埃拉托色尼Eratosthenes

埃拉托色尼出生在埃拉托色尼，该地现在位于北非的利比亚。他的老师包括学者Lysanias埃拉托色尼和希俄斯的希波克拉底的哲学家Ariston，后者曾师从斯多亚派哲学学派的创始人Zeno。埃拉托色尼还师从诗人和学者Callimachus，后者也出生在埃拉托色尼。埃拉托色尼随后在雅典学习了几年。

亚历山大图书馆是由克劳狄乌斯·托勒密一世Soter规划的，该项目在他的儿子托勒密二世Philadelphus统治下得以实现。该图书馆以亚里士多德图书馆中的著作副本为基础。托勒密二世Philadelphus任命埃拉托色尼的老师之一Callimachus为第二任图书馆员。当托勒密三世Euergetes于公元前245年继承其父时，他说服埃拉托色尼前往亚历山大担任其子Philopator的导师。大约公元前240年Callimachus去世后，埃拉托色尼成为亚历山大图书馆的第三任图书馆员，该图书馆位于一座名为Mouseion的缪斯神庙中。据说该图书馆藏有数十万卷纸莎草纸和羊皮纸卷轴。

尽管埃拉托色尼是一位顶尖的百科全书式学者，却被认为未能达到最高等级。托马斯·利特尔·希思写道[4]：-

[埃拉托色尼]确实被同时代人公认为在知识的所有分支中都有卓越成就的人，尽管在每一门学科中他都恰好未能达到最高地位。由于后一原因，他被称为Beta，另一个加在他身上的绰号Pentathlos也有同样的含义，因为它代表一位全能运动员，他不是赛跑或摔跤的第一名，而是在这些比赛以及其他比赛中都获得第二名。

当然，对于一个在许多不同领域中的成就今天不仅被铭记为具有历史重要性，而且在许多情况下仍然为现代科学方法提供基础的人来说，这是一个苛刻的绰号。

埃拉托色尼的重要著作之一是Platonicus，它论述了柏拉图哲学背后的数学。士麦那的塞翁在撰写Expositio rerum mathematicarum时大量使用了这部著作，而且，尽管Platonicus现已失传，士麦那的塞翁告诉我们埃拉托色尼的著作研究了几何学和算术的基本定义，并涵盖了音乐等主题。

关于 埃拉托色尼 的一个相当令人惊讶的信息来源是一封伪造的信件。在其对 阿基米德 的 Sphere and cylinder 第二卷命题1的评注中，阿什凯隆的欧托基奥斯 复制了一封据称由 埃拉托色尼 写给 托勒密 III Euergetes 的信。这封信描述了 duplication of the cube 问题的历史，特别是描述了 埃拉托色尼 发明的一种机械装置，用于找到线段 $x$ 和 $y$，使得对于给定的线段 $a$ 和 $b$，

$a : x = x : y = y : b$。

根据希俄斯的希波克拉底的著名结果，已知解决在一个数与其两倍之间求两个比例中项的问题等价于解决倍立方问题。尽管这封信是伪造的，但其中部分内容取自埃拉托色尼自己的著作。这封信在数学史上占有重要地位，在[14]中有详细讨论。这封信的一份原始阿拉伯文文本曾保存在贝鲁特圣约瑟夫大学的图书馆中。然而它现在已经消失，[14]中给出的细节来自该信消失前所拍摄的照片。

关于埃拉托色尼在Platonicus中所写内容的其它细节由士麦那的塞翁给出。他特别在那里描述了倍立方问题的历史（见托马斯·利特尔·希思[4]）：-

……当神通过神谕向提洛人宣告，为了摆脱瘟疫，他们应当建造一个体积为现有祭坛两倍的祭坛时，他们的工匠在努力发现如何使一个立体成为相似立体的两倍时陷入了极大的困惑；因此他们去询问柏拉图，他回答说，神谕的意思并不是神想要一个体积加倍的祭坛，而是他希望借此给他们布置任务，以使希腊人因忽视数学和轻视几何而感到羞愧。

埃拉托色尼在亚历山大里亚立了一根柱子，上面刻有一首铭文诗，涉及他自己对倍立方问题的机械解法[4]：-

如果，好朋友，你想从任何小立方体得到一个体积为其两倍的立方体，并恰当地将任何立体图形变为另一个，这就在你的能力范围内；你可以通过这种方法求出一块羊圈、一个坑或一个空心井的宽阔盆地的度量，也就是说，如果你这样在两个直尺之间截取两个比例中项，使其两端汇聚。你不要去寻求做阿尔库塔斯的圆柱体这一困难工作，也不要在梅内克穆斯的三元组中切割圆锥，也不要实现敬畏神的欧多克索斯所描述的那种曲线形式。不，你可以在这些 tablets 上，从一个小底开始，轻松找到无数个比例中项。你是幸福的，托勒密，因为，作为父亲，你在青春活力上与儿子相等，你自己已经给了他所有对缪斯和国王们珍贵的东西，并且将来，哦宙斯，天空之神，也可能从你手中接受权杖。但愿如此，并让任何看到这奉献物的人说“这是埃拉托色尼的礼物”。

埃拉托色尼还研究了素数。他因他的素数筛法，即‘埃拉托色尼筛法’而被铭记，这种筛法经修改后，仍然是数论研究中的重要工具。该筛法出现在尼科米迪斯的Introduction to arithmetic中。

埃拉托色尼写的另一本书是On means，虽然它现在已失传，但帕普斯提到它是几何学伟大著作之一。然而，在大地测量学领域，埃拉托色尼将永远因他对地球的测量而被铭记。

埃拉托色尼对地球周长做出了惊人准确的测量。细节在他现已失传的论著On the measurement of the Earth中给出。然而，这些计算的一些细节出现在其他作者如克莱奥迈季斯、士麦那的塞翁和斯特拉波的著作中。埃拉托色尼比较了夏至时赛伊尼（今埃及尼罗河上的阿斯旺）和亚历山大里亚之间的正午阴影。他假设太阳如此遥远，以至于其光线基本上是平行的，然后根据赛伊尼和亚历山大里亚之间距离的知识，他给出地球周长为250,000stadia。

当然，这个值有多准确取决于 stadion 的长度，学者们对此争论了很长时间。文章[11]讨论了学者们给出的 stadion 的各种值。确实，埃拉托色尼得到了一个好结果，如果像一些人从普林尼给出的值推断的那样取 stadion 为157.2米，甚至是一个非凡的结果。如果像 Gulbekian 在[11]中建议的那样，埃拉托色尼使用的值是166.7米，那就不那么好。

引用的几篇论文，例如[10]、[15]和[16]，讨论了埃拉托色尼结果的准确性。论文[15]特别有趣。在其中，Rawlins 令人信服地论证，埃拉托色尼在计算中自己所做的唯一测量是亚历山大里亚夏至至点的天顶距，并且他得到了7°12'的值。Rawlins 论证说，这有16'的误差，而埃拉托色尼使用的来自未知来源的其他数据则准确得多。

埃拉托色尼还测量了到太阳的距离为804,000,000 stadia，到月球的距离为780,000 stadia。他使用月食期间获得的数据计算了这些距离。克劳狄乌斯·托勒密告诉我们，埃拉托色尼非常准确地测量了地轴的倾斜，得到了$\large\frac{11}{83}\normalsize$ of 180°的值，即23° 51' 15"。

值$\large\frac{11}{83}\normalsize$令数学史家着迷，例如论文[9]和[17]就是为了考察这个值的来源而写的。也许最普遍的观点是，值$\large\frac{11}{83}\normalsize$归功于克劳狄乌斯·托勒密而非埃拉托色尼。托马斯·利特尔·希思[4]认为埃拉托色尼使用了24°，而180°的$\large\frac{11}{83}\normalsize$是克劳狄乌斯·托勒密的改进。Taisbak[17]同意将$\large\frac{11}{83}\normalsize$归功于克劳狄乌斯·托勒密，尽管他认为埃拉托色尼使用了180°的值$\large\frac{2}{15}\normalsize$。然而Rawlins[15]认为使用了连分数方法来计算值$\large\frac{11}{83}\normalsize$，而Fowler[9]提出使用了anthyphairesis（或欧几里得算法）方法（另见[3]）。

埃拉托色尼对科学进步还做出了许多其他重大贡献。他制定了一部包含闰年的历法，并且当他试图给出从特洛伊围城时期起的文学和政治事件的日期时，他为世界系统年代学奠定了基础。据说他还编纂了一部包含675颗恒星的星表。

埃拉托色尼对地理学做出了重大贡献。他相当准确地描绘了尼罗河到喀土穆的路线，标出了两条埃塞俄比亚支流。他还提出湖泊是这条河的源头。在埃拉托色尼之前，许多学者已经研究过尼罗河，他们试图解释这条河相当奇怪的行为，但大多数人像泰勒斯一样，其解释完全错误。埃拉托色尼是第一个给出本质上正确答案的人，他提出河流源头附近地区有时会下大雨，这可以解释下游的洪水。埃拉托色尼对地理学的另一贡献是他描述了“Eudaimon Arabia”地区，即现在的也门，居住着四个不同的种族。情况比埃拉托色尼所提出的要复杂一些，但今天埃拉托色尼提出的种族名称，即Minaeans、Sabaeans、Qatabanians和Hadramites，仍在使用。

埃拉托色尼的著作包括受天文学启发的诗作Hermes，以及关于戏剧和伦理学的文学作品，后者是希腊人喜爱的话题。据说埃拉托色尼晚年失明，并且有人声称他绝食自杀。

Eratosthenes was born in Cyrene which is now in Libya in North Africa. His teachers included the scholar Lysanias of Cyrene and the philosopher Ariston of Chios who had studied under Zeno, the founder of the Stoic school of philosophy. Eratosthenes also studied under the poet and scholar Callimachus who had also been born in Cyrene. Eratosthenes then spent some years studying in Athens.

The library at Alexandria was planned by Ptolemy I Soter and the project came to fruition under his son Ptolemy II Philadelphus. The library was based on copies of the works in the library of Aristotle. Ptolemy II Philadelphus appointed one of Eratosthenes' teachers Callimachus as the second librarian. When Ptolemy III Euergetes succeeded his father in 245 BC and he persuaded Eratosthenes to go to Alexandria as the tutor of his son Philopator. On the death of Callimachus in about 240 BC, Eratosthenes became the third librarian at Alexandria, in the library in a temple of the Muses called the Mouseion. The library is said to have contained hundreds of thousands of papyrus and vellum scrolls.

Despite being a leading all-round scholar, Eratosthenes was considered to fall short of the highest rank. Heath writes [4]:-

[Eratosthenes] was, indeed, recognised by his contemporaries as a man of great distinction in all branches of knowledge, though in each subject he just fell short of the highest place. On the latter ground he was called Beta, and another nickname applied to him, Pentathlos, has the same implication, representing as it does an all-round athlete who was not the first runner or wrestler but took the second prize in these contests as well as others.

Certainly this is a harsh nickname to give to a man whose accomplishments in many different areas are remembered today not only as historically important but, remarkably in many cases, still providing a basis for modern scientific methods.

One of the important works of Eratosthenes was Platonicus which dealt with the mathematics which underlie Plato's philosophy. This work was heavily used by Theon of Smyrna when he wrote Expositio rerum mathematicarum and, although Platonicus is now lost, Theon of Smyrna tells us that Eratosthenes' work studied the basic definitions of geometry and arithmetic, as well as covering such topics as music.

One rather surprising source of information concerning Eratosthenes is from a forged letter. In his commentary on Proposition 1 of Archimedes' Sphere and cylinder Book II, Eutocius reproduces a letter reputed to have been written by Eratosthenes to Ptolemy III Euergetes. The letter describes the history of the problem of the duplication of the cube and, in particular, it describes a mechanical device invented by Eratosthenes to find line segments $x$ and $y$ so that, for given segments $a$ and $b$,

$a : x = x : y = y : b$.

By the famous result of Hippocrates it was known that solving the problem of finding two mean proportionals between a number and its double was equivalent to solving the problem of duplicating the cube. Although the letter is a forgery, parts of it are taken from Eratosthenes' own writing. The letter, which occupies an important place in the history of mathematics, is discussed in detail in [14]. An original Arabic text of this letter was once kept in the library of the St Joseph University in Beirut. However it has now vanished and the details given in [14] come from photographs taken of the letter before its disappearance.

Other details of what Eratosthenes wrote in Platonicus are given by Theon of Smyrna. In particular he described there the history of the problem of duplicating the cube (see Heath [4]):-

... when the god proclaimed to the Delians through the oracle that, in order to get rid of a plague, they should construct an alter double that of the existing one, their craftsmen fell into great perplexity in their efforts to discover how a solid could be made the double of a similar solid; they therefore went to ask Plato about it, and he replied that the oracle meant, not that the god wanted an alter of double the size, but that he wished, in setting them the task, to shame the Greeks for their neglect of mathematics and their contempt of geometry.

Eratosthenes erected a column at Alexandria with an epigram inscribed on it relating to his own mechanical solution to the problem of doubling the cube [4]:-

If, good friend, thou mindest to obtain from any small cube a cube the double of it, and duly to change any solid figure into another, this is in thy power; thou canst find the measure of a fold, a pit, or the broad basin of a hollow well, by this method, that is, if thou thus catch between two rulers two means with their extreme ends converging. Do not thou seek to do the difficult business of Archytas's cylinders, or to cut the cone in the triads of Menaechmus, or to compass such a curved form of lines as is described by the god-fearing Eudoxus. Nay thou couldst, on these tablets, easily find a myriad of means, beginning from a small base. Happy art thou, Ptolemy, in that, as a father the equal of his son in youthful vigour, thou hast thyself given him all that is dear to muses and Kings, and may be in the future, O Zeus, god of heaven, also receive the sceptre at thy hands. Thus may it be, and let any one who sees this offering say "This is the gift of Eratosthenes of Cyrene".

Eratosthenes also worked on prime numbers. He is remembered for his prime number sieve, the 'Sieve of Eratosthenes' which, in modified form, is still an important tool in number theory research. The sieve appears in the Introduction to arithmetic by Nicomedes.

Another book written by Eratosthenes was On means and, although it is now lost, it is mentioned by Pappus as one of the great books of geometry. In the field of geodesy, however, Eratosthenes will always be remembered for his measurements of the Earth.

Eratosthenes made a surprisingly accurate measurement of the circumference of the Earth. Details were given in his treatise On the measurement of the Earth which is now lost. However, some details of these calculations appear in works by other authors such as Cleomedes, Theon of Smyrna and Strabo. Eratosthenes compared the noon shadow at midsummer between Syene (now Aswan on the Nile in Egypt) and Alexandria. He assumed that the sun was so far away that its rays were essentially parallel, and then with a knowledge of the distance between Syene and Alexandria, he gave the length of the circumference of the Earth as 250,000 stadia.

Of course how accurate this value is depends on the length of the stadium and scholars have argued over this for a long time. The article [11] discusses the various values scholars have given for the stadium. It is certainly true that Eratosthenes obtained a good result, even a remarkable result if one takes 157.2 metres for the stadium as some have deduced from values given by Pliny. It is less good if 166.7 metres was the value used by Eratosthenes as Gulbekian suggests in [11].

Several of the papers referenced, for example [10], [15] and [16], discuss the accuracy of Eratosthenes' result. The paper [15] is particularly interesting. In it Rawlins argues convincingly that the only measurement which Eratosthenes made himself in his calculations was the zenith distance on the summer solstice at Alexandria, and that he obtained the value of 7°12'. Rawlins argues that this is in error by 16' while other data which Eratosthenes used, from unknown sources, was considerably more accurate.

Eratosthenes also measured the distance to the sun as 804,000,000 stadia and the distance to the Moon as 780,000 stadia. He computed these distances using data obtained during lunar eclipses. Ptolemy tells us that Eratosthenes measured the tilt of the Earth's axis with great accuracy obtaining the value of $\large\frac{11}{83}\normalsize$ of 180°, namely 23° 51' 15".

The value $\large\frac{11}{83}\normalsize$ has fascinated historians of mathematics, for example the papers [9] and [17] are written just to examine the source of this value. Perhaps the most commonly held view is that the value $\large\frac{11}{83}\normalsize$ is due to Ptolemy and not to Eratosthenes. Heath [4] argues that Eratosthenes used 24° and that $\large\frac{11}{83}\normalsize$ of 180° was a refinement due to Ptolemy. Taisbak [17] agrees with attributing $\large\frac{11}{83}\normalsize$ to Ptolemy although he believes that Eratosthenes used the value $\large\frac{2}{15}\normalsize$ of 180°. However Rawlins [15] believes that a continued fraction method was used to calculate the value $\large\frac{11}{83}\normalsize$ while Fowler [9] proposes that the anthyphairesis (or Euclidean algorithm) method was used (see also [3]).

Eratosthenes made many other major contributions to the progress of science. He worked out a calendar that included leap years, and he laid the foundations of a systematic chronography of the world when he tried to give the dates of literary and political events from the time of the siege of Troy. He is also said to have compiled a star catalogue containing 675 stars.

Eratosthenes made major contributions to geography. He sketched, quite accurately, the route of the Nile to Khartoum, showing the two Ethiopian tributaries. He also suggested that lakes were the source of the river. A study of the Nile had been made by many scholars before Eratosthenes and they had attempted to explain the rather strange behaviour of the river, but most like Thales were quite wrong in their explanations. Eratosthenes was the first to give what is essentially the correct answer when he suggested that heavy rains sometimes fell in regions near the source of the river and that these would explain the flooding lower down the river. Another contribution that Eratosthenes made to geography was his description of the region "Eudaimon Arabia", now the Yemen, as inhabited by four different races. The situation was somewhat more complicated than that proposed by Eratosthenes, but today the names for the races proposed by Eratosthenes, namely Minaeans, Sabaeans, Qatabanians, and Hadramites, are still used.

Eratosthenes writings include the poem Hermes, inspired by astronomy, as well as literary works on the theatre and on ethics which was a favourite topic of the Greeks. Eratosthenes is said to have became blind in old age and it has been claimed that he committed suicide by starvation.