欧多克索斯Eudoxus of Cnidus

欧多克索斯是Aischines的儿子。至于他的老师，我们知道他前往现在意大利的塔兰托，在那里师从阿尔库塔斯，后者是毕达哥拉斯的追随者。倍立方的问题引起了阿尔库塔斯的兴趣，可以合理推测欧多克索斯对该问题的兴趣是由他的老师激发的。他可能从阿尔库塔斯那里学到的其他主题包括数论和音乐理论。

欧多克索斯还访问过西西里，在那里他与Philiston一起研究医学，之后在医生Theomedon的陪同下首次访问雅典。欧多克索斯在这次访问中在雅典待了两个月，他肯定参加了柏拉图和其他哲学家在科学院的哲学讲座，该学园刚建立不久。托马斯·利特尔·希思 [3]写到欧多克索斯在雅典做学生时：-

……他穷得住在比雷埃夫斯，每天步行往返雅典。

离开雅典后，他在埃及待了一年多，在赫利奥波利斯与祭司们一起研究天文学。此时欧多克索斯在一个位于赫利奥波利斯和Cercesura之间的天文台进行天文观测。从埃及，欧多克索斯前往小亚细亚西北部、马尔马拉海南岸的卡里普斯。在那里他建立了一所学校，这所学校非常受欢迎，他有许多追随者。

大约在公元前368年，欧多克索斯在众多追随者的陪同下第二次访问雅典。很难确切弄清他此时与柏拉图和科学院的关系。有一些证据表明，欧多克索斯对柏拉图的分析能力不太尊重，很容易看出原因，因为作为数学家，他的能力远远超过柏拉图。也有人认为，柏拉图看到欧多克索斯的学校变得如此成功并不完全高兴。当然，没有理由相信这两位哲学家对彼此的思想有很大影响。

欧多克索斯回到了他的故乡欧多克索斯，受到人民的赞誉，他们让他担任立法机构的重要角色。然而，他继续他的学术工作，写书并就神学、天文学和气象学进行演讲。

他在欧多克索斯上建了一个天文台，我们知道他从那里观测了老人星。在他的欧多克索斯天文台所做的观测，以及在赫利奥波利斯附近天文台所做的观测，构成了喜帕恰斯提到的两本书的基础。这些作品是Mirror和Phaenomena，一些学者认为它们是同一作品的修订版。喜帕恰斯告诉我们，这些作品涉及星座的升起和落下，但不幸的是，这些书，如同欧多克索斯的所有作品一样，已经失传。

他在这里建造了一个日晷。你可以在THIS LINK看到它的图片。

欧多克索斯对比例论做出了重要贡献，他给出一个定义，使得有可能以类似于今天所用交叉相乘的方法来比较无理数长度。到欧多克索斯时代，数学中已经出现了一个重大困难，即某些长度不可比较。通过寻找一个长度$t$使得$x = m \times t$和$y = n \times t$（其中$m$和$n$为整数）来比较两个长度$x$和$y$的方法，对于长度为1和√2的线段并不适用，正如毕达哥拉斯学派所表明的那样。

欧多克索斯发展的理论在欧几里得的Elements第五卷中阐述。该卷中的定义4被称为欧多克索斯公理，阿基米德将其归于他。该定义（在托马斯·利特尔·希思的译文[3]中）说：-

称两个量彼此具有比，如果其中一个量乘以某一倍数后能超过另一个量。

此处一段未译出，以下为英文原文 By this Eudoxus meant that a length and an area do not have a capable ratio. But a line of length √2 and one of length 1 do have a capable ratio since 1 × √2 > 1 and 2 × 1 > √2. Hence the problem of irrational lengths was solved in the sense that one could compare lines of any lengths, either 有理的 or irrational.

欧多克索斯接着说明了何时两个比相等。这出现在欧几里得的Elements第五卷定义5中，托马斯·利特尔·希思的译本[3]中写道：-

称四个量成相同的比，即第一个与第二个之比和第三个与第四个之比相同，如果取第一个和第三个的任何相同的倍数，以及第二个和第四个的任何相同的倍数，当前者按对应顺序与后者相比时，前者同时超过、同时等于或同时小于后者。

用现代记号来说，这表示a : b与c : d相等（其中$a, b, c, d$可能是无理数），如果对每一对可能的整数$m, n$

1. 如果$ma < nb$则$mc < nd$，
2. 如果$ma = nb$则$mc = nd$，
3. 如果$ma > nb$那么$mc > nd$。

Huxley在[1]中写道：-

很难夸大这一理论的重要性，因为它相当于对实数给出了严格的定义。在毕达哥拉斯学派发现无理数所导致的停滞之后，数论得以再次前进，这对所有后来的数学都带来了不可估量的益处。

许多作者讨论了欧多克索斯著作中关于实数的思想，并将他的思想与理查德·戴德金的思想进行了比较，特别是1872年给出的涉及“理查德·戴德金分割”的定义。理查德·戴德金本人强调，他的工作是受到欧多克索斯思想的启发。托马斯·利特尔·希思[3]写道，欧多克索斯关于相等比的定义：-

……恰好对应于由理查德·戴德金提出的现代无理数理论，并且它与卡尔·魏尔斯特拉斯关于相等数的定义逐字相同。

然而，一些历史学家持有相当不同的看法。例如，文章[15]（引自作者的摘要）：-

……首先分析了欧几里得的《几何原本》第五卷中所包含的、被归于欧多克索斯的比例论的历史意义。然后论证了理查德·戴德金基于有理数集合提出的实数定义相对于这一理论的彻底原创性。两个结论：（1）《几何原本》第五卷中不存在理查德·戴德金所察觉的缺陷；（2）不能恰当地说欧多克索斯的思想对理查德·戴德金的理论产生了‘影响’。

欧多克索斯对数学做出的另一项杰出贡献是他早期使用其穷竭法进行的积分工作。这项工作直接源于他在比例论方面的工作，因为他现在能够比较无理数了。它还基于更早的用安提丰逼近圆面积的思想，其中安提丰采用了边数不断增加的内接正多边形。欧多克索斯能够将安提丰的理论变为严格的理论，应用他的方法给出对最初由德谟克利特陈述的定理的严格证明，即

1. 棱锥的体积是同底等高的棱柱体积的三分之一；
   以及
2. 圆锥的体积是同底等高的圆柱体积的三分之一。

这些结果的证明被阿基米德在其著作On the sphere and cylinder中归于欧多克索斯，当然阿基米德继续使用欧多克索斯的穷竭法证明了一系列引人注目的定理。

我们知道欧多克索斯研究了立方体倍积这一经典问题。埃拉托色尼写过这个问题的历史，他说欧多克索斯用曲线解决了这个问题。阿什凯隆的欧托基奥斯写到了欧多克索斯的解法，但看来他面前有一份文献，尽管声称给出了欧多克索斯的解法，却必定是某个未能理解它的人所写。保罗·塔内里试图从极少的证据中重构欧多克索斯的证明，因此这只能停留在猜测的层面。Tannery的巧妙建议是，欧多克索斯在他的解法中使用了kampyle曲线，因此这条曲线现在被称为欧多克索斯的kampyle。然而，托马斯·利特尔·希思怀疑Tannery的建议[3]：-

在我看来，反对它的理由是它过于贴近地改编了阿尔库塔斯的思想……我认为，欧多克索斯作为数学家过于富有原创性，不会满足于仅仅改编阿尔库塔斯的解法。

我们还有待讨论欧多克索斯的行星理论，也许这是他最为著名的工作，他在现已失传的著作On velocities中发表了这一理论。也许值得作出的第一条评论是，欧多克索斯通过他的老师阿尔库塔斯深受毕达哥拉斯学派哲学的影响。因此，他发展出一套基于球体的体系，遵循毕达哥拉斯关于球体是最完美形状的信念，这并不令人意外。欧多克索斯提出的homocentric sphere system由若干旋转的球体组成，每个球体绕一条穿过地心的轴旋转。每个球体的旋转轴并非固定在空间中，而是对大多数球体而言，这条轴本身也在旋转，因为它由固定在另一个旋转球体上的点所决定。
插图：Eudoxus.gif ↗

如右图所示，假设我们有两个球$S_{1}$和$S_{2}$，$S_{1}$的轴$XY$是球$S_{2}$的一条直径。当$S_{2}$绕轴$AB$旋转时，$S_{1}$的轴$XY$随之旋转。如果这两个球以恒定但相反的角速度旋转，那么$S_{1}$赤道上的点$P$描出一条8字形曲线。这条曲线被称为马鞍形曲线（意为马绊）。

欧多克索斯用两个球体构造了hippopede，然后把行星视为沿该曲线运动的点$P$。他引入第三个球体来对应行星相对于背景恒星的一般运动，而沿hippopede的运动则产生观测到的周期性逆行运动。这个三球体子系统被置于第四个球体之中，后者给出恒星的周日旋转。

欧多克索斯的行星体系由亚里士多德在形而上学中描述，完整体系包含27个球体。西里西亚的辛普利修斯在大约公元540年撰写亚里士多德的注释时，也描述了欧多克索斯的球体。它们代表了一项宏伟的几何成就。正如托马斯·利特尔·希思所写[3]：-

……以这种理论方式，通过球体的叠加轴向旋转来产生逆行运动，是天才的非凡之举。在当时，论证这一假说的效果并非微不足道的几何成就；但与那种思辨能力相比，这就算不了什么了，正是这种能力使人能够发明出可以产生该效果的假说。

这一令人难以置信的数学成就毋庸置疑。但随后人们必然会问许多问题。欧多克索斯相信这些球体实际存在吗？他是否把它们发明为一种纯粹是计算工具的几何模型？这个模型是否准确代表了观测到的行星行为方式？欧多克索斯是否用观测证据检验过他的模型？

认为欧多克索斯只是把球体当作一种计算工具的一个论据是，他似乎既没有对球体的实质内容发表评论，也没有对其相互连接的方式发表评论。必须把欧多克索斯的观点与亚里士多德的观点区分开来，因为正如Huxley在[1]中所写：-

欧多克索斯可能只是把他的体系看作一个抽象的几何模型，但亚里士多德却把它当作对物理世界的描述……

欧多克索斯究竟是把他的球体看作几何还是物理实在，这个问题在有趣的论文[29]中得到了研究，该文主张欧多克索斯更感兴趣的是实际表示行星的路径，而不是预测天文现象。

当然，这个模型并没有以哪怕最简单的观测检验也能通过的精确度来表示——也许更重要的是，它不可能表示——行星的实际路径。至于欧多克索斯在验证自己的假说时在多大程度上依赖观测数据，奥托·纽格包尔在[7]中写道：-

……我们不仅没有证据表明在构造欧多克索斯的同心球时使用了数值数据，而且也很难说他的理论在与观测参数的比较中如何能够幸存下来。

也许，去追问欧多克索斯怎么能在没有用观测数据检验的情况下发展出如此错综复杂的理论，本身就是一种过于现代的思维方式。

许多早期评论者相信，柏拉图是欧多克索斯用其同心球体体系表示行星运动的灵感来源。这些观点至今仍相当流行，但文章[19]令人信服地论证说，情况并非如此，影响欧多克索斯提出其三维几何杰作的思想是毕达哥拉斯学派的，而不是来自柏拉图。

作为最后的评论，我们应当注意到欧多克索斯还写了一本名为Tour of the Earth的地理学著作，尽管该书已失传，但通过各类文献中约100条引文而颇为知名。这部著作共七卷，研究了欧多克索斯所知的地球上各民族，尤其考察了他们的政治制度、历史与背景。欧多克索斯以特别的权威性撰写了关于埃及及该国宗教的内容，显然他在那里度过的一年中对该国了解甚多。在第七卷中，欧多克索斯详细论述了意大利的毕达哥拉斯学派，对此他显然极为精通。

Eudoxus of Cnidus was the son of Aischines. As to his teachers, we know that he travelled to Tarentum, now in Italy, where he studied with Archytas who was a follower of Pythagoras. The problem of duplicating the cube was one which interested Archytas and it would be reasonable to suppose that Eudoxus's interest in that problem was stimulated by his teacher. Other topics that it is probable that he learnt about from Archytas include number theory and the theory of music.

Eudoxus also visited Sicily, where he studied medicine with Philiston, before making his first visit to Athens in the company of the physician Theomedon. Eudoxus spent two months in Athens on this visit and he certainly attended lectures on philosophy by Plato and other philosophers at the Academy which had only been established a short time before. Heath [3] writes of Eudoxus as a student in Athens:-

... so poor was he that he took up his abode at the Piraeus and trudged to Athens and back on foot each day.

After leaving Athens, he spent over a year in Egypt where he studied astronomy with the priests at Heliopolis. At this time Eudoxus made astronomical observations from an observatory which was situated between Heliopolis and Cercesura. From Egypt Eudoxus travelled to Cyzicus in northwestern Asia Minor on the south shore of the sea of Marmara. There he established a School which proved very popular and he had many followers.

In around 368 BC Eudoxus made a second visit to Athens accompanied by a number of his followers. It is hard to work out exactly what his relationship with Plato and the Academy were at this time. There is some evidence to suggest that Eudoxus had little respect for Plato's analytic ability and it is easy to see why that might be, since as a mathematician his abilities went far beyond those of Plato. It is also suggested that Plato was not entirely pleased to see how successful Eudoxus's School had become. Certainly there is no reason to believe that the two philosophers had much influence on each others ideas.

Eudoxus returned to his native Cnidus and there was acclaimed by the people who put him into an important role in the legislature. However he continued his scholarly work, writing books and lecturing on theology, astronomy and meteorology.

He had built an observatory on Cnidus and we know that from there he observed the star Canopus. The observations made at his observatory in Cnidus, as well as those made at the observatory near Heliopolis, formed the basis of two books referred to by Hipparchus. These works were the Mirror and the Phaenomena which are thought by some scholars to be revisions of the same work. Hipparchus tells us that the works concerned the rising and setting of the constellations but unfortunately these books, as all the works of Eudoxus, have been lost.

He constructed a sundial here. You can see a picture of it at THIS LINK.

Eudoxus made important contributions to the theory of proportion, where he made a definition allowing possibly irrational lengths to be compared in a similar way to the method of cross multiplying used today. A major difficulty had arisen in mathematics by the time of Eudoxus, namely the fact that certain lengths were not comparable. The method of comparing two lengths $x$ and $y$ by finding a length $t$ so that $x = m \times t$ and $y = n \times t$ for whole numbers $m$ and $n$ failed to work for lines of lengths 1 and √2 as the Pythagoreans had shown.

The theory developed by Eudoxus is set out in Euclid's Elements Book V. Definition 4 in that Book is called the Axiom of Eudoxus and was attributed to him by Archimedes. The definition states (in Heath's translation [3]):-

Magnitudes are said to have a ratio to one another which is capable, when a multiple of either may exceed the other.

By this Eudoxus meant that a length and an area do not have a capable ratio. But a line of length √2 and one of length 1 do have a capable ratio since 1 × √2 > 1 and 2 × 1 > √2. Hence the problem of irrational lengths was solved in the sense that one could compare lines of any lengths, either rational or irrational.

Eudoxus then went on to say when two ratios are equal. This appears as Euclid's Elements Book V Definition 5 which is, in Heath's translation [3]:-

Magnitudes are said to be of the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and the third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or are alike less than the latter equimultiples taken in corresponding order.

In modern notation, this says that a : b and c : d are equal (where $a, b, c, d$ are possibly irrational) if for every possible pair of integers $m, n$

1. if $ma < nb$ then $mc < nd$,
2. if $ma = nb$ then $mc = nd$,
3. if $ma > nb$ then $mc > nd$.

Huxley writes in [1]:-

It is difficult to exaggerate the significance of the theory, for it amounts to a rigorous definition of real number. Number theory was allowed to advance again, after the paralysis imposed on it by the Pythagorean discovery of irrationals, to the inestimable benefit of all subsequent mathematics.

A number of authors have discussed the ideas of real numbers in the work of Eudoxus and compared his ideas with those of Dedekind, in particular the definition involving 'Dedekind cuts' given in 1872. Dedekind himself emphasised that his work was inspired by the ideas of Eudoxus. Heath [3] writes that Eudoxus's definition of equal ratios:-

... corresponds exactly to the modern theory of irrationals due to Dedekind, and that it is word for word the same as Weierstrass's definition of equal numbers.

However, some historians take a rather different view. For example, the article [15] (quoting from the author's summary):-

... analyses, first, the historical significance of the theory of proportions contained in Book V of Euclid's "Elements" and attributed to Eudoxus. It then demonstrates the radical originality, relative to this theory, of the definition of real numbers on the basis of the set of rationals proposed by Dedekind. Two conclusions: (1) there are not in Book V of the "Elements" the gaps perceived by Dedekind; (2) one cannot properly speak of an 'influence' of Eudoxus's ideas on Dedekind's theory.

Another remarkable contribution to mathematics made by Eudoxus was his early work on integration using his method of exhaustion. This work developed directly out of his work on the theory of proportion since he was now able to compare irrational numbers. It was also based on earlier ideas of approximating the area of a circle by Antiphon where Antiphon took inscribed regular polygons with increasing numbers of sides. Eudoxus was able to make Antiphon's theory into a rigorous one, applying his methods to give rigorous proofs of the theorems, first stated by Democritus, that

1. the volume of a pyramid is one-third the volume of the prism having the same base and equal height;
   and
2. the volume of a cone is one-third the volume of the cylinder having the same base and height.

The proofs of these results are attributed to Eudoxus by Archimedes in his work On the sphere and cylinder and of course Archimedes went on to use Eudoxus's method of exhaustion to prove a remarkable collection of theorems.

We know that Eudoxus studied the classical problem of the duplication of the cube. Eratosthenes, who wrote a history of the problem, says that Eudoxus solved the problem by means of curved lines. Eutocius wrote about Eudoxus's solution but it appears that he had in front of him a document which, although claiming to give Eudoxus's solution, must have been written by someone who had failed to understand it. Paul Tannery tried to reconstruct Eudoxus's proof from very little evidence, so it must remain no more than a guess. Tannery's ingenious suggestion was that Eudoxus had used the kampyle curve in his solution and, as a consequence, the curve is now known as the kampyle of Eudoxus. Heath, however, doubts Tannery's suggestions [3]:-

To my mind the objection to it is that it is too close an adaptation of Archytas's ideas ... Eudoxus was, I think, too original a mathematician to content himself with a mere adaptation of Archytas's method of solution.

We have still to discuss Eudoxus's planetary theory, perhaps the work for which he is most famous, which he published in the book On velocities which is now lost. Perhaps the first comment that is worth making is that Eudoxus was greatly influenced by the philosophy of the Pythagoreans through his teacher Archytas. Therefore it is not surprising that he developed a system based on spheres following Pythagoras's belief that the sphere was the most perfect shape. The homocentric sphere system proposed by Eudoxus consisted of a number of rotating spheres, each sphere rotating about an axis through the centre of the Earth. The axis of rotation of each sphere was not fixed in space but, for most spheres, this axis was itself rotating as it was determined by points fixed on another rotating sphere.
插图：Eudoxus.gif ↗

As in the diagram on the right, suppose we have two spheres $S_{1}$ and $S_{2}$, the axis $XY$ of $S_{1}$ being a diameter of the sphere $S_{2}$. As $S_{2}$ rotates about an axis $AB$, then the axis $XY$ of $S_{1}$ rotates with it. If the two spheres rotate with constant, but opposite, angular velocity then a point $P$ on the equator of $S_{1}$ describes a figure of eight curve. This curve was called a hippopede (meaning a horse-fetter).

Eudoxus used this construction of the hippopede with two spheres and then considered a planet as the point $P$ traversing the curve. He introduced a third sphere to correspond to the general motion of the planet against the background stars while the motion round the hippopede produced the observed periodic retrograde motion. The three sphere subsystem was set into a fourth sphere which gave the daily rotation of the stars.

The planetary system of Eudoxus is described by Aristotle in Metaphysics and the complete system contains 27 spheres. Simplicius, writing a commentary on Aristotle in about 540 AD, also describes the spheres of Eudoxus. They represent a magnificent geometrical achievement. As Heath writes [3]:-

... to produce the retrogradations in this theoretical way by superimposed axial rotations of spheres was a remarkable stroke of genius. It was no slight geometrical achievement, for those days, to demonstrate the effect of the hypothesis; but this is nothing in comparison with the speculative power which enabled the man to invent the hypothesis which could produce the effect.

There is no doubting this incredible mathematical achievement. But there remain many questions which one must then ask. Did Eudoxus believe that the spheres actually existed? Did he invent them as a geometrical model which was purely a computational device? Did the model accurately represent the way the planets are observed to behave? Did Eudoxus test his model with observational evidence?

One argument in favour of thinking that Eudoxus believed in the spheres only as a computational device is the fact that he appears to have made no comment on the substance of the spheres nor on their mode of interconnection. One has to distinguish between Eudoxus's views and those of Aristotle for as Huxley writes in [1]:-

Eudoxus may have regarded his system simply as an abstract geometrical model, but Aristotle took it to be a description of the physical world...

The question of whether Eudoxus thought of his spheres as geometry or a physical reality is studied in the interesting paper [29] which argues that Eudoxus was more interested in actually representing the paths of the planets than in predicting astronomical phenomena.

Certainly the model does not represent, and perhaps more significantly could not represent, the actual paths of the planets with a degree of accuracy which would pass even the simplest of observational tests. As to the question of how much Eudoxus relied on observational data in verifying his hypothesis, Neugebauer writes in [7]:-

... not only do we not have evidence for numerical data in the construction of Eudoxus's homocentric spheres but it would also be difficult how his theory could have survived a comparison with observational parameters.

Perhaps it is just too modern a way of thinking to wonder how Eudoxus could have developed such an intricate theory without testing it out with observational data.

Many of the early commentators believed that Plato was the inspiration for Eudoxus's representation of planetary motion by his system of homocentric spheres. These view are still quite widely held but the article [19] argues convincingly that this is not so and that the ideas which influenced Eudoxus to come up with his masterpiece of 3-dimensional geometry were Pythagorean and not from Plato.

As a final comment we should note that Eudoxus also wrote a book on geography called Tour of the Earth which, although lost, is fairly well known through around 100 quotes in various sources. The work consisted of seven books and studied the peoples of the Earth known to Eudoxus, in particular examining their political systems, their history and background. Eudoxus wrote about Egypt and the religion of that country with particular authority and it is clear that he learnt much about that country in the year he spent there. In the seventh book Eudoxus wrote at length on the Pythagorean Society in Italy again about which he was clearly extremely knowledgeable.