# 《几何原本》 - 译自《希腊化时代的科学与文化》（2）

√(√a+√b)

《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。

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.

1】乔治·萨顿（George Sarton）与《希腊化时代的科学与文化》

2】张卜天译本，兰纪正、朱恩宽译本。

https://wap.sciencenet.cn/blog-2322490-1309309.html

## 全部精选博文导读

GMT+8, 2021-12-8 06:26