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《几何原本》 - 译自《希腊化时代的科学与文化》(2)

已有 549 次阅读 2021-10-25 04:39 |个人分类:解读哥德尔不完全性定理|系统分类:科研笔记

一,译文


我将欧几里得与荷马相比在另一个方面是有效的,正如每个人都知道《伊利亚特》和《奥德赛》一样,每个人也都知道《几何原本》。谁是荷马?《伊利亚特》的作者,谁是欧几里得?《几何原本》的作者。

我们无法知道这些伟人的生平,但我们有权研究和使用他们的作品最好的作品就像我们应得的那样。因此,让我们考察一下《几何原本》,这是流传下来的最早的关于几何学的详尽教科书,其重要性很快就被意识到了,因此,该文本被完整地传递给我们。它被分成十三卷,其内容可简要描述如下:


第一至第六卷:平面几何。当然,第一卷是基本的,包括定义和公设,涉及三角形、平行线、平行四边形等。第二卷的内容可以被称为几何代数。第三册是关于圆的几何。第四册讲述了正多边形。第五册给出了一个新的比例理论,适用于不可通约的量和可通约的量。第六册介绍了该理论在平面几何中的应用。


第七至第十卷:算术,数的理论,讨论了许多种类的数,素数或互为素数,最小公倍数,几何级数中的数等。第十卷是欧几里德的代表作,专门讨论无理数,可以用表达式表示的线段,如

√(√a+√b)

其中,ab是可通约的线,但√a√b 是相反的,并且彼此不可通约。


第十一至十三卷:立体几何学。第十一卷很像第一卷和第六卷,扩展到第三维。第十二卷将穷举法应用于圆、球、金字塔等的可测性上。 第十三卷涉及规则实体。


柏拉图的奇妙推测将规则多面体的理论提升到了一个高度的意义,因此,关于柏拉图体的良好知识被许多优秀的人认为是几何学的皇冠。普罗克洛斯(V-2)认为,欧几里得是一个柏拉图主义者,他建立其几何学丰碑的目的是为了解释柏拉图数。这显然是错误的,当然,欧几里德可能是柏拉图主义者,但他可能更喜欢另一种哲学,或者他可能小心翼翼地避免了哲学的含义。规则多面体的理论是实体几何学的自然顶峰,因此《几何原本》不能不以它作为结束。


然而,试图继续欧几里得努力的早期几何学家特别关注规则实体,这并不奇怪。无论欧几里得对这些实体的看法如何,它们,尤其是对新柏拉图主义者来说,是几何学中最迷人的项目。由于它们,几何学获得了宇宙的意义和神学的价值。


在《几何原本》中又增加了两本关于规则实体的书,称为第十四卷和第十五卷,包括在许多手稿或印刷品的版本和翻译中。所谓的第十四卷是公元前二世纪初由亚历山德里亚的Hypsicles创作的,是一部杰出的作品;另一部论文, “第十五卷,时间更晚,质量更差,是由米利托斯的Isidoros(圣索菲亚大教堂的建筑师,约532)的学生写的。


回到欧几里德,特别是他的主要作品十三卷的《几何原本》。在评价他的时候,我们应该避免两个相反的错误,这两个错误反复出现。第一个错误是把他说成是几何学的创始人,即几何学之父。正如我在谈到所谓的医学之父希波克拉底时已经解释过的,除了我们的天父之外,没有任何无人知晓的父亲。如果我们把埃及和巴比伦的努力也考虑进去,那么欧几里德的《几何原本》就是一千多年来思想的高峰。有人可能会反对,欧几里德被称为几何学之父是有另一个原因的。尽管在他之前有许多发现,但他难道不是第一个把别人和自己获得的所有知识综合起来,并把所有已知的命题放在一个强有力的逻辑秩序中的人吗?这种说法并不绝对正确,在欧几里德之前,命题已经被证明,命题链也已建立;此外,在他之前,希波克拉底(公元前五世纪)、莱昂(公元前四世纪)和马格尼西亚的忒迪奥(公元前四世纪)已经写了《几何原本》。欧几里德肯定很熟悉的Thedio streatise已经为学院准备好了,而且很可能在Lyceum使用过类似的版本。不管怎么说,亚里士多德对Eudoxos的比例理论和穷举法很感兴趣,欧几里德在《几何原本》第五、第六和第十二卷中对其进行了扩充。总之,无论你考虑的是特定的定理或方法还是《几何原本》的安排,欧几里德并不是一个完全的创新者,在更大程度上他比在他之前其他几何学家做得更好。


相反的错误是将欧几里德视为教科书的制造者,他没有任何发明,只是将其他人的发现按更好的顺序组合起来。今天,一个准备几何学初级读本的人很难被认为是一个有创造力的数学家;他是一个教科书的制作者(不是一个不光彩的职业,即使其目的往往是纯粹的功利性),但欧几里德不是。


《几何原本》中的许多命题可以归于早期的几何学家,我们可以假设,那些不能归于他人的命题是由欧几里德自己发现的,而且数量不少。至于排列方式,我们可以假定它在很大程度上是来自欧几里得自己的。他创造的这座纪念碑在对称性、内在美和清晰性方面与帕台农神庙一样令人惊叹,但却无比复杂和耐用。


对这一大胆声明的充分论证不可能在几段或几页中给出,要欣赏《几何原本》的丰富性和伟大性,就必须在像赫思这样有注释的翻译中研究它们。这里,除了强调几个要点外,不可能做更多的事情。请看第一卷,它解释了第一原理、定义、公设、公理、定理和问题。现在有可能做得更好,但二十二个世纪前有人能做得这么好,这几乎是不可思议的。



二,原文


The Elements 


My comparison with Homer is valid in another way. As every body knows the Iliad and the Odyssey, so does every body know the Elements. Who is Homer ? The author of the Iliad. Who is Euclid ? The author of the Elements.


We cannot know these great men as men, but we are privileged to study and use their works - the best of themselves - as much as we deserve to. Let us thus consider the Elements, the earliest elaborate textbook on geometry that has come down to us. Its importance was soon realized and, therefore, the text has been transmitted to us in its integrity. It is decided into thirteen books, the contents of which may be described briefly as follows :


Books I-VI : Plan geometry. Book I is, of course, fundamental ; it includes the definitions and postulates and deals with triangles, parallels, parallelograms, etc. The contents of book II might be called « geometric algebra ». Book III is on the geometry of the circle. Book IV rats regular polygons. Book V gives a new theory of proportion applied to incommensurable as well as commensurable quantities. Book VI os ap^lications of the theory to plan geometry.


Book VII-X : Arithmetic, theory of numbers. These books discuss numbers of many kinds, primes or prime to one another, least common multiples, numbers in geometric progression, and si on. Book X, which is Euclid’s masterpiece, is devoted to irrational lines, all the lines that can be represented by an expressions, such as 

√(√a+√b)

Wherein a and b are commensurable lines, but √a and √b are surds and incommensurable with one another.


Book XI-XIII : Solid geometry. Book XI is very much like Books I and VI extended to a third dimension. Book XII applied the method of exhaustion to the measurablemnt of circle’s, spheres, pyramids, ans so on Book XIII deals with regular solids.


Plato’s fantastic speculations had raised the theory of regular polyhedra to a high level of significance. Hence, a good knowledge of the « Platonic bodies » was considered by many good people as the crown of geometry. Proclos (V-2) suggested that Euclid was a Platonist and that he had built his geometric monument for the purpose of explaining the Platonic figures. That is obviously wrong. Euclid may have been a Platonist, go course, but he may have preferred another philosophy or he may have carefully avoided philosophic implications. The theory of regular polyhedra is the natural culmination of solid geometry and hence the Elements could not but end with it. 


It is not surprising, however, that the early geometers who tried to continue the Euclidean efforts devoted special attention to the regular solids. Whatever Euclid may have thought of these solids beyond mathematics, they were, especially for the Neoplatonists, the most fascinating items in geometry. Thanks to them, geometry obtained a cosmical meaning and a theological value. 


Two more books dealing with the regular solids were added to the Elements, called Books XIV and XV and included in many editions and translations, manuscript or printed. The so-called Book XIV was composed by Hypsicles of Alexxandria at the beginning of the second century B.C. and is a work of outstanding merit; the other treatise, « Book XV », of a much later time and inferior in quality, was written by a pupil of Isidoros of Miletos (the architect of Hagia Sophia, c. 532).


To return to Euclid, and especially to his main work, The thirteen books of the Elements when judging him, we should avoid two opposite mistakes, which have been made repeatedly. The first is to speak of him as if he were the originator, the father, of geometry. As I have already explained apropos of Hippocrates, the so-called « father of medicine », there are no unbegotten fathers except Our Father in Heaven. If we take Egyptian and Babylonian efforts into account, as we should, Euclid’s Elements is the climax of more than a thousand years of cogitations. One might object that Euclid deserves to be called the father of geometry for another reason. Granted that many discoveries were made before him, was he not the first to build a synthesis of all the knowledge obtained by others and himself and to put all the known propositions in a strong logical order ? That statement is not absolutely true. Propositions had been proved before Euclid and chains of propositions established ; moreover, « Elements » had been composed before him by Hippocrates of Chios (V B.C.), by Leon (IV-I B.C.), and finally by Theudios of Magnesia (IV-2 B.C.). Thedio  streatise, with which Euclid was certainly familiar, had been prepared for the Academy, and it is probable that a similar one was in use in the Lyceum. At any rate, Aristote knee Eudoxos’ theory of proportion and the method of exhaustion, which Euclid expanded in Books V, VI and XII of the Elements. In short, whether you consider particular theorems or methods or the arrangement of the Element, Euclid was seldom a complete innovator ; he did much better and on a larger scale what other geometry had done before him.


The opposite mistake is to consider Euclid as a « textbook maker » who invented nothing and simply put together in better order the discoveries of other people. It is velar that a schoolmaster preparing today an elementary book of geometry can hardly be considered a creative mathematician; he is a textbook maker (not a dishonourable calling, even if the purpose is more often than not purely meretricious), but Euclid was not.


Many propositions in the Elements can be ascribed to earlier geometers ; we may assume that those which cannot be ascribed to others were discovered by Euclid himself, and their number is considerable. As to the arrangement, it is safe to assume that it is to a large extent Euclid’s own. He created a monument that is as marvellous in its symmetry, inner beauty, and clearness as the Parthenon, but incomparably more complex and more durable. 


A full proof of this bold statement cannot be given in a few paragraphs or in a few pages. To appreciate the richness and greatness of the Elements one must study them in a well-annotated translation like Hearth’s. It is not possible to do more, here and now, than emphasise a few points. Consider Book I, which explains first principles, definitions, postulates, axioms, theorems, and problems. It is possible to do better at present, but it is almost unbelievable that anybody could have done it as well twenty-two centuries ago.


参考文献:

1】乔治·萨顿(George Sarton)与《希腊化时代的科学与文化》 http://blog.sciencenet.cn/blog-2322490-1292301.html

2】张卜天译本,兰纪正、朱恩宽译本。





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