克劳狄乌斯·托勒密Ptolemy

作为他那个时代最有影响力的希腊天文学家和地理学家之一，克劳狄乌斯·托勒密提出了一种地心说理论，其形式盛行了1400年。然而，在所有古希腊数学家中，可以公平地说，他的著作引发了比其他任何人都更多的讨论和争论。我们将在下面讨论这些争论，因为根据哪些是正确的，它们对托勒密的描绘截然不同。一些历史学家的论据表明托勒密是一位顶级数学家，另一些人的论据表明他不过是一位出色的阐述者，但更糟糕的是，有些人甚至声称他背叛了伦理学和他职业的诚信，从而对他的科学家同行犯下了罪行。

我们对托勒密的生平知之甚少。他在公元127-41年间从埃及的亚历山大进行了天文观测。事实上，我们能确切确定日期的第一次观测是由托勒密于127年3月26日进行的，而最后一次是在141年2月2日进行的。大约1360年，Theodore Meliteniotes声称托勒密出生在Hermiou（位于上埃及，而不是亚历山大所在的下埃及），但由于这一说法首次出现于托勒密生活的一千多年之后，因此必须将其视为相对不太可能为真。事实上，没有证据表明托勒密曾经去过亚历山大以外的任何地方。

他的名字，Claudius 托勒密，当然是希腊埃及的'托勒密'和罗马的'Claudius'的混合。这表明他是一位居住在埃及的希腊家庭的后裔，并且他是罗马公民，这可能是由于一位罗马皇帝将这一'奖赏'授予了托勒密的一位祖先。

我们确实知道托勒密使用了'数学家Theon'所做的观测，而这几乎可以肯定是士麦那的塞翁，他几乎可以肯定是他的老师。这当然说得通，因为士麦那的塞翁既是一位观测者，也是一位数学家，曾撰写过关于合、日食、掩星和中天等天文学主题的著作。托勒密的大部分早期著作都献给了Syrus，Syrus可能也是他在亚历山大的老师之一，但关于Syrus一无所知。

如果关于托勒密的老师的这些事实是正确的，那么可以肯定在士麦那的塞翁他没有遇到一位伟大的学者，因为士麦那的塞翁似乎并未深入理解他所描述的天文学工作。另一方面，亚历山大里亚有着学术传统，这意味着即使托勒密无法接触到最好的老师，他也能接触到图书馆，在那里他会找到他所充分利用的宝贵参考资料。

托勒密的主要著作流传了下来，我们将在本文中讨论它们。然而最重要的是AlmagestⓉ（主要论题：来自阿拉伯语“al-majisti”——希腊语“Mathematike Syntaxis”的阿拉伯语翻译，后来译为拉丁语“Magna Syntaxis”），这是一部十三卷的论著。我们应当直接说明，尽管这部著作现在几乎总是被称为Almagest，但那并非它最初的名字。它最初的希腊语标题翻译为The Mathematical Compilation，但这个标题很快被另一个希腊语标题取代，意为The Greatest Compilation。这个标题被翻译成阿拉伯语为“al-majisti”，由此当这部著作从阿拉伯语翻译成拉丁语时，被赋予了Almagest这个标题。

Almagest是托勒密最早的作品，详细给出了太阳、月亮和行星运动的数学理论。托勒密通过给出每颗行星运动的细节，做出了他最原创的贡献。直到尼古拉·哥白尼于1543年在De revolutionibus中提出他的日心说一个世纪之后，Almagest才被取代。Grasshoff在[8]中写道：-

托勒密的《天文学大成》与欧几里得的《几何原本》共享着作为使用时间最长的科学文本的荣耀。从公元二世纪它的构想直到文艺复兴晚期，这部著作将天文学确立为一门科学。在此期间，《天文学大成》不仅仅是一部天文学著作；这门学科被定义为《天文学大成》中所描述的内容。

托勒密非常清楚地描述了自己在写作这部著作时试图做什么（例如见[15]）：-

我们将尝试记下我们认为迄今所发现的一切；我们将尽可能简洁地这样做，并以那些已在该领域取得一些进展的人能够遵循的方式。为了我们论述的完整性，我们将按适当顺序列出对天体理论有用的一切，但为了避免不必要的冗长，我们只叙述古人已充分确立的内容。然而，那些我们的前辈完全没有处理过、或处理得不如本可以的那样有用的主题，我们将尽我们所能详细讨论。

托勒密首先为他基于亚里士多德所描述的地心体系的宇宙描述进行辩护。这是一种基于固定地球的世界观，恒星天球每天绕地球旋转，并带动太阳、月亮和行星的天球。托勒密使用几何模型来预测太阳、月亮和行星的位置，使用被称为epicycles的圆周运动的组合。建立了这个模型之后，托勒密接着描述他在其余著作中所需的数学。特别是他引入了基于弦函数Crd的三角学方法（弦函数通过$\sin a = \large\frac{1}{120}\normalsize$(Crd 2$a$)与正弦函数相关）。

托勒密设计了新的几何证明和定理。他利用圆的弦和一个内接360边形，得到了近似值

$\pi = 3\large\frac{17}{120}\normalsize = 3.14166$

并且，利用 √3 = 弦 60°，

$√3 = 1.73205.$

他使用了与我们的 $\sin(a + b)$ 和 $\sin(a - b)$ 公式类似的 Crd 函数公式，来制作一份以 $\large\frac{1}{2}\normalsize$ 度为间隔的 Crd 函数表。

他尤其以THIS LINK处的定理闻名，他用该定理推导出了上述加法公式。

这占据了 Almagest 十三卷中的前两卷，然后，再次引用导言，我们给出 托勒密 自己对他在该著作中打算如何展开其余数学天文学内容的描述（例如见 [15]）：-

[在引入数学概念之后]我们必须逐一讨论太阳和月球的运动，以及伴随这些运动的现象；因为如果不首先掌握这些事项，就不可能彻底考察恒星的理論。我们这种进路的最终任务是恒星的理论。在这里，也宜先处理所谓‘恒星’的球，然后再处理那五颗所谓的‘行星’。

在考察太阳理论时，托勒密 将他本人对 二分点 的观测与 喜帕恰斯 的观测以及公元前 432 年 Meton 的更早观测进行了比较。他确认 回归年 的长度为比 $365\large\frac{1}{4}\normalsize$ 日少 $\large\frac{1}{300}\normalsize$ 日，这正是 喜帕恰斯 所得到的精确值。由于正如 托勒密 本人所知，他其余数据的准确性在很大程度上取决于这个值，而真实值比 $365\large\frac{1}{4}\normalsize$ 日少 $\large\frac{1}{128}\normalsize$ 日这一事实确实在其余工作中产生了误差。我们将在下文更详细地讨论针对 托勒密 提出的指控，但这清楚地说明了这些指控的理由，因为 托勒密 必须在其分点观测中有 28 小时的误差才能产生这一误差，而即使考虑到古代仪器和方法所能期望的精度，他竟会犯下如此量级的误差，本质上也是令人难以置信的。对这一奇怪误差的良好讨论包含在优秀文章 [19] 中。

基于他对solstices和分点的观测，托勒密发现了季节的长度，并基于这些，他提出了一个简单的太阳模型，即均匀角速度的圆周运动，但地球不在圆的中心，而是在距此中心一个称为偏心距的距离处。这个太阳理论构成了Almagest第3卷的主题。

在第4卷和第5卷中，托勒密给出了他的月亮理论。在这里，他遵循喜帕恰斯，后者研究了三个可以与月亮运动相关联的不同周期。有月亮返回到相同经度所需的时间，返回到相同速度（异常）所需的时间，以及返回到相同纬度所需的时间。托勒密还像喜帕恰斯所做的那样，讨论了朔望月，即太阳和月亮连续对当关系之间的时间。在第4卷中，托勒密给出了喜帕恰斯的月亮运动本轮模型，但他注意到，正如事实上喜帕恰斯自己已经做的那样，模型与观测参数之间存在小的差异。尽管注意到了这些差异，喜帕恰斯似乎没有制定出更好的模型，但托勒密在第5卷中做到了这一点，在那里他给出的模型显著改进了喜帕恰斯提出的模型。关于托勒密月亮理论的有趣讨论见[24]。

在给出太阳和月球运动的理论之后，托勒密 已能够应用这些理论来获得食的理论，他在第 6 卷中就是这样做的。接下来的两卷讨论恒星，而在第 7 卷中 托勒密 利用他自己的观测以及 喜帕恰斯 的观测来证明他的信念：恒星彼此之间始终保持相同的位置。他写道（例如见 [15]）：-

如果将上述对齐方式与喜帕恰斯天球仪上构成星座的图样进行比对，他会发现，根据他所记录的，在喜帕恰斯时代所作的观测在天球仪上得出的相关恒星位置，与现在几乎相同。

在这两卷书中，托勒密还讨论了岁差，他将其发现归功于喜帕恰斯，但他的数字有些错误，主要是因为他所使用的回归年长度有误。第7卷和第8卷的大部分内容都用于介绍托勒密的星表，其中包含一千多颗恒星。

Almagest的最后五卷讨论行星理论。就原创性贡献而言，这必定是托勒密最伟大的成就，因为在Almagest之前，似乎没有任何令人满意的理论模型来解释五颗行星相当复杂的运动。托勒密结合了本轮和eccentric methods，给出了他的行星运动模型。因此，行星$P$的路径由本轮上的圆周运动组成，本轮的中心$C$绕一个圆运动，该圆的中心偏离地球。托勒密在这里真正巧妙的创新是使$C$的运动不是围绕它所绕圆的中心均匀，而是围绕一个称为等分点的点均匀，该点对称地放置在中心与地球相对的一侧。

托勒密在这里发展的行星理论是一部杰作。他创造了一个精密的数学模型来拟合观测数据，而在托勒密之前，这些数据还很稀少；他建立的模型虽然复杂，却相当好地表示了行星的运动。

Toomer在[1]中对Almagest总结如下：-

作为一部教学著作，《天文学大成》在清晰性和方法上是一部杰作，优于任何古代科学教科书，在任何时代都鲜有匹敌者。但它远不止于此。它并非像有时被描述的那样仅仅是对早期希腊天文学的‘系统化’，在许多方面它是一部原创性著作。

在简要评论托勒密的其他著作之后，我们将回头讨论一些针对他的指责。他将散布于Almagest各处的表格单独出版，题为Handy Tables。然而，这些表格并非仅仅从Almagest中摘录而来，托勒密在它们的呈现方式、易用性方面做了大量改进，他甚至改进了基本参数以提高精度。我们仅通过亚历山大的席恩的评注才知道Handy Tables的细节，但在[76]中，作者表明需要谨慎，因为亚历山大的席恩并未完全了解托勒密的方法。

托勒密还做了许多撰写深奥科学著作的作家曾经做过、并且仍在做的事：以Planetary Hypothesis为题写了一个通俗的成果介绍。这部著作分两卷，同样遵循读者所需数学技能较低的常见路子。托勒密做得相当巧妙，他用机械理论代替抽象的几何理论。托勒密还写了一部关于占星术的著作。对现代读者来说，一个写出如此优秀科学著作的人竟然写占星术，可能显得奇怪。然而，托勒密的看法颇为不同，因为他声称Almagest使人能够求出天体的位置，而他把自己的占星术著作看作一部姊妹作，描述天体对人生活的影响。

在一本名为Analemma的书中，他讨论了构造日晷所需角度的求法，这涉及天球上点的投影。在Planisphaerium中，他关注的是天球在平面上的球极投影。这在[48]中有所讨论，其中写道：-

在托勒密于⟦E1⟧中处理的球极投影中，天球通过从南极投影映射到赤道平面上。托勒密没有证明球面上的圆在平面上成为圆这一重要性质。

托勒密的主要著作Geography共八卷，试图绘制已知世界的地图，给出主要地点的纬度和经度坐标。托勒密给出的地图在许多地方相当不准确，这并不奇怪，因为不能指望他做得比利用现有数据更多，而这些数据对于罗马帝国以外的任何地方来说质量都非常差，甚至罗马帝国的部分地区也严重失真。在[19]中，托勒密被描述为：-

……一个在没有发达理论支持的情况下工作[于地图制作]的人，但身处数学传统之中，并由他对问题适宜性的感觉所引导。

另一部关于Optics的著作共五卷，在其中托勒密研究了颜色、反射、折射以及各种形状的镜子。Toomer在[1]中评论道：-

通过实验建立理论，常常通过构造特殊仪器，是托勒密的《光学》最显著的特征。无论主题内容主要是派生还是原创，《光学》都是一个令人印象深刻的例子，展示了数学科学的发展对物理数据的适当关注，并且配得上《天文学大成》的作者。

在[14]中给出了一种英译，试图消除糟糕的阿拉伯语翻译中引入的不准确之处，而该阿拉伯语翻译是我们Optics的唯一来源。

第一个对托勒密提出指责的是第谷·布拉赫。他发现星表中的恒星经度存在一度左右的系统误差，并声称，尽管托勒密说这代表了他自己的观测，它只不过是把一份归于喜帕恰斯的星表按岁差改正到托勒密的时代。当然，比较两份星表存在确定的问题，其中一份我们还有副本，而另一份已经失传。

在皮埃尔·西蒙·拉普拉斯和拉朗德的评论之后，下一个猛烈攻击托勒密的是让·巴蒂斯特·约瑟夫·德朗布尔。他提出，错误可能来自喜帕恰斯，而托勒密可能除了未能根据二分点和二至点之间的时间校正喜帕恰斯的数据之外，没有犯更严重的错误。然而让·巴蒂斯特·约瑟夫·德朗布尔接着说道（见[8]）：-

人们可以用一种不那么有利但更简单的方式解释一切：否认托勒密观测了恒星和二分点，并声称他从喜帕恰斯那里吸收了一切，使用了后者关于岁差运动的最小值。

托勒密绝非没有支持者，进一步的分析使人们相信让·巴蒂斯特·约瑟夫·德朗布尔对托勒密的指控是错误的。Boll在1894年写道[4]：-

从一切表象来看，人们将不得不归功于托勒密，因为他在其杰出的前辈之后，对希腊天穹给出了本质上更丰富的图景。

Vogt在其重要论文[77]中清楚地表明，通过考虑喜帕恰斯的Commentary on Aratus and Eudoxus，并作出合理假设，即其中给出的数据与喜帕恰斯的星表一致，那么托勒密的星表不可能根据喜帕恰斯给出的恒星位置产生，除了少数几颗恒星，托勒密似乎确实从喜帕恰斯获取了数据。Vogt写道：-

这使我们能够将恒星表视为他自己制作的，正如托勒密本人强烈声明的那样。

最近针对托勒密的伪造指控来自艾萨克·牛顿，见[12]。他在本书开头明确陈述了自己的观点：-

这是一个科学犯罪的故事。……我指的是一个科学家对同行科学家和学者犯下的罪行，是对其职业伦理和诚信的背叛，永远剥夺了人类关于天文学和历史一个重要领域的基本信息。

在结尾处，牛顿声称证明了托勒密在Almagest中所声称的每一项观测都是伪造的，他写道[12]：-

[托勒密]发展了某些天文学理论，并发现它们与观测不一致。他没有放弃这些理论，而是故意根据理论伪造观测，以便声称观测证明了他的理论的有效性。在所有已知的科学或学术环境中，这种做法被称为欺诈，是对科学和学术的犯罪。

尽管第谷·布拉赫、让·巴蒂斯特·约瑟夫·德朗布尔、牛顿等人提供的证据确实表明托勒密的错误并非随机，但我[EFR]认为，[12]中的最后一段引文是对托勒密的犯罪（用牛顿自己的话来说）。[8]一书旨在研究这些指控的有效性，我坚信这部著作给出了正确的解释。Grasshoff写道：-

……人们必须假设，托勒密星表中相当大一部分是基于那些喜帕恰斯观测，而喜帕恰斯已经将这些观测用于编纂其《阿拉托斯评注》的第二部分。尽管不能排除星表中包含来自真正的托勒密观测的坐标，但它们不可能超过星表的一半。

……对喜帕恰斯观测的吸收已不能再放在剽窃的视角下讨论。托勒密的意图是发展一套关于天象的全面理论，他无法使用以算术平均来评估数据的方法，而现代天文学家正是用这种方法从一组各不相同的测量结果中得出检验一个假说所需的那一个代表值。因此，出于方法论上的原因，托勒密不得不从一组测量值中选出与他不得不视为最可靠数据最为相符的那一个值。当凭直觉在数据中作出选择已不再可能时……托勒密不得不把那些能够被理论预测所证实的值视为“观测到的”值。

作为最后的评论，我们引用一首被许多学者认为出自托勒密本人之手的短诗，它出现在Almagest第1卷目录之后（例如见[11]）：——

我深知自己是凡人，只活一日。
但若我的心灵追随星辰的蜿蜒轨迹，
我的双足便不再停驻于大地，而是立于
宙斯身旁，饱饮神馔，那不朽的仙食。

One of the most influential Greek astronomers and geographers of his time, Ptolemy propounded the geocentric theory in a form that prevailed for 1400 years. However, of all the ancient Greek mathematicians, it is fair to say that his work has generated more discussion and argument than any other. We shall discuss the arguments below for, depending on which are correct, they portray Ptolemy in very different lights. The arguments of some historians show that Ptolemy was a mathematician of the very top rank, arguments of others show that he was no more than a superb expositor, but far worse, some even claim that he committed a crime against his fellow scientists by betraying the ethics and integrity of his profession.

We know very little of Ptolemy's life. He made astronomical observations from Alexandria in Egypt during the years AD 127-41. In fact the first observation which we can date exactly was made by Ptolemy on 26 March 127 while the last was made on 2 February 141. It was claimed by Theodore Meliteniotes in around 1360 that Ptolemy was born in Hermiou (which is in Upper Egypt rather than Lower Egypt where Alexandria is situated) but since this claim first appears more than one thousand years after Ptolemy lived, it must be treated as relatively unlikely to be true. In fact there is no evidence that Ptolemy was ever anywhere other than Alexandria.

His name, Claudius Ptolemy, is of course a mixture of the Greek Egyptian 'Ptolemy' and the Roman 'Claudius'. This would indicate that he was descended from a Greek family living in Egypt and that he was a citizen of Rome, which would be as a result of a Roman emperor giving that 'reward' to one of Ptolemy's ancestors.

We do know that Ptolemy used observations made by 'Theon the mathematician', and this was almost certainly Theon of Smyrna who almost certainly was his teacher. Certainly this would make sense since Theon was both an observer and a mathematician who had written on astronomical topics such as conjunctions, eclipses, occultations and transits. Most of Ptolemy's early works are dedicated to Syrus who may have also been one of his teachers in Alexandria, but nothing is known of Syrus.

If these facts about Ptolemy's teachers are correct then certainly in Theon he did not have a great scholar, for Theon seems not to have understood in any depth the astronomical work he describes. On the other hand Alexandria had a tradition for scholarship which would mean that even if Ptolemy did not have access to the best teachers, he would have access to the libraries where he would have found the valuable reference material of which he made good use.

Ptolemy's major works have survived and we shall discuss them in this article. The most important, however, is the Almagest Ⓣ which is a treatise in thirteen books. We should say straight away that, although the work is now almost always known as the Almagest that was not its original name. Its original Greek title translates as The Mathematical Compilation but this title was soon replaced by another Greek title which means The Greatest Compilation. This was translated into Arabic as "al-majisti" and from this the title Almagest was given to the work when it was translated from Arabic to Latin.

The Almagest is the earliest of Ptolemy's works and gives in detail the mathematical theory of the motions of the Sun, Moon, and planets. Ptolemy made his most original contribution by presenting details for the motions of each of the planets. The Almagest was not superseded until a century after Copernicus presented his heliocentric theory in the De revolutionibus of 1543. Grasshoff writes in [8]:-

Ptolemy's "Almagest" shares with Euclid's "Elements" the glory of being the scientific text longest in use. From its conception in the second century up to the late Renaissance, this work determined astronomy as a science. During this time the "Almagest" was not only a work on astronomy; the subject was defined as what is described in the "Almagest".

Ptolemy describes himself very clearly what he is attempting to do in writing the work (see for example [15]):-

We shall try to note down everything which we think we have discovered up to the present time; we shall do this as concisely as possible and in a manner which can be followed by those who have already made some progress in the field. For the sake of completeness in our treatment we shall set out everything useful for the theory of the heavens in the proper order, but to avoid undue length we shall merely recount what has been adequately established by the ancients. However, those topics which have not been dealt with by our predecessors at all, or not as usefully as they might have been, will be discussed at length to the best of our ability.

Ptolemy first of all justifies his description of the universe based on the earth-centred system described by Aristotle. It is a view of the world based on a fixed earth around which the sphere of the fixed stars rotates every day, this carrying with it the spheres of the sun, moon, and planets. Ptolemy used geometric models to predict the positions of the sun, moon, and planets, using combinations of circular motion known as epicycles. Having set up this model, Ptolemy then goes on to describe the mathematics which he needs in the rest of the work. In particular he introduces trigonometrical methods based on the chord function Crd (which is related to the sine function by $\sin a = \large\frac{1}{120}\normalsize$(Crd 2$a$).

Ptolemy devised new geometrical proofs and theorems. He obtained, using chords of a circle and an inscribed 360-gon, the approximation

$\pi = 3\large\frac{17}{120}\normalsize = 3.14166$

and, using √3 = chord 60°,

$√3 = 1.73205.$

He used formulae for the Crd function which are analogous to our formulae for $\sin(a + b)$ and $\sin(a - b)$ to create a table of the Crd function at intervals of $\large\frac{1}{2}\normalsize$ a degree.

He is known in particular for the theorem at THIS LINK which he used to deduce the above addition formulae.

This occupies the first two of the 13 books of the Almagest and then, quoting again from the introduction, we give Ptolemy's own description of how he intended to develop the rest of the mathematical astronomy in the work (see for example [15]):-

[After introducing the mathematical concepts] we have to go through the motions of the sun and of the moon, and the phenomena accompanying these motions; for it would be impossible to examine the theory of the stars thoroughly without first having a grasp of these matters. Our final task in this way of approach is the theory of the stars. Here too it would be appropriate to deal first with the sphere of the so-called 'fixed stars', and follow that by treating the five 'planets', as they are called.

In examining the theory of the sun, Ptolemy compares his own observations of equinoxes with those of Hipparchus and the earlier observations Meton in 432 BC. He confirmed the length of the tropical year as $\large\frac{1}{300}\normalsize$ of a day less than $365\large\frac{1}{4}\normalsize$ days, the precise value obtained by Hipparchus. Since, as Ptolemy himself knew, the accuracy of the rest of his data depended heavily on this value, the fact that the true value is $\large\frac{1}{128}\normalsize$ of a day less than $365\large\frac{1}{4}\normalsize$ days did produce errors in the rest of the work. We shall discuss below in more detail the accusations which have been made against Ptolemy, but this illustrates clearly the grounds for these accusations since Ptolemy had to have an error of 28 hours in his observation of the equinox to produce this error, and even given the accuracy that could be expected with ancient instruments and methods, it is essentially unbelievable that he could have made an error of this magnitude. A good discussion of this strange error is contained in the excellent article [19].

Based on his observations of solstices and equinoxes, Ptolemy found the lengths of the seasons and, based on these, he proposed a simple model for the sun which was a circular motion of uniform angular velocity, but the earth was not at the centre of the circle but at a distance called the eccentricity from this centre. This theory of the sun forms the subject of Book 3 of the Almagest.

In Books 4 and 5 Ptolemy gives his theory of the moon. Here he follows Hipparchus who had studied three different periods which one could associate with the motion of the moon. There is the time taken for the moon to return to the same longitude, the time taken for it to return to the same velocity (the anomaly) and the time taken for it to return to the same latitude. Ptolemy also discusses, as Hipparchus had done, the synodic month, that is the time between successive oppositions of the sun and moon. In Book 4 Ptolemy gives Hipparchus's epicycle model for the motion of the moon but he notes, as in fact Hipparchus had done himself, that there are small discrepancies between the model and the observed parameters. Although noting the discrepancies, Hipparchus seems not to have worked out a better model, but Ptolemy does this in Book 5 where the model he gives improves markedly on the one proposed by Hipparchus. An interesting discussion of Ptolemy's theory of the moon is given in [24].

Having given a theory for the motion of the sun and of the moon, Ptolemy was in a position to apply these to obtain a theory of eclipses which he does in Book 6. The next two books deal with the fixed stars and in Book 7 Ptolemy uses his own observations together with those of Hipparchus to justify his belief that the fixed stars always maintain the same positions relative to each other. He wrote (see for example [15]):-

If one were to match the above alignments against the diagrams forming the constellations on Hipparchus's celestial globe, he would find that the positions of the relevant stars on the globe resulting from the observations made at the time of Hipparchus, according to what he recorded, are very nearly the same as at present.

In these two book Ptolemy also discusses precession, the discovery of which he attributes to Hipparchus, but his figure is somewhat in error mainly because of the error in the length of the tropical year which he used. Much of Books 7 and 8 are taken up with Ptolemy's star catalogue containing over one thousand stars.

The final five books of the Almagest discuss planetary theory. This must be Ptolemy's greatest achievement in terms of an original contribution, since there does not appear to have been any satisfactory theoretical model to explain the rather complicated motions of the five planets before the Almagest. Ptolemy combined the epicycle and eccentric methods to give his model for the motions of the planets. The path of a planet $P$ therefore consisted of circular motion on an epicycle, the centre $C$ of the epicycle moving round a circle whose centre was offset from the earth. Ptolemy's really clever innovation here was to make the motion of $C$ uniform not about the centre of the circle around which it moves, but around a point called the equant which is symmetrically placed on the opposite side of the centre from the earth.

The planetary theory which Ptolemy developed here is a masterpiece. He created a sophisticated mathematical model to fit observational data which before Ptolemy's time was scarce, and the model he produced, although complicated, represents the motions of the planets fairly well.

Toomer sums up the Almagest in [1] as follows:-

As a didactic work the "Almagest" is a masterpiece of clarity and method, superior to any ancient scientific textbook and with few peers from any period. But it is much more than that. Far from being a mere 'systemisation' of earlier Greek astronomy, as it is sometimes described, it is in many respects an original work.

We will return to discuss some of the accusations made against Ptolemy after commenting briefly on his other works. He published the tables which are scattered throughout the Almagest separately under the title Handy Tables. These were not merely lifted from the Almagest however but Ptolemy made numerous improvements in their presentation, ease of use and he even made improvements in the basic parameters to give greater accuracy. We only know details of the Handy Tables through the commentary by Theon of Alexandria but in [76] the author shows that care is required since Theon was not fully aware of Ptolemy's procedures.

Ptolemy also did what many writers of deep scientific works have done, and still do, in writing a popular account of his results under the title Planetary Hypothesis. This work, in two books, again follows the familiar route of reducing the mathematical skills needed by a reader. Ptolemy does this rather cleverly by replacing the abstract geometrical theories by mechanical ones. Ptolemy also wrote a work on astrology. It may seem strange to the modern reader that someone who wrote such excellent scientific books should write on astrology. However, Ptolemy sees it rather differently for he claims that the Almagest allows one to find the positions of the heavenly bodies, while his astrology book he sees as a companion work describing the effects of the heavenly bodies on people's lives.

In a book entitled Analemma he discussed methods of finding the angles need to construct a sundial which involves the projection of points on the celestial sphere. In Planisphaerium he is concerned with stereographic projection of the celestial sphere onto a plane. This is discussed in [48] where it is stated:-

In the stereographic projection treated by Ptolemy in the "Planisphaerium" the celestial sphere is mapped onto the plane of the equator by projection from the south pole. Ptolemy does not prove the important property that circles on the sphere become circles on the plane.

Ptolemy's major work Geography, in eight books, attempts to map the known world giving coordinates of the major places in terms of latitude and longitude. It is not surprising that the maps given by Ptolemy were quite inaccurate in many places for he could not be expected to do more than use the available data and this was of very poor quality for anything outside the Roman Empire, and even parts of the Roman Empire are severely distorted. In [19] Ptolemy is described as:-

... a man working [on map-construction] without the support of a developed theory but within a mathematical tradition and guided by his sense of what is appropriate to the problem.

Another work on Optics is in five books and in it Ptolemy studies colour, reflection, refraction, and mirrors of various shapes. Toomer comments in [1]:-

The establishment of theory by experiment, frequently by constructing special apparatus, is the most striking feature of Ptolemy's "Optics". Whether the subject matter is largely derived or original, "The Optics" is an impressive example of the development of a mathematical science with due regard to physical data, and is worthy of the author of the "Almagest".

An English translation, attempting to remove the inaccuracies introduced in the poor Arabic translation which is our only source of the Optics is given in [14].

The first to make accusations against Ptolemy was Tycho Brahe. He discovered that there was a systematic error of one degree in the longitudes of the stars in the star catalogue, and he claimed that, despite Ptolemy saying that it represented his own observations, it was merely a conversion of a catalogue due to Hipparchus corrected for precession to Ptolemy's date. There is of course definite problems comparing two star catalogues, one of which we have a copy of while the other is lost.

After comments by Laplace and Lalande, the next to attack Ptolemy vigorously was Delambre. He suggested that perhaps the errors came from Hipparchus and that Ptolemy might have done nothing more serious than to have failed to correct Hipparchus's data for the time between the equinoxes and solstices. However Delambre then goes on to say (see [8]):-

One could explain everything in a less favourable but all the simpler manner by denying Ptolemy the observation of the stars and equinoxes, and by claiming that he assimilated everything from Hipparchus, using the minimal value of the latter for the precession motion.

However, Ptolemy was not without his supporters by any means and further analysis led to a belief that the accusations made against Ptolemy by Delambre were false. Boll writing in 1894 says [4]:-

To all appearances, one will have to credit Ptolemy with giving an essentially richer picture of the Greek firmament after his eminent predecessors.

Vogt showed clearly in his important paper [77] that by considering Hipparchus's Commentary on Aratus and Eudoxus and making the reasonable assumption that the data given there agreed with Hipparchus's star catalogue, then Ptolemy's star catalogue cannot have been produced from the positions of the stars as given by Hipparchus, except for a small number of stars where Ptolemy does appear to have taken the data from Hipparchus. Vogt writes:-

This allows us to consider the fixed star catalogue as of his own making, just as Ptolemy himself vigorously states.

The most recent accusations of forgery made against Ptolemy came from Newton in [12]. He begins this book by stating clearly his views:-

This is the story of a scientific crime. ... I mean a crime committed by a scientist against fellow scientists and scholars, a betrayal of the ethics and integrity of his profession that has forever deprived mankind of fundamental information about an important area of astronomy and history.

Towards the end Newton, having claimed to prove every observation claimed by Ptolemy in the Almagest was fabricated, writes [12]:-

[Ptolemy] developed certain astronomical theories and discovered that they were not consistent with observation. Instead of abandoning the theories, he deliberately fabricated observations from the theories so that he could claim that the observations prove the validity of his theories. In every scientific or scholarly setting known, this practice is called fraud, and it is a crime against science and scholarship.

Although the evidence produced by Brahe, Delambre, Newton and others certainly do show that Ptolemy's errors are not random, this last quote from [12] is, I [EFR] believe, a crime against Ptolemy (to use Newton's own words). The book [8] is written to study validity of these accusations and it is a work which I strongly believe gives the correct interpretation. Grasshoff writes:-

... one has to assume that a substantial proportion of the Ptolemaic star catalogue is grounded on those Hipparchan observations which Hipparchus already used for the compilation of the second part of his "Commentary on Aratus". Although it cannot be ruled out that coordinates resulting from genuine Ptolemaic observations are included in the catalogue, they could not amount to more than half the catalogue.

... the assimilation of Hipparchan observations can no longer be discussed under the aspect of plagiarism. Ptolemy, whose intention was to develop a comprehensive theory of celestial phenomena, had no access to the methods of data evaluation using arithmetical means with which modern astronomers can derive from a set of varying measurement results, the one representative value needed to test a hypothesis. For methodological reason, then, Ptolemy was forced to choose from a set of measurements the one value corresponding best to what he had to consider as the most reliable data. When an intuitive selection among the data was no longer possible ... Ptolemy had to consider those values as 'observed' which could be confirmed by theoretical predictions.

As a final comment we quote the epigram which is accepted by many scholars to have been written by Ptolemy himself, and it appears in Book 1 of the Almagest, following the list of contents (see for example [11]):-

Well do I know that I am mortal, a creature of one day.
But if my mind follows the winding paths of the stars
Then my feet no longer rest on earth, but standing by
Zeus himself I take my fill of ambrosia, the divine dish.