阿里斯塔克斯Aristarchus of Samos

直到近代，阿里斯塔克斯似乎才得到了数学史家应有的关注。例如，托马斯·利特尔·希思在其希腊数学史第二卷的开头写道[5]：-

数学史家通常对阿里斯塔克斯关注太少。原因无疑在于他是一位天文学家，因此人们可能会认为他的工作对数学家没有足够的吸引力。希腊人更清楚；他们称他为“数学家阿里斯塔克斯”。

然而，他作为天文学家而非数学家而闻名的事实，被奥托·纽格包尔关于其工作[6]的说法所抵消：-

……这是一项纯数学练习，与实用天文学……关系不大……

Zhitomirskii在[14]中陈述：-

阿里斯塔克斯是尼古拉·哥白尼的一位鲜为人知但常被引用的先驱。关于他的所有信息都来自古典作家零散的少量提及，以及他的一篇未提及日心说的短论。因此，历史学家经常提到他，引用一两个事实，然后转向另一个主题——在提供几句解释之后，这些解释揭示了历史学家的许多偏见。

论文14随后认真尝试弥补作者认为其他历史学家的不足之处。让我们在本文中尝试做的不仅仅是“提及一两个事实”，而是既指出阿里斯塔克斯成就的规模和独创性，也指出他在数学天文学发展中的作用。

阿里斯塔克斯 无疑既是数学家又是天文学家，他最著名的是第一个提出日心宇宙。他还因开创性地尝试确定太阳和月亮的大小与距离而闻名。下面我们将考察这两项成就。

阿里斯塔克斯是兰普萨库斯的斯特拉托的学生，而斯特拉托是亚里士多德的吕克昂学园的负责人。然而，人们并不认为阿里斯塔克斯是在雅典跟随斯特拉托学习，而是认为他在亚历山大港跟随斯特拉托学习。公元前287年，斯特拉托成为亚历山大港吕克昂学园的负责人，据认为阿里斯塔克斯在那之后不久就开始在那里跟随他学习。

阿里斯塔克斯被维特鲁威（公元前1世纪）提及，后者作为罗马建筑师和工程师而闻名。维特鲁威是重要论著De architectura（《论建筑》）的作者，在这部著作中他列出了在各个科学分支中都有渊博知识的人（例如参见[3]、[4]或[5]）：-

这类人很罕见，例如过去时代的阿里斯塔克斯、塔伦图姆的菲洛劳斯和阿尔库塔斯、阿波罗尼奥斯的阿波罗尼奥斯、西奥多罗斯的埃拉托色尼、阿基米德和锡拉库萨的Scopinas，他们给后世留下了许多机械和日晷装置，这些是他们发明并根据数学原理加以解释的。

当然，直接的问题是阿里斯塔克斯发明了什么，维特鲁威解释说，他发明了一种半球碗形状的日晷，碗中央放置一根投下阴影的指针。

关于阿里斯塔克斯对heliocentric system的信念的起源，现有证据很少。我们不知道此前有这种类型的假说，但实际上该理论未被希腊人接受，因此显然从未流行。我们只知道阿里斯塔克斯的理论，是因为阿基米德的The Sand-Reckoner中有一句概述性陈述，以及普鲁塔克的类似提及。阿基米德写道（例如参见[3]、[4]或[5]，或参见[1]以获取更短的引文）：-

格隆王，你知道大多数天文学家把“宇宙”这个名称给予这样一个球体：其中心是地球的中心，而其半径等于太阳中心与地球中心之间的直线。这是你从天文学家那里听到的通常说法。但阿里斯塔克斯出版了一部由某些假设组成的书，其中作为所作假设的结果表明，宇宙比刚才提到的“宇宙”大许多倍。他的假设是：恒星和太阳保持不动，地球沿一个圆周绕太阳旋转，太阳位于轨道的中央，并且恒星所在的球体与太阳处于同一中心，它如此之大，以至于他设想地球在其上旋转的那个圆与恒星距离之比，如同球心与球面之比。

阿基米德在报告了阿里斯塔克斯的观点后，批评这些观点给出了在数学上无意义的比例。事实上，根据托马斯·利特尔·希思，阿里斯塔克斯表达其比例的方式与希腊著作中出现的其他表达方式相似，并表明阿里斯塔克斯认为恒星天球的半径与地球轨道相比是无限大的。当然，阿里斯塔克斯不得不做出某种这样的假设，否则视差效应就会可见。

普鲁塔克给了我们一点额外的信息，因为他报告说，阿里斯塔克斯跟随本都的赫拉克利德斯·彭提乌斯，相信恒星的每日视旋转是由于地球绕其轴旋转所致。

阿里斯塔克斯唯一存世的作品On the Sizes and Distances of the Sun and Moon并非基于日心说，不幸的是，阿基米德提到的他关于日心说的著作已经失传。On the Sizes and Distances of the Sun and Moon详细说明了他基于观察的非凡几何论证，据此他确定太阳与地球的距离约为月球与地球距离的20倍，太阳的大小约为月球的20倍。这两个估计都小了一个数量级，但错误在于阿里斯塔克斯缺乏精确仪器，而不在于他正确的推理方法。
插图：Aristarchus.gif ↗
该图显示了阿里斯塔克斯使用的一个论证。他知道月球靠反射阳光发光，因此他论证说，如果在月球恰好被照亮一半时测量月球与太阳之间的角度，那么就可以计算它们距离之比。阿里斯塔克斯估计半明时该角为87°，因此距离之比为sin 3°。当然，我们已将其转换为现代记号，因为阿里斯塔克斯既不使用度，三角学也尚未发明，所以他无法使用正弦函数。然而这实际上就是他所做的计算，原则上正确，但在实践中几乎不可能观测，因为确定月球半明发生的时刻只能非常不准确地找到。

阿里斯塔克斯随后面临计算用我们的记号表示为sin 3°的近似值的问题。他得到了不等式

$\large\frac{1}{18}\normalsize > \sin 3° > \large\frac{1}{20}\normalsize$

并推断出太阳距离是月亮距离的18到20倍。事实上在半明半暗的时刻，月亮与太阳之间的角度实际上是89° 50'，而太阳实际上比月亮远约400倍。

相当奇怪的是，阿里斯塔克斯 将太阳和月亮所张角的值取为 2°。这个数值很不准确，因为它大了四倍。他正确地利用日食和月食的证据指出太阳和月亮所张的角相同。然而，阿基米德 引用了太阳所张角的一个值为 $\large\frac{1}{2}\normalsize °$，并将这个数值归于 阿里斯塔克斯。我们只能假设 阿里斯塔克斯 在职业生涯早期写了 On the Sizes and Distances of the Sun and Moon，后来他采纳了以太阳为中心的宇宙假说，并计算出了太阳所张角的一个准确得多的值。人们不得不假设 阿里斯塔克斯 能够在职业生涯后期研制出仪器来进行精确的天文测量。

奥托·纽格包尔在[6]中论证说，阿里斯塔克斯对精确的天文数据不感兴趣（因为如果他感兴趣，他本可以轻松做得更好得多）。相反，奥托·纽格包尔暗示，阿里斯塔克斯只对寻找距离和直径背后的数学理论感兴趣。他在表明这样的测量可以进行，并且由于他成功地表明了这一点，他的工作具有重大重要性。正如奥托·纽格包尔在[6]中所写：-

……数学方法解决天文问题的力量已被极大地展示出来，其意义与一个世纪前的欧多克索斯相同，后者构建了可以与行星运动相关联的电影模型，而没有解决任何一个具体问题。

还有一两处关于阿里斯塔克斯工作的其他引用，最近已被研究。例如在[7]中，作者们解释：-

……公元2世纪一篇用希腊文写成的匿名评注中有一段难解的段落，评注针对的是荷马《奥德赛》第20卷。……[匿名]作者引用阿里斯塔克斯，而阿里斯塔克斯又引泰勒斯和Heraclitus，以支持他关于日食的论点……[他]关于日食可能发生时间的论点，依据的是对希腊和埃及历法惯例的分析，而不是诉诸对日食的观测。

Aristarchus of Samos does not seem to have had the attention from historians of mathematics which he deserved until recent times. For example Heath begins Volume II of his history of Greek mathematics with the words [5]:-

Historians of mathematics have, as a rule, given too little attention to Aristarchus of Samos. The reason is no doubt that he was an astronomer, and therefore it might be supposed that his work would have no sufficient interest for the mathematician. The Greeks knew better; they called him 'Aristarchus the mathematician'.

However the fact that he was known as an astronomer rather than a mathematician is rather countered by Neugebauer's claim that his work [6]:-

... is a purely mathematical exercise which has ... little to do with practical astronomy ...

Zhitomirskii, in [14], states:-

Aristarchus of Samos is a little-known but often cited precursor of Copernicus. All information about him derives from a handful of scattered references in Classical writers, plus a short treatise of his which does not mention heliocentrism. Accordingly historians often mention him, cite one or two facts and move on to another subject - after providing a few words of explanation that reveal much about the historians' biases.

The paper [14] then makes a serious attempt to remedy what the author considers to be the shortcomings of other historians. Let us try in this article to do more than 'mention one or two facts' and to indicate both the magnitude and originality of Aristarchus's achievements and also his role in the development of mathematical astronomy.

Aristarchus was certainly both a mathematician and astronomer and he is most celebrated as the first to propose a sun-centred universe. He is also famed for his pioneering attempt to determine the sizes and distances of the sun and moon. We shall look at these two achievements below.

Aristarchus was a student of Strato of Lampsacus, who was head of Aristotle's Lyceum. However, it is not thought that Aristarchus studied with Strato in Athens but rather that he studied with him in Alexandria. Strato became head of the Lyceum at Alexandria in 287 BC and it is thought that Aristarchus studied with him there starting his studies shortly after that date.

Aristarchus is mentioned by Vitruvius (1st century BC) who was famous as a Roman architect and engineer. Vitruvius was the author of the important treatise De architectura (On Architecture) and in this work he lists men who have been knowledgeable across all branches of science (see for example [3], [4], or [5]):-

Men of this type are rare, men such as were, in past times, Aristarchus of Samos, Philolaus and Archytas of Tarentum, Apollonius of Perga, Eratosthenes of Cyrene, Archimedes and Scopinas of Syracuse, who left to posterity many mechanical and gnomonic appliances which they invented and explained on mathematical principles.

Of course there is the immediate question of what Aristarchus invented, and Vitruvius explains that he invented a sundial in the shape of a hemispherical bowl with a pointer to cast shadows placed in the middle of the bowl.

There is little existing evidence concerning the origin of Aristarchus's belief in a heliocentric system. We know of no earlier hypothesis of this type but in fact the theory was not accepted by the Greeks so apparently never had any popularity. We only know of Aristarchus's theory because of a summary statement made in Archimedes' The Sand-Reckoner and a similar reference by Plutarch. Archimedes wrote (see for example [3], [4], or [5], or see [1] for a shorter quote):-

You King Gelon are aware the 'universe' is the name given by most astronomers to the sphere the centre of which is the centre of the earth, while its radius is equal to the straight line between the centre of the sun and the centre of the earth. This is the common account as you have heard from astronomers. But Aristarchus has brought out a book consisting of certain hypotheses, wherein it appears, as a consequence of the assumptions made, that the universe is many times greater than the 'universe' just mentioned. His hypotheses are that the fixed stars and the sun remain unmoved, that the earth revolves about the sun on the circumference of a circle, the sun lying in the middle of the orbit, and that the sphere of fixed stars, situated about the same centre as the sun, is so great that the circle in which he supposes the earth to revolve bears such a proportion to the distance of the fixed stars as the centre of the sphere bears to its surface.

Archimedes having reported the views of Aristarchus, criticised those views as giving mathematically meaningless proportions. In fact the way that Aristarchus expresses his proportions is, according to Heath, similar to other expressions which occur in Greek writings and indicated that Aristarchus considered that the radius of the sphere of the fixed stars was infinitely large compared with the orbit of the earth. Of course, Aristarchus had to make some such assumption, for otherwise parallax effects would be visible.

Plutarch gives us a little extra information, for he reports that Aristarchus followed Heraclides of Pontus in believing that the apparent daily rotation of the fixed stars was due to the rotation of the earth on its axis.

The only surviving work of Aristarchus, On the Sizes and Distances of the Sun and Moon, is not based on the sun centred theory and unfortunately his work on that sun centred theory referred to by Archimedes has been lost. On the Sizes and Distances of the Sun and Moon provides the details of his remarkable geometric argument, based on observation, whereby he determined that the Sun was about 20 times as distant from the Earth as the Moon, and 20 times the Moon's size. Both these estimates were an order of magnitude too small, but the fault was in Aristarchus's lack of accurate instruments rather than in his correct method of reasoning.
插图：Aristarchus.gif ↗
The diagram shows an argument used by Aristarchus. He knew that the moon shines by reflected sunlight, so he argued, if one measured the angle between the moon and sun when the moon is exactly half illuminated then one could compute the ratio of their distances. Aristarchus estimated that the angle at the time of half illumination was 87° so the ratio of the distances is sin 3°. Of course, we have translated this into modern notation for Aristarchus did not use degrees nor had trigonometry been invented so he did not have the sine function at his disposal. However this is in effect the calculation he made, correct in principle yet almost impossibly difficult to observe in practice since determining the moment at which half illumination of the moon occurs can only be very inaccurately found.

Aristarchus was then faced with calculating an approximation for what is in our notation sin 3°. He obtained the inequality

$\large\frac{1}{18}\normalsize > \sin 3° > \large\frac{1}{20}\normalsize$

and deduced that the sun was between 18 to 20 times as far away as the moon. In fact at the moment of half illumination the angle between the moon and the sun is actually 89° 50' and the sun is actually about 400 times further away than the moon.

Rather strangely Aristarchus uses values for the angle subtended by the sun and moon to be 2°. This figure is quite inaccurate as it is four times too large. He correctly uses the evidence of eclipses to state that the sun and moon subtend the same angle. However, Archimedes quotes a value of $\large\frac{1}{2}\normalsize °$ for the angle subtended by the sun and attributes this figure to Aristarchus. We can only assume that Aristarchus wrote On the Sizes and Distances of the Sun and Moon early in his career, then later on he adopted his hypothesis of a sun centred universe and computed a much more accurate value of the angle subtended by the sun. One has to assume Aristarchus was able to develop instruments to make accurate astronomical measurements later in his career.

Neugebauer argues in [6] that Aristarchus was not interested in accurate astronomical data (since he could have easily done very much better had he been interested). Rather Neugebauer suggests, Aristarchus was only interested in the mathematical theory behind finding the distances and diameters. He was showing that such measurements could be made and, since he succeeds in showing this, his work is of major importance. As Neugebauer writes in [6]:-

... the power of the mathematical approach to astronomical problems has been drastically demonstrated, in the same sense as Eudoxus, a century earlier, constructed cinematic models which could be related to planetary motions without solving a single specific problem.

There are one or two other references to work of Aristarchus which have been investigated recently. For example in [7] the authors interpret:-

... a difficult passage in an anonymous commentary written in Greek during the 2nd century AD on Book 20 of Homer's Odyssey. ... the [anonymous] author quotes Aristarchus of Samos citing Thales and Heraclitus in order to support his thesis [of] solar eclipses ... [His] thesis concerning the times when solar eclipses may occur rests on an analysis of Greek and Egyptian calendrical conventions, rather than on an appeal to observation of solar eclipses.