IN THE previous chapter we pointed out that lengths are measurable in terms of some unit length, areas in term of a unit area, and volumes in terms of a unit volume.
在上一章中，我们指出，长度是通过某个单位长度来衡量的，面积是通过单位面积来衡量的，体积是通过单位体积来衡量的。
GEOMETRY, LIKE the rest of mathematics, is abstract. In it the properties of the shapes and relative positions of things are studied. But we do not need to consider who is observing the things, or whether he becomes acquainted with them by sight or touch or hearing. In short, we ignore all particular sensations. Furthermore, particular things such as the Houses of Parliament, or the terrestrial globe are ignored. Every proposition refers to any things with such and such geometrical properties. Of course it helps our imagination to look at particular examples of spheres and cones and triangles and squares, But the propositions do not merely apply to the actual figures printed in the book, but to any such figures.
几何学和数学的其他分支一样，是抽象的。在几何学中，我们研究物体的形状特征以及它们之间的位置关系。但我们并不需要考虑是谁在观察这些物体，或者他是通过视觉、触觉还是听觉来认识它们。简而言之，我们忽略了一切具体的感官体验。
THE INVENTION of the differential calculus marks a crisis in the history of mathematics. The progress of science is divided between periods characterized by a slow accumulation of ideas and periods, when, owing to the new material for thought thus patiently collected, some genius by the invention of a new method or a new point of view, suddenly transforms the whole subject on to a higher level. These contrasted periods in the progress of the history of thought are compared by Shelley to the formation of an avalanche.
微积分的发明标志着数学史上的一次变革。科学的进步可以分为两种时期：一种是思想的缓慢积累期，另一种则是由于前者耐心收集的思想材料，一位天才通过发明一种新方法或提出新的观点，突然将整个学科提升到更高的层次。这种思想史上对比鲜明的时期，被雪莱比作雪崩的形成过程。
No PART of Mathematics suffers more from the triviality of its initial presentation to beginners than the great subject of series. Two minor examples of series, namely arithmetic and geometric series, are considered; these examples are important because they are the simplest examples of an important general theory. But the general ideas are never disclosed and thus the examples, which exemplify noting, are reduced to silly trivialities.
没有哪部分数学比级数这一伟大课题更容易在向初学者介绍时显得琐碎。两个小的级数例子——即算术级数和几何级数——被考虑在内；这些例子之所以重要，是因为它们是一个重要一般理论的最简单例子。但是，普遍的理念从未被揭示出来，因此这些例子——本应展示重要内容——反而沦为无聊的琐事。
TRIGONOMETRY DID not take its rise from the general consideration of the periodicity of nature. In this respect its history is analogous to that of conic sections, which also had their origin in very particular ideas. Indeed, a comparison of the histories of the two sciences yields some very instructive analogies and contrasts. Trigonometry, like conic sections, had its origin among the Greeks. Its inventor was Hipparchus (born about 160 B.C), a Greek astronomer, who made his observations at Rhodes. His services to astronomy were very great, and it left his hands a truly scientific subject with important results established, and the right method of progress indicated. Perhaps the invention of trigonometry was not the least of these services to the main science of his study. The next man who extended trigonometry was Ptolemy, the great Alexandrian astronomer, whom we have already mentioned. We now see at once the great contrast between conic sections and trigonometry. The origin of trigonometry was practical; it was invented because it was necessary for astronomical research. The origin of conic sections was purely theoretical. The only reason for its initial study was the abstract interest of the ideas involved. Characteristically enough conic sections were invented about 150 years earlier that trigonometry, during the very best period of Greek thought. But the importance of trigonometry, both to theory and the application of mathematics, is only one of innumerable instances of the fruitful ideas which the general science has gained from its practical applications.
三角学的起源并非源自对自然周期性的普遍考察。在这一点上，它的历史类似于圆锥曲线的历史，后者同样起源于非常特定的概念。事实上，对比这两门科学的发展历程，可以得出一些非常有启发性的类比和对比。三角学与圆锥曲线一样，起源于希腊。它的发明者是希腊天文学家喜帕恰斯（约公元前160年出生），他在罗得岛进行天文观测。他对天文学的贡献极为重大，使其成为一门真正的科学，确立了重要的研究成果，并指明了正确的研究方法。或许，三角学的发明正是他对其主要研究领域——天文学——所作贡献中最重要的一项。接下来扩展三角学的是我们之前提到的伟大的亚历山大天文学家托勒密。至此，我们可以清楚地看到圆锥曲线与三角学之间的显著对比。三角学的起源是实用的——它的发明是天文研究的必然需求；而圆锥曲线的起源则是纯理论的，它最初受到研究的唯一原因是其中所蕴含的抽象数学思想的趣味。很有代表性的是，圆锥曲线大约在三角学发明前150年问世，正是在希腊思想最鼎盛的时期。然而，三角学在数学理论和应用中的重要性，仅仅是数学这门学科从实践应用中获得的无数富有成效的思想之一。
THE WHOLE life of Nature is dominated by the existence of periodic events, that is, by the existence of successive events so analogous to each other that, without any straining of language, they may be termed recurrences of the same event. The rotation of earth produces the successive days. It is true that each day is different from the preceding days, however abstractly we define the meaning of a day, so as to exclude casual phenomena. But with a sufficiently abstract definition of a day, the distinction in properties between two days becomes faint and remote from practical interest; and each day may then be conceived as a recurrence of the phenomenon of one rotation of the earth. Again the path of the earth round the sun leads to the yearly recurrence of the seasons, and imposes another periodicity on all the operations of nature. Another less fundamental periodicity is provided by the phases of the moon. In modern civilized life, with its artificial light, these phases are of slight importance, but in ancient times, in climates where the days are burning and the skies clear, human life was apparently largely influenced by the existence of moonlight. Accordingly our divisions into weeks and months, with their religious associations, have spread over the European races from Syria and Mesopotamia, though independent observances following the moon's phases are found amongst most nations. It is, however, through the tides, and not through its phases of light and darkness, that the moon's periodicity has chiefly influenced the history of the earth.
大自然的整个生命是由周期性事件的存在主导的，也就是说，主导着一系列彼此极为相似的连续事件，以至于在语言上毫不勉强地，它们可以被称为同一事件的重复。地球的自转产生了连续的日夜。虽然我们如何抽象地定义一天，排除偶然现象，每一天与前一天是不同的，但通过一个足够抽象的定义，两个日期之间的属性区别变得微弱且与实际兴趣无关；此时，每一天可以被看作是地球自转现象的重复。再者，地球绕太阳的轨迹导致了季节的年度循环，并为自然界的所有运作带来了另一个周期性。月亮的周期性则提供了另一个较为次要的周期性。在现代文明生活中，人工光源使得这些月相的重要性微乎其微，但在古代，尤其在白天气温酷热、天空晴朗的气候条件下，月光的存在显然在很大程度上影响了人类的生活。因此，我们对周和月的划分及其宗教意义，已从叙利亚和美索不达米亚传播到欧洲各族群，尽管大多数民族中也有独立的习俗遵循月相变化。然而，月亮的周期性主要通过潮汐影响了地球的历史，而不是通过光明和黑暗的变化。
THE MATHEMATICAL use of the term function has been adopted also in
common life. For example, 'His temper is a function of his digestion,'
uses the term exactly in this mathematical sense. It means that a rule
can be assigned which will tell you what his temper will be when you
know how his digestion is working. Thus the idea of a 'function' is
simple enough, we only have to see how it is applied in mathematics to
variable numbers. Let us think first of some concrete examples: If a
train has been travelling at the rate of twenty miles per hour, the
distance (
“数学中‘函数’一词的使用也被引入到日常生活中。例如，‘他的脾气是他消化情况的函数’，在这里‘函数’一词正是以数学意义被使用。它的意思是，可以设定一个规则，通过知道他的消化状况，来预测他的脾气。因此，‘函数’的概念其实很简单，我们只需要看它在数学中如何应用于变量数值。首先让我们考虑一些具体的例子：如果一列火车以每小时20英里的速度行驶，那么经过若干小时后的行驶距离（
In 2006 I was asked to give the sixth Sir David Williams Lecture at the University of Cambridge. This is an annual lecture established in honor (not, happily, in memory ) of a greatly respected legal scholar, leader and college head in that university. The organizers generously offered me a free choice of subject. Such an offer always poses a problem to unimaginative people like myself. We become accustomed at school and university to being given a subject title for our weekly essay, and it was rather the same in legal practice: clients came with a specific problem which they wanted answered, or appeared before the judge with a specific issue which they wanted (or in some cases did not want) resolved. There was never a free choice of subject matter.
2006年，我受邀在剑桥大学做第六届戴维·威廉姆斯爵士讲座。这是为纪念（幸运的是，不是为了悼念）一位在该大学备受尊敬的法律学者、领袖和学院院长而设立的年度讲座。组织者慷慨地为我提供了自由选择主题的机会。对于像我这样缺乏想象力的人来说，这种提议总是一个问题。在学校和大学里，我们习惯了每周写作时会被指定一个题目，而在法律实践中也是如此：客户带着一个具体问题来寻求解答，或者出现在法官面前，带着一个需要（或有时不需要）解决的具体问题。永远没有自由选择主题的机会。
THE METHODS and ideas of co-ordinate geometry have already been employed in the previous chapters. It is now time for us to consider them more closely for their own sake; and in doing so we shall strengthen our hold on other ideas to which we have attained. In the present and succeeding chapters we will go back to the idea of the positive and negative real numbers and will ignore the imaginaries which were introduced in the last two chapters.