1. 挑选好一个确定得研究对象，锲而不舍。你可能永远达不到终点，但是一路上准可以发现一些有趣的东西。——克莱因

1. Choose a definite purpose to work towards, and persist with it. You may never reach the end, but you will get a lot of interesting things along the way. — Klein

2. 思维的户外形式通常是这样的：有意识的研究——潜意识的活动——有意识的研究。——庞加莱

2. The outdoor form of thought is usually like this: conscious study – subconscious activity – conscious study. – Poincaré

3. 观察可能导致发现。观察将揭示某种规律模式或定律。——波利亚

3. Observation may lead to discovery. Observation will reveal some regular pattern or law. — Polya

4. 数学中的一些美丽定理具有这样的特性：它们极易从事实中归纳出来，但证明却隐藏的极深。——高斯

4. Some beautiful theorems in mathematics have the characteristic that they can be easily归纳 from facts, but their proofs are hidden deeply. - Gauss

5. 整数的简单构成，若干世纪以来一直是使数学获得新生的源泉。——伯克霍夫

5. The simple structure of integers has been a source of new life for mathematics for several centuries. — Birkhoff

6. 一个数学家越超脱越好。

6. The more transcendent a mathematician is, the better.

7. 在数学的天地里，重要的不是我们知道什么，而是我们怎么知道什么。——毕达哥拉斯

7. In the realm of mathematics, what is important is not what we know, but how we know it. — Pythagoras

8. 一个国家的科学水平能够用它消耗的数学来度量——拉奥

8. A country's level of science can be measured by the mathematics it consumes — Rao

9. 也许我可以并非不适当地要求获得数学上亚当这一称号，因为我相信数学理性创造物由我命名（已经流行通用）比起同时代其它数学家加在一起还要多。——西尔维斯特

9. Perhaps I may not inappropriately claim the title of Mathematical Adam, as I believe that the mathematical creations of reason named by me (already popular and widespread) exceed those of all other contemporary mathematicians combined. — Sylvester

10. 只要一门科学分支能提出大量的问题，它就充满着生命力，而问题缺乏则预示独立发展的终止或衰亡。——希尔伯特

10. As long as a branch of science can pose a large number of questions, it is full of vitality; while the lack of questions预示s the termination or decline of independent development. - Hilbert

11. 纯数学这门科学再其现代发展阶段，能够说是人类精神之最具独创性的创造。——怀德海

11. Pure mathematics, in its modern stage of development, can be said to be the most original creation of the human spirit. - Whitehead

12. 数学——科学不可动摇的基石，促进人类事业进步的丰富源泉。——巴罗

12. Mathematics – the unshakable foundation of science, a rich source promoting the progress of human endeavors. – Barrow

13. 数论是人类知识最古老的一个分支，然而他的一些最深奥的秘密与其最平凡的真理是密切相连的。——史密斯

13. Number theory is one of the oldest branches of human knowledge, yet some of its deepest secrets are closely connected to its most ordinary truths. — Smith

14. 无限！再也没有其他问题如此深刻地打动过人类的心灵。

14. Infinity! There is no other question that has so deeply touched the hearts of humanity.

15. 一门科学，只有当它成功地运用数学时，才能达到真正完善的地步。——马克思

15. A science is only truly perfected when it successfully applies mathematics. - Marx

16. 如果没有数所制造的关於宇宙的永恒的仿造品，则人类将不能继续生存。——尼采

16. Without the eternal copies of the universe that number has created, humanity could not continue to exist. — Nietzsche

17. 学习数学要多做习题，边做边思索。先知其然，然后知其所以然。

17. To learn mathematics, one should do plenty of exercises and think while doing them. First, know what is to be done, and then understand why it is to be done that way.

18. 一个没有几分诗人气的数学家永远成不了一个完全的数学家。

18. A mathematician without a bit of poet's spirit can never become a complete mathematician.

19. 一个国家的科学水平可以用它消耗的数学来度量。——拉奥

19. A country's level of scientific achievement can be measured by the mathematics it consumes. — Rao

20. 尺有所短；寸有所长。物有所不足；智有所不明。

20. Every inch has its shortcoming; every foot has its long. Every thing has its shortcomings; every intelligence has its blind spots.

21. 数学是科学的女王，而数论是数学的女王。——高斯

21. Mathematics is the queen of sciences, and number theory is the queen of mathematics. — Gauss

22. 我决心放弃那个仅仅是抽象的几何。这就是说，不再去考虑那些仅仅是用来练思想的问题。我这样做，是为了研究另一种几何，即目的在于解释自然现象的几何。——笛卡儿

22. I resolve to abandon that geometry which is merely abstract. This means no longer to consider those problems that are simply used to train the mind. I do this in order to study another geometry, which aims to explain natural phenomena. — Descartes

23. 在学习中要敢于做减法，就是减去前人已经解决的部分，看看还有那些问题没有解决，需要我们去探索解决。——华罗庚

23. In learning, we should dare to subtract, that is, to subtract the parts that have already been solved by predecessors, and see what other problems have not been solved and need us to explore and solve them. — Hua Loo-keng

24. 一道好题的价值之一在于它能产生其他一些好题。

24. One of the values of a good question lies in its ability to generate other good questions.

25. 非数学归纳法在数学的研究中，起着不可缺少的作用。

25. Non-mathematical induction plays an indispensable role in the research of mathematics.

26. 用心智的全部力量，来选取咱们应遵循的道路。——笛卡儿

26. Use all the power of your mind to choose the path we should follow. — Descartes

27. 攀登科学高峰，就像登山运动员攀登珠穆朗玛峰一样，要克服无数艰难险阻，懦夫和懒汉是不可能享受到胜利的喜悦和幸福的。——陈景润

27. Climbing the peak of science is like a mountaineer scaling Mount Everest, requiring the overcoming of countless hardships and dangers. Cowards and idlers are unable to enjoy the joy and happiness of victory. — Chen Jingrun

28. 一个数学家的目的，是要了解数学。历史上数学的进展不外两途：增加对于已知材料的了解，和推广范围。——陈省身

28. A mathematician's purpose is to understand mathematics. Historically, the progress of mathematics has taken two paths: to increase understanding of known materials and to expand its scope. — Chen Jingrun

29. 自然这一巨举是用数学符号写成的。——伽里略

29. Nature's grand design is written in the mathematical language. - Galileo

30. 天才是不足恃的，聪明是不可靠的，要想顺手拣来的伟大科学发明是不可想象的。——华罗庚

30. Talent is not enough to rely on, intelligence is not reliable, and it is unimaginable to pick up great scientific inventions conveniently. - Hua Luogeng

31. 无限！再也没有其他问题如此深刻地打动过人类的心灵。——D·希尔伯特

31. Infinity! No other question has so deeply touched the human heart. — D. Hilbert

32. 我把数学看成是一件有意思的工作，而不是想为自我建立什么纪念碑。能够肯定地说，我对别人的工作比自我的更钟爱。我对自我的工作总是不满意。——拉格朗日

32. I view mathematics as an interesting pursuit, not as a means to establish some monument to myself. I can confidently say that I have a greater fondness for others' work than for my own. I am always dissatisfied with my own work. — Lagrange

33. 数学家实际上是一个著迷者，不迷就没有数学。——诺瓦利斯

33. Mathematicians are actually devotees; without fascination, there is no mathematics. - Novalis

34. 在数学中，我们发现真理的主要工具是归纳和模拟。

34. In mathematics, we find that the main tools for discovering truth are induction and simulation.

35. 攀登科学，就像登山运动员攀登珠穆朗玛峰一样，要克服无数艰难险阻，懦夫和懒汉是不可能享受到胜利的喜悦和幸福的。——陈景润

35. Climbing the peak of science is like a mountaineer ascending Mount Everest; it requires overcoming countless difficulties and dangers. Cowards and lazy people are unable to enjoy the joy and happiness of victory. — Chen Jingrun

36. 如果别人思考数学的真理像我一样深入持久，他也会找到我的发现。——高斯

36. If someone else thinks deeply and persistently about the truths of mathematics as I do, they will also find my discoveries. - Gauss

37. 一个人就好像一个分数，他的实际才能好比分子，而他对自我的估价好比分母。分母越大，则分数的值就越小。——托尔斯泰

37. A person is like a fraction, with his actual abilities as the numerator and his self-evaluation as the denominator. The larger the denominator, the smaller the value of the fraction. - Tolstoy

38. 有时候，你一开始未能得到一个最简单，最美妙的证明，但正是这样的证明才能深入到高等算术真理的奇妙联系中去。这是我们继续研究的动力，并且最能使我们有所发现。——高斯

38. Sometimes, at first, you may not get the simplest and most wonderful proof, but it is precisely such proofs that can delve into the wonderful connections of higher arithmetic truths. This is our driving force for continued research, and the most likely to lead us to discoveries. - Gauss

39. 不管数学的任一分支是多么抽象，总有一天会应用在这实际世界上。

39. No matter how abstract any branch of mathematics may be, it will eventually find application in this real world.

40. 发现每一个新的群体在形式上都是数学的，因为我们不可能有其他的指导。——达尔文

40. We find that every new group is formally mathematical, for we cannot have any other guidance.—Darwin

41. 天才=1％的灵感+99％的血汗。——爱迪生

41. Genius = 1% inspiration + 99% perspiration. - Thomas Edison

42. 上帝创造了整数，所有其余的数都是人造的。——克隆内克

42. God created the integers, all the rest are the work of man. — Kronecker

43. 数学，科学的女皇；数论，数学的女皇。——高斯

43. Mathematics, the queen of sciences; number theory, the queen of mathematics. - Gauss

44. 石可破也，而不可夺坚；丹可磨也，而不可夺赤。

44. Stones can be broken, but they cannot be deprived of their hardness; cinnabar can be ground, but it cannot be stripped of its redness.

45. 纯数学这门科学再其现代发展阶段，可以说是人类精神之最具独创性的创造。——怀德海

45. Pure mathematics as a science in its modern stage of development can be said to be the most original creation of the human spirit. — Whitehead

46. 宇宙之大，粒子之微，火箭之速，化工之巧，地球之变，生物之谜，日用之繁，无处不用数学。

46. From the vastness of the universe to the minuteness of particles, from the speed of rockets to the intricacy of chemical engineering, from the changes of the Earth to the mysteries of biology, from the complexity of everyday life to the simplicity of mathematics, mathematics is employed everywhere.

47. 迟序之数，非出神怪，有形可检，有数可推。——祖冲之

47. The numbers of delayed sequence are not from supernatural beings; they can be inspected in form and can be deduced with numbers. — Zu Chongzhi

48. 用心智的全部力量，来选择我们应遵循的道路。——笛卡儿

48. Use the full force of the mind to choose the path we should follow. — Descartes

49. 把抽象的数学知识建立在形象的感知之上，把静态的教学理论建立在动态的体验、感悟和思考之上。

49. Establish abstract mathematical knowledge on the basis of concrete perception, and build static teaching theories on the basis of dynamic experiences, insights, and reflections.

50. 发现每一个新的群体在形式上都是数学的，因为我们不可能有其他的指导。

50. We find that every new group is formally mathematical, because we cannot have any other guidance.

51. 数学是一门演绎的学问，从一组公设，经过逻辑的推理，获得结论。——陈省身

51. Mathematics is a deductive discipline, deriving conclusions from a set of axioms through logical reasoning. — Chen Xingshen

52. 整数的简单构成，若干世纪以来一直是使数学获得新生的源泉。——GD伯克霍夫

52. The simple composition of integers has been a source of renewal for mathematics for several centuries. — G.D. Birkhoff

53. 直接向大师们而不是他们得的学生学习。——阿贝尔

53. Learn directly from the masters rather than from their students. — Abel

54. 可以数是属统治着整个量的世界，而算数的四则运算则可以看作是数学家的全部装备。——麦克斯韦

54. Numbers are what dominate the world of quantity, and the four operations of arithmetic can be seen as the whole equipment of mathematicians. — Maxwell

55. 我之所以比笛卡儿看得远些，是因为我站在巨人的肩上。——牛顿

55. The reason why I see further than Descartes is that I stand on the shoulders of giants. — Newton

56. 特殊化与一般化是有用的辅助问题的重要源泉。

56. Specialization and generalization are important sources of useful auxiliary questions.

57. 数学是一切知识中的最高形式。

57. Mathematics is the highest form of all knowledge.

58. 我把数学看成是一件有意思的工作，而不是想为自己建立什么纪念碑。可以肯定地说，我对别人的工作比自己的更喜欢。我对自己的工作总是不满意。——拉格朗日

58. I consider mathematics to be an interesting occupation, rather than something to build a monument to myself. It can be said with certainty that I prefer the work of others to my own. I am always dissatisfied with my own work. — Lagrange

59. 数学方法渗透并支配着一切自然科学的理论分支。它愈来愈成为衡量科学成就的主要标志了。

59. Mathematical methods permeate and dominate all branches of theoretical natural sciences. They are increasingly becoming the main criterion for measuring scientific achievements.

60. 以我一生最好的时光追寻那个目标……书已经写成了。现代人读或后代读都无关紧要，也许要等一百年才有一个读者。

60. Chasing that goal with the best years of my life... The book has been written. Whether it is read by modern people or by future generations is irrelevant; perhaps it will be a hundred years before it finds a reader.

61. 异常抽象的问题，必须讨论得异常清楚。——笛卡儿

61. Exceptionally abstract problems must be discussed with exceptional clarity. - Descartes

62. 数学是研究现实生活中数量关系和空间形式的数学。

62. Mathematics is the study of quantitative relationships and spatial forms in real life.

63. 精巧的论证常常不是一蹴而就的，而是人们长期切磋积累的成果。我也是慢慢学来的，而且还要继续不断的学习。——阿贝尔

63. Subtle arguments are often not achieved overnight, but rather the accumulated results of long-term discussion and accumulation. I have also learned slowly, and I still need to continue learning continuously. — Abel

64. 不发生作用的东西是不会存在的。——莱布尼茨

64. That which does not act does not exist. - Leibniz

65. 这是一个可靠的规律，当数学或哲学著作的作者以模糊深奥的话写作时，他是在胡说八道。——A·N·怀德海

65. This is a reliable rule: when the author of a mathematical or philosophical work writes in vague and abstruse language, he is talking nonsense. — A.N. Whitehead

66. 事类相推，各有攸归，故枝条虽分而同本干知，发其一端而已。又所析理以辞，解体用图，庶亦约而能周，通而不黩，览之者思过半矣。——刘徽

66. By analogy, each thing has its proper place, so even though the branches are divided, they all know the same trunk. This is just the tip of the iceberg. Furthermore, the analysis of principles is done through words, and the structure and function are explained with diagrams. Perhaps this method is concise yet comprehensive, clear yet not pedantic, and those who read it will understand more than half of it. —— Liu Hui

67. 无限！再也没有其他问题如此深刻地打动过人类的心灵。——希尔伯特

67. Infinity! No other question has ever struck the human heart so profoundly. - Hilbert

68. 读读欧拉，读读欧拉，他是咱们大家的老师。——拉普拉斯

68. Read Euler, read Euler, he is our teacher for all.—Laplace

69. 一个没有几分诗人才能的数学家决不会成为一个完全的数学家。——魏尔斯特拉斯

69. A mathematician without a bit of poetic talent will never be a complete mathematician. - Weierstrass

70. 数学不可比拟的永久性和万能性及他对时间和文化背景的独立行是其本质的直接后果。

70. The unparalleled permanence and versatility of mathematics, as well as its independence from time and cultural context, are direct consequences of its essence.

71. 一个人就好像一个分数，他的实际才能好比分子，而他对自己的估价好比分母。分母越大，则分数的值就越小。——托尔斯泰

71. A person is like a fraction, with their actual abilities being the numerator and their self-assessment being the denominator. The larger the denominator, the smaller the value of the fraction. - Tolstoy

72. 一个国家只有数学蓬勃的发展，才能展现它国立的强大。数学的发展和至善和国家繁荣昌盛密切相关。

72. A country can only demonstrate the strength of its national power through the vigorous development of mathematics. The development and perfection of mathematics are closely related to the prosperity and well-being of the nation.

73. 数学中的一些美丽定理具有这样的特性：它们极易从事实中归纳出来，但证明却隐藏的极深。数学是科学之王。——高斯

73. Some beautiful theorems in mathematics have the characteristic that they are极易 from facts, but their proofs are hidden deeply. Mathematics is the king of sciences. - Gauss

74. 没有任何问题可以向无穷那样深深的触动人的情感，很少有别的观念能像无穷那样激励理智产生富有成果的思想，然而也没有任何其他的概念能向无穷那样需要加以阐明。——希尔伯特

74. There is no question that can touch the human heart as deeply as infinity, and few other ideas can inspire the intellect to produce fruitful thoughts as much as infinity does. However, there is no other concept that requires as much clarification as infinity. — Hilbert

75. 数学是一种理性的精神，使人类的思维得以运用到最完善的程度。

75. Mathematics is a rational spirit that allows human thought to be applied to the highest degree of perfection.

76. 宇宙的伟大建筑是现在开始以纯数学家的面目出现了。

76. The grand architecture of the universe is now beginning to appear in the face of the pure mathematician.

77. 数学发明创造的动力不是推理，而是想象力的发挥。——德摩根

77. The driving force behind mathematical inventions is not reasoning, but the exercise of imagination. — De Morgan

78. 时间是个常数，但对勤奋者来说，是个‘变数’。用‘分’来计算时间的人比用‘小时’来计算时间的人时间多59倍。——雷巴柯夫

78. Time is a constant, but for the diligent, it is a 'variable'. The person who calculates time in 'minutes' has 59 times more time than the person who calculates time in 'hours'。——Rybakov

79. 会用数学公式，并不说明你会数学。

79. Knowing how to use mathematical formulas does not necessarily mean you understand mathematics.