大的數學家，不像笛卡爾那麼全能：
人家雖然業餘，但是只熱愛純數學：During his long vagabondage in Holland Descartes occupied himself with a number of studies in addition to his philosophy and mathematics. Optics, chemistry, physics, anatomy, embryology, medicine, astronomical observations, and meteorology, including a study of the rainbow, all claimed their share of his restless activity. Any man today spreading his effort over so diversified a miscellany would write himself down a fiddling dilettante.（引自前一章）
對於一個母語是法語的傢伙來說，懂得拉丁文跟西班牙文也許並沒有十分了不起，但是能用這兩種語言寫詩仍然是值得稱道的：As for Descartes and Fermat, each of them, entirely independently of the other, invented analytical geometry. [...] Fermat seems never to have been tempted, as both Descartes and Pascal were, by the insidious seductiveness of philosophizing about God, man, and the universe as a whole ...
公務員是業餘數學家的合適主業；附註，天朝的就算樂：His knowledge of the chief European languages and literatures of Continental Europe was wide and accurate, and Greek and Latin philology are indebted to him for several important corrections. In the composition of Latin, French, and Spanish verses, one of the gentlemanly accomplishments of his day, he showed great skill and a fine taste.
丟番圖的《算術》一書的裝幀留白看來不小，一共讓費馬寫出了47個定理。並且已經全部被後來者一一證實。在他擁有的全套定理中，既有重要的，也有僅僅是趣味性的。關於甚麼樣的定理是重要的，感謝作者，為我們這些外行讀者提供了一個解釋，並針對費馬給出了很好的範例：Fermat's work as a King's councillor was an aid rather than a detriment to his intellectual activities. Unlike other public servants  in the army for instance  parliamentary councillors were expected to hold themselves aloof from their fellow townsmen and to abstain from unnecessary social activities lest they be corrupted by bribery or otherwise in the discharge of their office. Thus Fermat found plenty of leisure.
在那47個定理中，最有名的還是他的最後定理，畢竟沒幾個定理能困擾住全世界所有的數學家三百年之久的（當然，也有可能是因為並非所有好腦袋都跑去搞數論樂。當今數學界不是有這麼個說法麼：“如果你在一個問題上卡住樂，其中一個辦法是讓陶哲軒對它感興趣。”）。對於那些僅僅因為腦子還不錯就認為自己好了不起的人來說，如果不能分清楚厚積薄發於天賦異稟之間的關係，作者建議他們去試一試費馬最後定理（雖然我認為最終摘得桂冠的那一位也並非天才，後詳）：It is difficult if not impossible to state why some theorems in arithmetic are considered 'important' while others, equally difficult to prove, are dubbed trivial. One criterion, although not necessarily conclusive, is that the theorem shall be of use in other fields of mathematics. Another is that it shall suggest researches in arithmetic or in mathematics generally, and a third that it shall be in some respect universal. Fermat's theorem just stated satisfies all of these somewhat arbitrary demands: it is of indispensable use in many departments of mathematics, including the theory of groups, which in turn is at the root of the theory of algebraic equations; it has suggested many investigations, of which the entire subject of primitive roots may be recalled to mathematical readers as an important instance; and finally it is universal in the sense that it states a property of all prime numbers  such general statements are extremely difficult to find and very few are known.
此書成書年代頗早（1937年），費馬定理1994年被證出來了，可以肯定，作者如果現在重寫這章，那麼長度大約是老版本的兩至三倍，有人用樂一本書的篇幅才把這個故事講完（http://book.douban.com/subject/1322358/）。不過這則八卦在那本書里倒沒有出現過：Something rarer than grubby patience or the greatly overrated 'infinite capacity for taking pains' is needed to find a way through the wilderness. Those who imagine genius is nothing more than the ability to be a good bookkeeper may be recommended to exert their infinite patience on Fermat's Last Theorem.
拿到這市值一分錢獎金的人名為安德魯懷爾斯，專業數學教授。八年苦心耕耘，運用幾百年後最先進的技巧，寫掉100多頁的公式加術語，終於把費馬最後定理給證了出來，然後，這份證明報告還是濃縮版的（《算術》的空白地方確實寫不下）。費馬如果真的如他自己所說，有一個“十分美妙的證明”，唯一可以確定的是那絕不是懷爾斯給出的那一種。哪一種更美妙，答案顯而易見——In 1908 the late Professor Paul Wolfskehl (German) left 100,000 marks to be awarded to the first person giving a complete proof of Fermat's Last Theorem. The inflation after the World War reduced this prize to a fraction of a cent, which is what the mercenary will now get for a proof.
這段高斯發表於1874年的言論，對120年後問世的費馬最後定理的首例冗長證明來說，恰好是一個絕妙的評述。我想我就在這裡結束。：）This seems to be an appropriate place to quote some famous remarks of Gauss concerning the favourite field of Fermat's interests and his own. [...] 'The higher arithmetic presents us with an inexhaustible store of interesting truths  of truths, too, which are not isolated, but stand in a close internal connexion, and between which, as our knowledge increases, we are continually discovering new and sometimes wholly unexpected ties. A great part of its theories derives an additional charm from the peculiarity that important propositions, with the impress of simplicity upon them, are often easily discoverable by induction, and yet are of so profound a character that we cannot find their demonstration till after many vain attempts; and even then, when we do succeed, it is often by some tedious and artificial process, while the simpler methods may long remain concealed.'
回应 20111202 11:36 
中譯本把所有作為形容詞的gay都翻譯成了“放蕩的”。 Descartes gambled with enthusiasm  and some success. Whatever he undertook he did with his whole soul.雖然人家不是搞概率的⋯⋯ November 10th, 1619, then, is the official birthday of analytic geometry and therefore also of modern mathematics. Eighteen years were to pass before the method was published. In the meantime Descartes went on wi...
20111129 16:07
中譯本把所有作為形容詞的gay都翻譯成了“放蕩的”。
雖然人家不是搞概率的⋯⋯Descartes gambled with enthusiasm  and some success. Whatever he undertook he did with his whole soul.
這寫法！November 10th, 1619, then, is the official birthday of analytic geometry and therefore also of modern mathematics. Eighteen years were to pass before the method was published. In the meantime Descartes went on with his soldiering. On his behalf mathematics may thank Mars that no halfspent shot knocked his head off at the battle of Prague. A score or so of promising young mathematicians a few years short of three centuries later were less lucky, owing to the advance of that science which Descartes, dream inspired.
腦中不禁浮現出一坨圓錐曲線方程組成的二次元人形生物舉著劍把一群大塊頭水手逼到甲板盡頭的詭異畫面⋯⋯... Unfortunately for their plans, Descartes understood their language. Whipping out his sword he compelled them to row him back to the shore, and once again analytical geometry escaped the accidents of battle, murder, and sudden death.
... as he dryly remarks, he soon discovered that the number of those who understand man is negligible in comparison with the number of those who think they understand geometry.
網上流傳的那個有關笛卡爾臨終前為瑞典女王創造樂個心形曲線當情書段子實在是有夠胡扯。克莉絲汀是恐怖的女人！！！不過依此書來看，其智力並沒有得到笛卡兒的高度認可，好吧他老人家是拿自己作的參考系：This somewhat masculine young woman was then nineteen, already a capable ruler, reputedly a good classicist (of this, more later), a wiry athlete with the physical endurance of Satan himself, a ruthless huntress, an expert horsewoman who thought nothing of ten hours in the saddle without once getting off, and finally a tough morsel of femininity who was as hardened to cold as a Swedish lumberjack. With all this she combined a certain thick obtuseness toward the frailties of less thickskinned beings. Her own meals were sparing; so were those of her courtiers. Like a hibernating frog she could sit for hours in an unheated library in the middle of a Swedish winter; her hangerson begged her through their chattering teeth to throw all the windows wide open and let the merry snow in. Her cabinet, she noted without a qualm, always agreed with her. She knew everything there was to be known; her ministers and tutors told her so. As she got along on only five hours' sleep she kept her toadies hopping through the hoop nineteen hours a day. ... had not the obtuse Christine got it into her immovable head that five o'clock in the morning was the proper hour for a busy, hardboiled young woman like herself to study philosophy. ... Christine appears to have lacked a normal human skin as well as nerves. ... He (Descartes) tried to make up his rest by lying down in the afternoons. She soon broke him of that.
由於短短幾個月內笛卡兒所受的各種折騰，作者將笛卡兒的死大半歸咎於瑞典女王：... He (Descartes) had chanced to interrupt one of the lessons in Greek. To his amazement Descartes learned that the vaunted classicist Christine was struggling over grammatical puerilities which, he says, he had mastered by himself when he was a little boy. His opinion of her mentality thereafter appears to have been respectful but low.
女王陛下那時候才二十多歲吧，一天睡5個小時就夠的學習狂，要體諒從小愛賴床的笛老人家確實有點困難。Thus he died on 11 February 1650, aged 54, a sacrifice to the overweening vanity of a headstrong girl.
即使翻譯成中文也不減其色的類比。Jacques Hadamard:... we shall quote Jacques Hadamard. He remarks first that the mere invention of coordinates was not Descartes' greatest merit, because that had already been done 'by the ancients'  a statement which is exact only if we read the unexpressed intention into the unaccomplished deed. Hell is paved with the halfbaked ideas of 'the ancients' which they could never quite cook through with their own steam.
It is quite another thing to recognize [as in the use of coordinates] a general method and to follow to the end the idea which it represents. It is exactly this merit, whose importance every real mathematician knows, that was preeminently Descartes' in geometry; it was thus that he was led to what ... is his truly great discovery in the matter; namely, the application of the method of coordinates not only to translate into equations curves already defined geometrically, but, looking at the question from an exactly opposite point of view, to the a priori definition of more and more complicated curves and, hence, more and more general ... Directly, with Descartes himself, later, indirectly, in the return which the following century made in the opposite direction, it is the entire conception of the object of mathematical science that was revolutionized. Descartes indeed understood thoroughly the significance of what he had done, and he was right when he boasted that he had as far surpassed all geometry before him as Cicero's rhetoric surpasses the ABC.
回应 20111129 16:07 
中譯開頭引了Edgar Poe的詩句，“光榮歸於希臘，輝煌歸於羅馬”，英文版中沒有。引得還算確切吧，尤其參照本章節最後一段來看：As Whitehead has observed, 'No Roman lost his life because he was absorbed in the contemplation of a mathematical diagram.'說的當然是阿基米德。 Of all the ancients Archimedes is the only one who habitually thought with the unfettered freedom that the greater mathematicians perm... (1回应)
20111129 13:14
中譯開頭引了Edgar Poe的詩句，“光榮歸於希臘，輝煌歸於羅馬”，英文版中沒有。引得還算確切吧，尤其參照本章節最後一段來看：
說的當然是阿基米德。As Whitehead has observed, 'No Roman lost his life because he was absorbed in the contemplation of a mathematical diagram.'
（阿基米德的穿越性）Of all the ancients Archimedes is the only one who habitually thought with the unfettered freedom that the greater mathematicians permit themselves today with all the hardwon gains of twentyfive centuries to smooth their way, for he alone of all the Greeks had sufficient stature and strength to stride clear over the obstacles thrown in the path of mathematical progress by frightened geometers who had listened to the philosophers. [...] Had the Greek mathematicians and scientists followed Archimedes rather than Euclid, Plato, and Aristotle, they might easily have anticipated the age of modern mathematics, which began with Descartes (1596~1650) and Newton in the seventeenth century, and the age of modern physical science inaugurated by Galileo (1564~1642) in the same century, by 2,000 years.
（畢達哥拉斯無理數發現的各種表述版本）... the common whole numbers 1,2,3 ... are insufficient for the construction of mathematics even in the rudimentary form in which he (Pythagoras) knew it. [...] This was what knocked his theory flat: it is impossible to find two whole numbers such that the square of one of them is equal to twice the square of the other. [...] Actually Pythagoras found his stumblingblock in geometry: the ratio of the side of a square to one of its diagonals cannot be expressed as the ratio of any two whole numbers. This is equivalent to the statement above about squares of whole numbers. In another form we would say that the square root of 2 is irrational, that is, is not equal to any whole number or decimal fraction, or sum of the two, got by dividing one whole number by another.
（在自己塗滿橄欖油的皮膚上畫圖很萌，說實話，很有點香豔的場景，橄欖油甚麼的）Archimedes made his own occasions. Sitting before the fire he would rake out the ashes and draw in them. After stepping from the bath he would anoint himself with olive oil, according to the custom of the time, and then, instead of putting on his clothes, proceed to lose himself in the diagrams which he traced with a fingernail on his own oily skin.
此處首度印證前述某段（於Introduction中），一併引來：In short he (Archimedes) used his mechanics to advance his mathematics. This is one of his titles to a modern mind: he used anything and everything that suggested itself as a weapon to attack his problems.
It must not be imagined that the sole function of mathematics  'the handmaiden of the sciences'  is to serve science. Mathematics has also been called 'the Queen of the Sciences.' If occasionally the Queen has seemed to beg from the sciences she has been a very proud sort of beggar, neither asking nor accepting favours from any of her more affluent sister sciences. What she gets she pays for.
比費馬最後定理還誇張，雖然沒它有名。幾乎忘記尺規作圖中的“尺”是指不帶刻度的直邊了。接前：... 'the three problems of antiquity': to trisect an angle; to construct a cube having double the volume of a given cube; to construct a square equal to a circle. None of these problems is possibly with only straightedge and compass, although it is hard to prove that the third is not, and the impossibility was finally proved only in 1882.
說流毒無窮也不為過，竟然！此處也印證了之前談到歐多克斯時作者的觀點之一，即：All constructions effected with other implements were dubbed 'mechanical' and, as such, for some mystical reason known only to Plato and his geometrizing God, were considered shockingly vulgar and were rigidly taboo in respectable geometry. Not till Descartes, 1,985 years after the death of Plato, published his analytical geometry, did geometry escape from its Platonic straightjacket
緊隨其後：As we shall see, his (Plato's) total influence on the development of mathematics was probably baneful.
柏拉圖，嫉妒⋯⋯哇。But he did recognize what Eudoxus was and became his devoted friend until he began to exhibit something like jealousy towards his brilliant protégé.
1回应 20111129 13:14 
After all, the whole purpose of science is not technology  God knows we have gadgets enough already; science also explores depths of a universe that will never, by any stretch of the imagination, be visited by human beings or affect our material existence. So we shall attend also to some of the things which the great mathematicians have considered worthy of loving understanding for their intrinsi...
20111129 10:38
After all, the whole purpose of science is not technology  God knows we have gadgets enough already; science also explores depths of a universe that will never, by any stretch of the imagination, be visited by human beings or affect our material existence. So we shall attend also to some of the things which the great mathematicians have considered worthy of loving understanding for their intrinsic beauty.
Lagrange believed that a mathematician has not thoroughly understood his own work till he has made it so clear that he can go out and explain it effectively to the first man he meets on the street.
（此處中文版翻譯有誤。）... the first time something new is studied the details seem too numerous and hopelessly confused, and no coherent impression of the whole is left on the mind. Then, on returning after a rest, it is found that everything has fallen into place with its proper emphasis  like the development of a photographic film.
It is probably correct to say that as a class they (mathematicians) have tended slightly to the left in their political opinions.
An impartial account of western mathematics, including the award to each man and to each nation of its just share in the intricate development, could be written only by a Chinese historian. He alone would have the patience and the detached cynicism necessary for disentangling the curiously perverted pattern to discover whatever truth may be concealed in our variegated occidental boasting.
... the nineteenth century, prolonged into the twentieth, was, and is, the greatest age of mathematics the world has ever known. Compared to what glorious Greece did in mathematics the nineteenth century is a bonfire beside a penny candle.
（離散和連續）From the earliest times two opposing tendencies, sometimes helping one another, have governed the whole involved development of mathematics. Roughly these are the discrete and the continuous.
Geometry partakes of both the continuous and the discrete. A major task of mathematics today is to harmonize the continuous and the discrete, to include them in one comprehensive mathematics, and to eliminate obscurity from both.
Bertrand Russell:If we care to inspect the symbols in nature's great book through the critical eyes of modern science we soon perceive that we ourselves did the writing, and that we used the particular script we did because we invented it to fit our own understanding. Some day we may find a more expressive shorthand than mathematics for correlating our experiences of the physical universe  unless we accept the creed of the scientific mystics that everything is mathematics and is not merely described for our convenience in mathematical language. If 'Number rules the universe' as Pythagoras asserted, Number is merely our delegate to the throne, for we rule Number.
（以上，雕塑般冷峻而嚴厲的美）'Mathematics, rightly viewed, possesses not only truth but supreme beauty  a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.'
The nineteenth century, on this scale, contributed to mathematical knowledge about five times as much as was done in the whole of preceding history.
（以上，巴比倫人首度發現了“證明”之於數學的重要性，而非希臘人。）More important than the technical algebra of these ancient Babylonians is their recognition  as shown by their work  of the necessity for proof in mathematics. Until recently it had been supposed that the Greeks were the first to recognize that proof is demanded for mathematical propositions.
Mathematics then has had four great ages: the Babylonian, the Greek, the Newtonian (to give the period around 1700 a name), and the recent, beginning about 1800 and continuing to the present day. Competent judges have called the last the Golden Age of Mathematics.
回应 20111129 10:38

費馬經常被認為是十七世紀最偉大的數學家，不像笛卡爾那麼全能：During his long vagabondage in Holland Descartes occupied himself with a number of studies in addition to his philosophy and mathematics. Optics, chemistry, physics, anatomy, embryology, medicine, astronomical observations, and meteorology, including a study of the rainbow, all claimed their share of his restless activity. Any man today ...
20111202 11:36
費馬經常被認為是十七世紀最偉大的數學家，不像笛卡爾那麼全能：
人家雖然業餘，但是只熱愛純數學：During his long vagabondage in Holland Descartes occupied himself with a number of studies in addition to his philosophy and mathematics. Optics, chemistry, physics, anatomy, embryology, medicine, astronomical observations, and meteorology, including a study of the rainbow, all claimed their share of his restless activity. Any man today spreading his effort over so diversified a miscellany would write himself down a fiddling dilettante.（引自前一章）
對於一個母語是法語的傢伙來說，懂得拉丁文跟西班牙文也許並沒有十分了不起，但是能用這兩種語言寫詩仍然是值得稱道的：As for Descartes and Fermat, each of them, entirely independently of the other, invented analytical geometry. [...] Fermat seems never to have been tempted, as both Descartes and Pascal were, by the insidious seductiveness of philosophizing about God, man, and the universe as a whole ...
公務員是業餘數學家的合適主業；附註，天朝的就算樂：His knowledge of the chief European languages and literatures of Continental Europe was wide and accurate, and Greek and Latin philology are indebted to him for several important corrections. In the composition of Latin, French, and Spanish verses, one of the gentlemanly accomplishments of his day, he showed great skill and a fine taste.
丟番圖的《算術》一書的裝幀留白看來不小，一共讓費馬寫出了47個定理。並且已經全部被後來者一一證實。在他擁有的全套定理中，既有重要的，也有僅僅是趣味性的。關於甚麼樣的定理是重要的，感謝作者，為我們這些外行讀者提供了一個解釋，並針對費馬給出了很好的範例：Fermat's work as a King's councillor was an aid rather than a detriment to his intellectual activities. Unlike other public servants  in the army for instance  parliamentary councillors were expected to hold themselves aloof from their fellow townsmen and to abstain from unnecessary social activities lest they be corrupted by bribery or otherwise in the discharge of their office. Thus Fermat found plenty of leisure.
在那47個定理中，最有名的還是他的最後定理，畢竟沒幾個定理能困擾住全世界所有的數學家三百年之久的（當然，也有可能是因為並非所有好腦袋都跑去搞數論樂。當今數學界不是有這麼個說法麼：“如果你在一個問題上卡住樂，其中一個辦法是讓陶哲軒對它感興趣。”）。對於那些僅僅因為腦子還不錯就認為自己好了不起的人來說，如果不能分清楚厚積薄發於天賦異稟之間的關係，作者建議他們去試一試費馬最後定理（雖然我認為最終摘得桂冠的那一位也並非天才，後詳）：It is difficult if not impossible to state why some theorems in arithmetic are considered 'important' while others, equally difficult to prove, are dubbed trivial. One criterion, although not necessarily conclusive, is that the theorem shall be of use in other fields of mathematics. Another is that it shall suggest researches in arithmetic or in mathematics generally, and a third that it shall be in some respect universal. Fermat's theorem just stated satisfies all of these somewhat arbitrary demands: it is of indispensable use in many departments of mathematics, including the theory of groups, which in turn is at the root of the theory of algebraic equations; it has suggested many investigations, of which the entire subject of primitive roots may be recalled to mathematical readers as an important instance; and finally it is universal in the sense that it states a property of all prime numbers  such general statements are extremely difficult to find and very few are known.
此書成書年代頗早（1937年），費馬定理1994年被證出來了，可以肯定，作者如果現在重寫這章，那麼長度大約是老版本的兩至三倍，有人用樂一本書的篇幅才把這個故事講完（http://book.douban.com/subject/1322358/）。不過這則八卦在那本書里倒沒有出現過：Something rarer than grubby patience or the greatly overrated 'infinite capacity for taking pains' is needed to find a way through the wilderness. Those who imagine genius is nothing more than the ability to be a good bookkeeper may be recommended to exert their infinite patience on Fermat's Last Theorem.
拿到這市值一分錢獎金的人名為安德魯懷爾斯，專業數學教授。八年苦心耕耘，運用幾百年後最先進的技巧，寫掉100多頁的公式加術語，終於把費馬最後定理給證了出來，然後，這份證明報告還是濃縮版的（《算術》的空白地方確實寫不下）。費馬如果真的如他自己所說，有一個“十分美妙的證明”，唯一可以確定的是那絕不是懷爾斯給出的那一種。哪一種更美妙，答案顯而易見——In 1908 the late Professor Paul Wolfskehl (German) left 100,000 marks to be awarded to the first person giving a complete proof of Fermat's Last Theorem. The inflation after the World War reduced this prize to a fraction of a cent, which is what the mercenary will now get for a proof.
這段高斯發表於1874年的言論，對120年後問世的費馬最後定理的首例冗長證明來說，恰好是一個絕妙的評述。我想我就在這裡結束。：）This seems to be an appropriate place to quote some famous remarks of Gauss concerning the favourite field of Fermat's interests and his own. [...] 'The higher arithmetic presents us with an inexhaustible store of interesting truths  of truths, too, which are not isolated, but stand in a close internal connexion, and between which, as our knowledge increases, we are continually discovering new and sometimes wholly unexpected ties. A great part of its theories derives an additional charm from the peculiarity that important propositions, with the impress of simplicity upon them, are often easily discoverable by induction, and yet are of so profound a character that we cannot find their demonstration till after many vain attempts; and even then, when we do succeed, it is often by some tedious and artificial process, while the simpler methods may long remain concealed.'
回应 20111202 11:36 
中譯本把所有作為形容詞的gay都翻譯成了“放蕩的”。 Descartes gambled with enthusiasm  and some success. Whatever he undertook he did with his whole soul.雖然人家不是搞概率的⋯⋯ November 10th, 1619, then, is the official birthday of analytic geometry and therefore also of modern mathematics. Eighteen years were to pass before the method was published. In the meantime Descartes went on wi...
20111129 16:07
中譯本把所有作為形容詞的gay都翻譯成了“放蕩的”。
雖然人家不是搞概率的⋯⋯Descartes gambled with enthusiasm  and some success. Whatever he undertook he did with his whole soul.
這寫法！November 10th, 1619, then, is the official birthday of analytic geometry and therefore also of modern mathematics. Eighteen years were to pass before the method was published. In the meantime Descartes went on with his soldiering. On his behalf mathematics may thank Mars that no halfspent shot knocked his head off at the battle of Prague. A score or so of promising young mathematicians a few years short of three centuries later were less lucky, owing to the advance of that science which Descartes, dream inspired.
腦中不禁浮現出一坨圓錐曲線方程組成的二次元人形生物舉著劍把一群大塊頭水手逼到甲板盡頭的詭異畫面⋯⋯... Unfortunately for their plans, Descartes understood their language. Whipping out his sword he compelled them to row him back to the shore, and once again analytical geometry escaped the accidents of battle, murder, and sudden death.
... as he dryly remarks, he soon discovered that the number of those who understand man is negligible in comparison with the number of those who think they understand geometry.
網上流傳的那個有關笛卡爾臨終前為瑞典女王創造樂個心形曲線當情書段子實在是有夠胡扯。克莉絲汀是恐怖的女人！！！不過依此書來看，其智力並沒有得到笛卡兒的高度認可，好吧他老人家是拿自己作的參考系：This somewhat masculine young woman was then nineteen, already a capable ruler, reputedly a good classicist (of this, more later), a wiry athlete with the physical endurance of Satan himself, a ruthless huntress, an expert horsewoman who thought nothing of ten hours in the saddle without once getting off, and finally a tough morsel of femininity who was as hardened to cold as a Swedish lumberjack. With all this she combined a certain thick obtuseness toward the frailties of less thickskinned beings. Her own meals were sparing; so were those of her courtiers. Like a hibernating frog she could sit for hours in an unheated library in the middle of a Swedish winter; her hangerson begged her through their chattering teeth to throw all the windows wide open and let the merry snow in. Her cabinet, she noted without a qualm, always agreed with her. She knew everything there was to be known; her ministers and tutors told her so. As she got along on only five hours' sleep she kept her toadies hopping through the hoop nineteen hours a day. ... had not the obtuse Christine got it into her immovable head that five o'clock in the morning was the proper hour for a busy, hardboiled young woman like herself to study philosophy. ... Christine appears to have lacked a normal human skin as well as nerves. ... He (Descartes) tried to make up his rest by lying down in the afternoons. She soon broke him of that.
由於短短幾個月內笛卡兒所受的各種折騰，作者將笛卡兒的死大半歸咎於瑞典女王：... He (Descartes) had chanced to interrupt one of the lessons in Greek. To his amazement Descartes learned that the vaunted classicist Christine was struggling over grammatical puerilities which, he says, he had mastered by himself when he was a little boy. His opinion of her mentality thereafter appears to have been respectful but low.
女王陛下那時候才二十多歲吧，一天睡5個小時就夠的學習狂，要體諒從小愛賴床的笛老人家確實有點困難。Thus he died on 11 February 1650, aged 54, a sacrifice to the overweening vanity of a headstrong girl.
即使翻譯成中文也不減其色的類比。Jacques Hadamard:... we shall quote Jacques Hadamard. He remarks first that the mere invention of coordinates was not Descartes' greatest merit, because that had already been done 'by the ancients'  a statement which is exact only if we read the unexpressed intention into the unaccomplished deed. Hell is paved with the halfbaked ideas of 'the ancients' which they could never quite cook through with their own steam.
It is quite another thing to recognize [as in the use of coordinates] a general method and to follow to the end the idea which it represents. It is exactly this merit, whose importance every real mathematician knows, that was preeminently Descartes' in geometry; it was thus that he was led to what ... is his truly great discovery in the matter; namely, the application of the method of coordinates not only to translate into equations curves already defined geometrically, but, looking at the question from an exactly opposite point of view, to the a priori definition of more and more complicated curves and, hence, more and more general ... Directly, with Descartes himself, later, indirectly, in the return which the following century made in the opposite direction, it is the entire conception of the object of mathematical science that was revolutionized. Descartes indeed understood thoroughly the significance of what he had done, and he was right when he boasted that he had as far surpassed all geometry before him as Cicero's rhetoric surpasses the ABC.
回应 20111129 16:07 
中譯開頭引了Edgar Poe的詩句，“光榮歸於希臘，輝煌歸於羅馬”，英文版中沒有。引得還算確切吧，尤其參照本章節最後一段來看：As Whitehead has observed, 'No Roman lost his life because he was absorbed in the contemplation of a mathematical diagram.'說的當然是阿基米德。 Of all the ancients Archimedes is the only one who habitually thought with the unfettered freedom that the greater mathematicians perm... (1回应)
20111129 13:14
中譯開頭引了Edgar Poe的詩句，“光榮歸於希臘，輝煌歸於羅馬”，英文版中沒有。引得還算確切吧，尤其參照本章節最後一段來看：
說的當然是阿基米德。As Whitehead has observed, 'No Roman lost his life because he was absorbed in the contemplation of a mathematical diagram.'
（阿基米德的穿越性）Of all the ancients Archimedes is the only one who habitually thought with the unfettered freedom that the greater mathematicians permit themselves today with all the hardwon gains of twentyfive centuries to smooth their way, for he alone of all the Greeks had sufficient stature and strength to stride clear over the obstacles thrown in the path of mathematical progress by frightened geometers who had listened to the philosophers. [...] Had the Greek mathematicians and scientists followed Archimedes rather than Euclid, Plato, and Aristotle, they might easily have anticipated the age of modern mathematics, which began with Descartes (1596~1650) and Newton in the seventeenth century, and the age of modern physical science inaugurated by Galileo (1564~1642) in the same century, by 2,000 years.
（畢達哥拉斯無理數發現的各種表述版本）... the common whole numbers 1,2,3 ... are insufficient for the construction of mathematics even in the rudimentary form in which he (Pythagoras) knew it. [...] This was what knocked his theory flat: it is impossible to find two whole numbers such that the square of one of them is equal to twice the square of the other. [...] Actually Pythagoras found his stumblingblock in geometry: the ratio of the side of a square to one of its diagonals cannot be expressed as the ratio of any two whole numbers. This is equivalent to the statement above about squares of whole numbers. In another form we would say that the square root of 2 is irrational, that is, is not equal to any whole number or decimal fraction, or sum of the two, got by dividing one whole number by another.
（在自己塗滿橄欖油的皮膚上畫圖很萌，說實話，很有點香豔的場景，橄欖油甚麼的）Archimedes made his own occasions. Sitting before the fire he would rake out the ashes and draw in them. After stepping from the bath he would anoint himself with olive oil, according to the custom of the time, and then, instead of putting on his clothes, proceed to lose himself in the diagrams which he traced with a fingernail on his own oily skin.
此處首度印證前述某段（於Introduction中），一併引來：In short he (Archimedes) used his mechanics to advance his mathematics. This is one of his titles to a modern mind: he used anything and everything that suggested itself as a weapon to attack his problems.
It must not be imagined that the sole function of mathematics  'the handmaiden of the sciences'  is to serve science. Mathematics has also been called 'the Queen of the Sciences.' If occasionally the Queen has seemed to beg from the sciences she has been a very proud sort of beggar, neither asking nor accepting favours from any of her more affluent sister sciences. What she gets she pays for.
比費馬最後定理還誇張，雖然沒它有名。幾乎忘記尺規作圖中的“尺”是指不帶刻度的直邊了。接前：... 'the three problems of antiquity': to trisect an angle; to construct a cube having double the volume of a given cube; to construct a square equal to a circle. None of these problems is possibly with only straightedge and compass, although it is hard to prove that the third is not, and the impossibility was finally proved only in 1882.
說流毒無窮也不為過，竟然！此處也印證了之前談到歐多克斯時作者的觀點之一，即：All constructions effected with other implements were dubbed 'mechanical' and, as such, for some mystical reason known only to Plato and his geometrizing God, were considered shockingly vulgar and were rigidly taboo in respectable geometry. Not till Descartes, 1,985 years after the death of Plato, published his analytical geometry, did geometry escape from its Platonic straightjacket
緊隨其後：As we shall see, his (Plato's) total influence on the development of mathematics was probably baneful.
柏拉圖，嫉妒⋯⋯哇。But he did recognize what Eudoxus was and became his devoted friend until he began to exhibit something like jealousy towards his brilliant protégé.
1回应 20111129 13:14 
After all, the whole purpose of science is not technology  God knows we have gadgets enough already; science also explores depths of a universe that will never, by any stretch of the imagination, be visited by human beings or affect our material existence. So we shall attend also to some of the things which the great mathematicians have considered worthy of loving understanding for their intrinsi...
20111129 10:38
After all, the whole purpose of science is not technology  God knows we have gadgets enough already; science also explores depths of a universe that will never, by any stretch of the imagination, be visited by human beings or affect our material existence. So we shall attend also to some of the things which the great mathematicians have considered worthy of loving understanding for their intrinsic beauty.
Lagrange believed that a mathematician has not thoroughly understood his own work till he has made it so clear that he can go out and explain it effectively to the first man he meets on the street.
（此處中文版翻譯有誤。）... the first time something new is studied the details seem too numerous and hopelessly confused, and no coherent impression of the whole is left on the mind. Then, on returning after a rest, it is found that everything has fallen into place with its proper emphasis  like the development of a photographic film.
It is probably correct to say that as a class they (mathematicians) have tended slightly to the left in their political opinions.
An impartial account of western mathematics, including the award to each man and to each nation of its just share in the intricate development, could be written only by a Chinese historian. He alone would have the patience and the detached cynicism necessary for disentangling the curiously perverted pattern to discover whatever truth may be concealed in our variegated occidental boasting.
... the nineteenth century, prolonged into the twentieth, was, and is, the greatest age of mathematics the world has ever known. Compared to what glorious Greece did in mathematics the nineteenth century is a bonfire beside a penny candle.
（離散和連續）From the earliest times two opposing tendencies, sometimes helping one another, have governed the whole involved development of mathematics. Roughly these are the discrete and the continuous.
Geometry partakes of both the continuous and the discrete. A major task of mathematics today is to harmonize the continuous and the discrete, to include them in one comprehensive mathematics, and to eliminate obscurity from both.
Bertrand Russell:If we care to inspect the symbols in nature's great book through the critical eyes of modern science we soon perceive that we ourselves did the writing, and that we used the particular script we did because we invented it to fit our own understanding. Some day we may find a more expressive shorthand than mathematics for correlating our experiences of the physical universe  unless we accept the creed of the scientific mystics that everything is mathematics and is not merely described for our convenience in mathematical language. If 'Number rules the universe' as Pythagoras asserted, Number is merely our delegate to the throne, for we rule Number.
（以上，雕塑般冷峻而嚴厲的美）'Mathematics, rightly viewed, possesses not only truth but supreme beauty  a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.'
The nineteenth century, on this scale, contributed to mathematical knowledge about five times as much as was done in the whole of preceding history.
（以上，巴比倫人首度發現了“證明”之於數學的重要性，而非希臘人。）More important than the technical algebra of these ancient Babylonians is their recognition  as shown by their work  of the necessity for proof in mathematics. Until recently it had been supposed that the Greeks were the first to recognize that proof is demanded for mathematical propositions.
Mathematics then has had four great ages: the Babylonian, the Greek, the Newtonian (to give the period around 1700 a name), and the recent, beginning about 1800 and continuing to the present day. Competent judges have called the last the Golden Age of Mathematics.
回应 20111129 10:38
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最近改名叫《数学大师》，上海科教出版社  来自cydong  2 回应  20091005 
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0 有用 zzllrr 20091109
太老的书
0 有用 [已注销] 20150521
中文版已经读过，这本当做收藏。也是amazon只付邮费就能得到系列....
0 有用 wbw 20130321
本书有中文版，感觉翻译一般，原版的很好。
0 有用 堂唐 20090304
只是关于人，历史。。。
0 有用 Nise 20160614
读毕
0 有用 Nise 20160614
读毕
0 有用 大龄考生郭根宝 20151129
Professor Bell's like my favorite Klein, this work of his is really melded by humanistic concerns and exquisite clarity, without the eccentric arrogance of general mathematicians
0 有用 [已注销] 20150521
中文版已经读过，这本当做收藏。也是amazon只付邮费就能得到系列....
0 有用 Larenopfer 20130430
为什么我们不说数体教? 因为维尔斯特拉斯曾长期担任中学体育老师
0 有用 wbw 20130321
本书有中文版，感觉翻译一般，原版的很好。