喜帕恰斯Hipparchus

关于喜帕恰斯的生平所知甚少，但已知他出生在比提尼亚的狄奥多西的尼西亚。尼西亚镇现在叫伊兹尼克，位于土耳其西北部。尼西亚建于公元前4世纪，位于伊兹尼克湖的东岸。相当合理的是，喜帕恰斯常被称为尼西亚的喜帕恰斯或比提尼亚的狄奥多西的喜帕恰斯，他被斯特拉波列为比提尼亚的狄奥多西的著名人物之一，斯特拉波是希腊地理学家和历史学家，生活在大约公元前64年到公元24年。有来自尼西亚的硬币描绘喜帕恰斯坐着看地球仪，他的形象出现在公元138年至253年间五位不同罗马皇帝统治下铸造的硬币上。

这似乎牢固地将喜帕恰斯置于尼西亚，确实克劳狄乌斯·托勒密描述喜帕恰斯在比提尼亚的狄奥多西进行观测，人们自然会认为他实际上是在尼西亚观测。然而，在据说由喜帕恰斯进行的观测中，有些是在罗得岛北部进行的，还有几次（尽管只有一次肯定是喜帕恰斯本人所做的）是在亚历山大进行的。如果这些确实如它们所显示的那样，我们可以肯定地说，喜帕恰斯在公元前146年在亚历山大，在公元前127年和公元前126年他职业生涯接近尾声时在罗得岛。

关于希腊数学家生平细节很少并不太不寻常，但对于喜帕恰斯，情况有点不寻常，因为尽管喜帕恰斯是一位极其重要的数学家和天文学家，但关于他的工作，我们令人失望地几乎没有确定的细节。喜帕恰斯只有一部作品留存下来，即Commentary on Aratus and Eudoxus，这肯定不是他的主要作品之一。然而，它很重要，因为它为我们提供了喜帕恰斯自己著作的唯一来源。

我们关于喜帕恰斯工作的绝大部分信息来自克劳狄乌斯·托勒密的Almagest Ⓣ（主要论题：源自阿拉伯语“al-majisti”——希腊语“Mathematike Syntaxis”的阿拉伯语译本，后来译为拉丁语“Magna Syntaxis”），但正如Toomer在[1]中所写：-

……尽管克劳狄乌斯·托勒密显然仔细研读过喜帕恰斯的著作，并对其工作怀有深深的敬意，但他的主要关切并非将其传诸后世，而是利用它，并在可能的情况下加以改进，以构建他自己的天文学体系。

人们或许希望在克劳狄乌斯·托勒密的AlmagestⓉ（主要论题：源自阿拉伯语“al-majisti”——希腊语“Mathematike Syntaxis”的阿拉伯语译本，后来译为拉丁语“Magna Syntaxis”）的评注中找到关于喜帕恰斯的更多信息。特别是有两部评注，分别由亚历山大的杰出评注者亚历山大的席恩和帕普斯所著，但不幸的是，这些评注相当紧密地遵循克劳狄乌斯·托勒密的文本，未能添加关于喜帕恰斯的预期信息。由于当克劳狄乌斯·托勒密提到喜帕恰斯的结果时，他往往以一种晦涩的方式这样做，至少他似乎假定读者能够接触到喜帕恰斯的原著，而且令人惊讶的是，亚历山大的席恩和帕普斯都没有填补细节。人们只能假定，他们两人都没有接触到关于喜帕恰斯的信息，而我们本来希望他们能报告这些信息。

让我们先总结喜帕恰斯的主要贡献，然后再更详细地考察它们。他对三角学做出了早期贡献，制作了一张弦表，这是三角表的早期例子；事实上，有些历史学家甚至说三角学是他发明的。这张弦表的目的是提供一种解三角形的方法，避免从基本原理出发解每个三角形。他还将圆周分为360度引入希腊。

喜帕恰斯将一年的长度计算到6.5分钟以内，并发现了岁差。喜帕恰斯的岁差值为46"，与现代值50.26"相比很好，也比近300年后克劳狄乌斯·托勒密将得到的36"好得多。我们相信喜帕恰斯的星表包含约850颗恒星，可能没有以系统的坐标系统列出，而是使用各种不同的方式来指定恒星的位置。他的星表，可能完成于公元前129年，据称被克劳狄乌斯·托勒密用作他自己星表的基础。然而，Vogt在其重要论文[26]中清楚地表明，通过考虑Commentary on Aratus and Eudoxus并合理假设那里给出的数据与他的星表一致，那么克劳狄乌斯·托勒密的星表不可能根据喜帕恰斯给出的恒星位置制作出来。

最后一点表明，在对喜帕恰斯成就的任何详细讨论中，我们必须更深入地探究，而不是仅仅假设克劳狄乌斯·托勒密的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，后来翻译成拉丁语为'Magna Syntaxis'）中他不声称是自己的所有内容都归功于喜帕恰斯。这种观点被采纳了很多年，但自Vogt 1925年的论文[26]以来，已经做了很多研究，试图确切地确定喜帕恰斯取得了什么成就。因此，我们对喜帕恰斯的理解发生了重大转变，首先假设他的发现都由克劳狄乌斯·托勒密阐述，然后一旦意识到并非如此，就有一种感觉，认为永远不可能详细了解他的成就，但现在我们处于第三阶段，人们意识到可以获得对他工作的良好了解，但需要付出很多努力和研究。

让我们从查看唯一幸存的作品开始，详细描述喜帕恰斯的成就。喜帕恰斯的Commentary on Aratus and Eudoxus分三本书写成，作为对三部不同著作的评论。首先有一篇欧多克索斯的论文（不幸的是现已丢失），其中他命名并描述了星座。Aratus写了一首名为Phaenomena的诗，基于欧多克索斯的论文，并被证明是一部非常受欢迎的作品。这首诗幸存下来，我们有它的文本。第三，有Attalus of Rhodes对Aratus的评论，写于喜帕恰斯时代前不久。

当然不幸的是，在喜帕恰斯的所有著作中，正是这一部流传了下来，因为喜帕恰斯为之撰写评注的三部书并不包含数学天文学。因此，喜帕恰斯选择在第一卷以及三卷本中第二卷的大部分内容里，以同样的定性水平来写作。然而，在第二卷接近末尾处，并一直延续到第三卷全篇，喜帕恰斯给出了他自己关于星座升落的论述。在第三卷接近末尾处，喜帕恰斯给出了一份始终可见的亮星列表，目的是使夜间时间能够被精确确定。正如我们上面所指出的，喜帕恰斯并未使用单一一致的坐标系来表示恒星位置，而是混合使用了不同的坐标。他使用了一些赤道坐标，尽管方式往往相当奇怪，例如说一颗恒星（见[1]）：-

……沿其平行圈占据狮子座的三度……

因此，他把每个平行于赤道的小圆分成12个30°的部分，这意味着引文中提到的该恒星的赤经是123°。Commentary on Aratus and Eudoxus中的数据已被许多作者分析过。特别是[15]的作者们认为，喜帕恰斯使用了一个可移动的天球，上面绘有球面上的恒星。他们声称，这些数据取自约公元前140年编制的一份星表，该星表基于精确到三分之一度甚至更好的观测。在同一批作者较早的著作[16]中，他们提出这些观测是在北纬36°15'进行的，这与北罗得西亚的纬度相对应。这倾向于证实喜帕恰斯的这项工作是在他职业生涯末期完成的。正如Toomer在[1]中所写：-

远非像人们经常描述的那样是一部“青年时代的作品”，对Aratus的评注表明，喜帕恰斯已经积累了大量的观测，发明了解决球面天文学问题的方法，并提出了用数学方法确定恒星位置这一极为重要的思想……

当然，这里讨论的许多问题并没有一致意见。例如，Maeyama在[13]中看到Commentary on Aratus and Eudoxus中的数据（据称写于约公元前140年）与喜帕恰斯的星表（据称制作于约公元前130年）在精度上存在重大差异。Maeyama写道[13]：-

……喜帕恰斯的“评注”包含了他自己对恒星位置的观测，数量众多但操作不精确，尽管他有进行精确观测的一切能力。……他两个不同时期的观测精度毫无共同之处，仿佛它们涉及的是两个不同的观测者。在10年的间隔内，一切都可能发生，尤其对于像喜帕恰斯这样的人来说。那些认为喜帕恰斯在两个不同时期的天文活动相似的观点是完全没有根据的。

也许喜帕恰斯最著名的发现是岁差，这是由于地球自转轴方向的缓慢变化造成的。这项工作源于喜帕恰斯试图高精度地计算一年的长度。对于“年”有两种不同的定义，因为人们可以取太阳返回到恒星中同一位置所需的时间，或者可以取季节重复之前的时长，这是通过考虑二分点来定义的时长。前者称为恒星年，而后者称为回归年。

当然，喜帕恰斯计算这两种不同年长度所需的数据，并非他通过几年观测就能找到的。Swerdlow[20]认为，喜帕恰斯利用巴比伦数据计算了回归年的长度，得出比$365\large\frac{1}{4}\normalsize$天少$\large\frac{1}{300}\normalsize$天的值。然后他将其与二分点观测以及solstices进行核对，包括他自己的数据以及公元前280年阿里斯塔克斯和公元前432年Meton的数据。喜帕恰斯还计算了恒星年的长度，同样使用了更古老的巴比伦数据，得出了比$365\large\frac{1}{4}\normalsize$天长$\large\frac{1}{144}\normalsize$天的高度精确数值。由此得出他的岁差率为每世纪1°。

喜帕恰斯还仔细研究了月球的运动。在这项研究中有一些困难的问题，因为可以确定三个不同的周期。有月球回到相同经度所需的时间，回到相同速度（近点角）所需的时间，以及回到相同纬度所需的时间。此外还有朔望月，即太阳和月球连续对当关系之间的时间。22写道：-

为了建立他的月球理论，喜帕恰斯需要确定月球在经度、近点角和纬度上的平均运动。他能获得的最佳数据是巴比伦参数。但他并不满足于仅仅接受它们：他想用经验来检验它们，因此他（纯算术地）构造了126007天1小时的食周期，然后在他可获得的观测材料中寻找成对的食，以确认这确实是一个食周期。因此，观测确实发挥了真正的作用，但这个作用不是发现，而是确认。

在计算月球距离时，喜帕恰斯不仅出色地运用了数学技术和观测技术，而且还给出了一个范围，他计算出真实距离必定位于其中。尽管喜帕恰斯的论著On sizes and distances没有留存下来，但克劳狄乌斯·托勒密、帕普斯等人给出的细节使我们能够重建他的方法和结果。

对喜帕恰斯 技术的重建在 [24] 中得到了精美的呈现，作者在其中表明喜帕恰斯 的计算基于公元前190年3月14日发生的一次食。喜帕恰斯 的计算使他得出月球距离在59到67个地球半径之间的值，这相当了不起（正确的距离是60个地球半径）。他给出一个范围的主要原因是，他无法确定太阳的视差，只能给出一个上限值。喜帕恰斯 似乎知道，月球距离的67个地球半径来自太阳视差的这个上限，而59个地球半径的较低值对应于太阳位于无穷远。

喜帕恰斯 不仅给出了月球的观测数据，使他能够精确计算各种周期，而且还基于epicycles 发展了月球运动的理论模型。他表明他的模型与观测并不完全一致，但似乎是克劳狄乌斯·托勒密 第一个修正模型以将这些差异考虑在内。喜帕恰斯 还能够给出太阳运动的均轮-本轮模型（这更容易），但他没有尝试给出行星运动的均轮-本轮模型。

最后让我们考察喜帕恰斯对三角学所做的贡献。托马斯·利特尔·希思在6中写道：-

即使不是他发明的，喜帕恰斯也是我们有文献证据表明系统使用三角学的第一人。

文献证据来自亚历山大的克劳狄乌斯·托勒密和亚历山大的席恩，后者明确说喜帕恰斯写了一部关于弦的著作，共12卷。然而，奥托·纽格包尔7指出：-

……这个数字显然是无稽之谈，因为13卷就足以容纳整部Ⓣ（主要论题：源自阿拉伯语“al-majisti”——希腊语“Mathematike Syntaxis”的阿拉伯语译本，后来译为拉丁语“Magna Syntaxis”）或欧几里得的《几何原本》……

Toomer（[1]或[23]）重构了喜帕恰斯的弦表，以及喜帕恰斯计算该表所用的数学方法。该表基于一个分为360度的圆，每度分为60分。圆的半径则为$360\times \Large\frac{60}{2\pi}\normalsize = 3438$分，而喜帕恰斯的弦函数Crd与正弦函数的关系为

$\large\frac{1}{2}\normalsize$（Crd $2a) = 3438 \sin a$。

Toomer声称喜帕恰斯以7.5°的间隔（即圆的$\large\frac{1}{48}\normalsize$）定义了他的Crd函数，并使用线性插值来求中间点的值。他接着表明，该表可以由喜帕恰斯所知晓的一些基本公式计算得出，其中之一是补角定理，本质上就是毕达哥拉斯的定理，以及半角定理。喜帕恰斯的表格唯一留存下来的痕迹是在印度的表格中，这些表格被认为是以喜帕恰斯的表格为基础的。

Toomer在[1]中总结喜帕恰斯在这一领域的贡献时写道：-

……看来极有可能的是，喜帕恰斯是第一个构造弦表的人，从而为三角学问题提供了一般解法。其推论是，在喜帕恰斯之前，并不存在基于希腊几何方法的天文表。如果情况如此，那么喜帕恰斯不仅是三角学的创始人，也是将希腊天文学从纯理论转变为实用预测科学的人。

Little is known of Hipparchus's life, but he is known to have been born in Nicaea in Bithynia. The town of Nicaea is now called Iznik and is situated in north-western Turkey. Founded in the 4th Century BC, Nicaea lies on the eastern shore of Lake Iznik. Reasonably enough Hipparchus is often referred to as Hipparchus of Nicaea or Hipparchus of Bithynia and he is listed among the famous men of Bithynia by Strabo, the Greek geographer and historian who lived from about 64 BC to about 24 AD. There are coins from Nicaea which depict Hipparchus sitting looking at a globe and his image appears on coins minted under five different Roman emperors between 138 AD and 253 AD.

This seems to firmly place Hipparchus in Nicaea and indeed Ptolemy does describe Hipparchus as observing in Bithynia, and one would naturally assume that in fact he was observing in Nicaea. However, of the observations which are said to have been made by Hipparchus, some were made in the north of the island of Rhodes and several (although only one is definitely due to Hipparchus himself) were made in Alexandria. If these are indeed as they appear we can say with certainty that Hipparchus was in Alexandria in 146 BC and in Rhodes near the end of his career in 127 BC and 126 BC.

It is not too unusual to have few details of the life of a Greek mathematician, but with Hipparchus the position is a little unusual for, despite Hipparchus being a mathematician and astronomer of major importance, we have disappointingly few definite details of his work. Only one work by Hipparchus has survived, namely Commentary on Aratus and Eudoxus and this is certainly not one of his major works. It is however important in that it gives us the only source of Hipparchus's own writings.

Most of the information which we have about the work of Hipparchus comes from Ptolemy's Almagest Ⓣ but, as Toomer writes in [1]:-

... although Ptolemy obviously had studied Hipparchus's writings thoroughly and had a deep respect for his work, his main concern was not to transmit it to posterity but to use it and, where possible, improve upon it in constructing his own astronomical system.

Where one might hope for more information about Hipparchus would be in the commentaries on Ptolemy's Almagest Ⓣ. There are two in particular by the excellent commentators Theon of Alexandria and by Pappus, but unfortunately these follow Ptolemy's text fairly closely and fail to add the expected information about Hipparchus. Since when Ptolemy refers to results of Hipparchus he does so often in an obscure way, at least he seems to assume that the reader will have access to the original writings by Hipparchus, and it is certainly surprising that neither Theon or Pappus fills in the details. One can only assume that neither of them had access to the information about Hipparchus on which we would have liked them to report.

Let us first summarise the main contribution of Hipparchus and then examine them in more detail. He made an early contribution to trigonometry producing a table of chords, an early example of a trigonometric table; indeed some historians go so far as to say that trigonometry was invented by him. The purpose of this table of chords was to give a method for solving triangles which avoided solving each triangle from first principles. He also introduced the division of a circle into 360 degrees into Greece.

Hipparchus calculated the length of the year to within 6.5 minutes and discovered the precession of the equinoxes. Hipparchus's value of 46" for the annual precession is good compared with the modern value of 50.26" and much better than the figure of 36" that Ptolemy was to obtain nearly 300 years later. We believe that Hipparchus's star catalogue contained about 850 stars, probably not listed in a systematic coordinate system but using various different ways to designate the position of a star. His star catalogue, probably completed in 129 BC, has been claimed to have been used by Ptolemy as the basis of his own star catalogue. However, Vogt shows clearly in his important paper [26] that by considering the Commentary on Aratus and Eudoxus and making the reasonable assumption that the data given there agreed with his star catalogue, then Ptolemy's star catalogue cannot have been produced from the positions of the stars as given by Hipparchus.

This last point shows that in any detailed discussion of the achievements of Hipparchus we have to delve more deeply than just assuming that everything in the Ptolemy's Almagest Ⓣ which he does not claim as his own must be due to Hipparchus. This view was taken for many years but since Vogt's 1925 paper [26] there has been much research done trying to ascertain exactly what Hipparchus achieved. So major shifts have taken place in our understanding of Hipparchus, first it was assumed that his discoveries were all set out by Ptolemy, then once it was realised that this was not so there was a feeling that it would be impossible to ever have detailed knowledge of his achievements, but now we are in a third stage where it is realised that it is possible to gain a good knowledge of his work but only with much effort and research.

Let us begin our detailed description of Hipparchus's achievements by looking at the only work which has survived. Hipparchus's Commentary on Aratus and Eudoxus was written in three books as a commentary on three different writings. Firstly there was a treatise by Eudoxus (unfortunately now lost) in which he named and described the constellations. Aratus wrote a poem called Phaenomena which was based on the treatise by Eudoxus and proved to be a work of great popularity. This poem has survived and we have its text. Thirdly there was commentary on Aratus by Attalus of Rhodes, written shortly before the time of Hipparchus.

It is certainly unfortunate that of all of the writings of Hipparchus this was the one to survive since the three books on which Hipparchus was writing a commentary contained no mathematical astronomy. As a result of this Hipparchus chose to write at the same qualitative level in the first book and also for much of the second of his three book. However towards the end of the second book, continuing through the whole of the third book, Hipparchus gives his own account of the rising and setting of the constellations. Towards the end of Book 3 Hipparchus gives a list of bright stars always visible for the purpose of enabling the time at night to be accurately determined. As we noted above Hipparchus does not use a single consistent coordinate system to denote stellar positions, rather using a mixture of different coordinates. He uses some equatorial coordinates, although often in a rather strange way as for example saying that a star (see [1]):-

... occupies three degrees of Leo along its parallel circle...

He has therefore divided each small circle parallel to the equator into 12 portions of 30° each and this means that the right ascension of the star referred to in the quotation is 123°. The data in the Commentary on Aratus and Eudoxus has been analysed by many authors. In particular the authors of [15] argue that Hipparchus used a mobile celestial sphere with the stars pictured on the sphere. They claim that the data was taken from on a star catalogue constructed around 140 BC based on observations accurate to a third of a degree or even better. In the earlier work [16] by the same authors, they suggest that the observations were made at a latitude of 36° 15' which corresponds to that of northern Rhodes. This would tend to confirm that this work by Hipparchus was done near the end of his career. As Toomer writes in [1]:-

Far from being a "work of his youth", as it is frequently described, the commentary on Aratus reveals Hipparchus as one who had already compiled a large number of observations, invented methods for solving problems in spherical astronomy, and developed the highly significant idea of mathematically fixing the positions of the stars...

There is of course no agreement on many of the points discussed here. For example Maeyama in [13] sees major differences between the accuracy of the data in Commentary on Aratus and Eudoxus (claimed to be written around 140 BC) and Hipparchus's star catalogue (claimed to be produced around 130 BC). Maeyama writes [13]:-

... Hipparchus's "Commentary" contains his own observations of the stellar positions, great in number but inaccurate in operation, despite all his ability for accurate observations. ... the observational accuracy [of] his two different epochs have nothing in common, as if they dealt with two different observers. Within an interval of 10 years everything can happen, particularly in the case of a man like Hipparchus. Those views which consider Hipparchus's astronomical activities at his two different epochs as similar are completely unfounded.

Perhaps the discovery for which Hipparchus is most famous is the discovery of precession which is due to the slow change in direction of the axis of rotation of the earth. This work came from Hipparchus's attempts to calculate the length of the year with a high degree of accuracy. There are two different definitions of a 'year' for one might take the time that the sun takes to return to the same place amongst the fixed stars or one could take the length of time before the seasons repeated which is a length of time defined by considering the equinoxes. The first of these is called the sidereal year while the second is called the tropical year.

Of course the data needed by Hipparchus to calculate the length of these two different years was not something that he could find over a few years of observations. Swerdlow [20] suggests that Hipparchus calculated the length of the tropical year using Babylonian data to arrive at the value of $\large\frac{1}{300}\normalsize$ of a day less than $365\large\frac{1}{4}\normalsize$ days. He then checked this against observations of equinoxes and solstices including his own data and those of Aristarchus in 280 BC and Meton in 432 BC. Hipparchus also calculated the length of the sidereal year, again using older Babylonian data, and arrived at the highly accurate figure of $\large\frac{1}{144}\normalsize$ days longer than $365\large\frac{1}{4}\normalsize$ days. This gives his rate of precession of 1° per century.

Hipparchus also made a careful study of the motion of the moon. There are difficult problems in such a study for there are three different periods which one could determine. 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. In addition there is the synodic month, that is the time between successive oppositions of the sun and moon. Toomer [22] writes:-

For his lunar theory [Hipparchus] needed to establish the mean motions of the Moon in longitude, anomaly and latitude. The best data available to him were the Babylonian parameters. But he was not content merely to accept them: he wanted to test them empirically, and so he constructed (purely arithmetically) the eclipse period of 126007 days 1 hour, then looked in the observational material available to him for pairs of eclipses which would confirm that this was indeed an eclipse period. The observations thus played a real role, but that role was not discovery, but confirmation.

In calculating the distance of the moon, Hipparchus not only made excellent use of both mathematical techniques and observational techniques but he also gave a range of values within which be calculated that the true distance must lie. Although Hipparchus's treatise On sizes and distances has not survived details given by Ptolemy, Pappus, and others allow us to reconstruct his methods and results.

The reconstruction of Hipparchus's techniques is beautifully presented in [24] where the author shows that Hipparchus based his calculations on an eclipse which occurred on 14 March 190 BC. Hipparchus's calculations led him to a value for the distance to the moon of between 59 and 67 earth radii which is quite remarkable (the correct distance is 60 earth radii). The main reason for his range of values was that he was unable to determine the parallax of the sun, only managing to give an upper value. Hipparchus appears to know that 67 earth radii for the distance of the moon comes from this upper limit of solar parallax, while the lower value of 59 earth radii corresponds to the sun being at infinity.

Hipparchus not only gave observational data for the moon which enabled him to compute accurately the various periods, but he developed a theoretical model of the motion of the moon based on epicycles. He showed that his model did not agree totally with observations but it seems to be Ptolemy who was the first to correct the model to take these discrepancies into account. Hipparchus was also able to give an epicycle model for the motion of the sun (which is easier), but he did not attempt to give an epicycle model for the motion of the planets.

Finally let us examine the contributions which Hipparchus made to trigonometry. Heath writes in [6]:-

Even if he did not invent it, Hipparchus is the first person whose systematic use of trigonometry we have documentary evidence.

The documentary evidence comes from Ptolemy and Theon of Alexandria who explicitly says that Hipparchus wrote a work on chords in 12 books. However, Neugebauer [7] points out that:-

... this number is obvious nonsense since 13 books sufficed for the whole of the "Almagest" Ⓣ or of Euclid's "Elements"...

Toomer ([1] or [23]) reconstructs Hipparchus's table of chords, and the mathematical means by which Hipparchus calculated it. The table was based on a circle divided into 360 degrees with each degree divided into 60 minutes. The radius of the circle is then $360\times \Large\frac{60}{2\pi}\normalsize = 3438$ minutes and the chord function Crd of Hipparchus is related to the sine function by

$\large\frac{1}{2}\normalsize$(Crd $2a) = 3438 \sin a$.

Toomer claims that Hipparchus defined his Crd function at 7.5° intervals ($\large\frac{1}{48}\normalsize$ of the circle) and used linear interpolation to find the value at intermediate points. He then goes on to show that the table can be computed from some basic formulae which would be known to Hipparchus, one of which is the supplementary angle theorem, essentially Pythagoras's theorem, and the half-angle theorem. The only trace of Hipparchus's tables that survives is in Indian tables which are thought to have been based on that of that of Hipparchus.

Toomer summarises the contributions of Hipparchus in this area when he writes in [1]:-

... it seems highly probable that Hipparchus was the first to construct a table of chords and thus provide a general solution for trigonometrical problems. A corollary of this is that, before Hipparchus, astronomical tables based on Greek geometrical methods did not exist. If this is so, Hipparchus was not only the founder of trigonometry but also the man who transformed Greek astronomy from a purely theoretical into a practical predictive science.