海伦Hero of Alexandria

有时被称为海伦，海伦是一位重要的几何学家和力学工作者。也许第一个值得评论的是，海伦这个名字在当时是多么常见，在数学史上，要确定哪些提到海伦的文献是指本文所描述的数学家，哪些是指同名的其他人，是一个困难的问题。我们将在下面讨论其他识别问题。

关于海伦的一个主要困难是确定他生活的年代。对此有两种主要观点，一种认为他生活在公元前150年左右，第二种认为他生活在公元250年左右。第一种主要基于海伦没有引用任何晚于阿基米德的著作。第二种基于一个论证，旨在表明他生活在克劳狄乌斯·托勒密之后，并且，由于帕普斯提到了海伦，所以在帕普斯之前。

这两个论点都已被证明是错误的。有人提出了第三个日期，其依据是相信海伦与Columella是同时代人。Columella是一位罗马士兵和农民，他写了大量关于农业和类似主题的文章，希望培养人们对农业的热爱和对简单生活的喜爱。Columella在大约公元62年写的一篇文本中[5]：-

……给出了平面图形的测量值，这些测量值与海伦使用的公式一致，特别是等边三角形、正六边形（在这种情况下，不仅公式而且实际数字都与海伦的一致）以及小于半圆的圆弓形……

然而，大多数历史学家认为Columella和海伦都使用了更早的来源，并声称相似性并不能证明任何依赖关系。我们现在知道，那些相信海伦生活在Columella时代附近的人实际上是对的，因为奥托·纽格包尔在1938年发现海伦在他的其中一部作品中提到了一次最近的日食，根据海伦提供的信息，他能够将其与海伦在62年3月13日23时发生的一次日食联系起来。

从海伦的著作中可以合理推断，他曾在海伦的博物馆任教。他的作品看起来像是他在那里讲授数学、物理学、气体力学和力学的课程讲义。有些显然是教科书，而另一些可能是尚未整理成最终学生教科书形式的讲义草稿。

帕普斯在他的Mathematical Collection第八卷中描述了海伦的贡献。帕普斯写道（例如见[8]）：-

海伦学派的机械学家说，力学可以分为理论部分和手工部分；理论部分由几何学、算术、天文学和物理学组成，手工部分则涉及金属加工、建筑、木工和绘画以及任何涉及手部技能的技艺。

……古人也把奇迹创造者描述为机械学家，其中一些人通过气动力学工作，如海伦在其《气动力学》中所述，一些人通过使用绳索和绳子，试图模仿生物的运动，如海伦在其《自动机》和《平衡》中所述，……或者通过使用水来报时，如海伦在其《水钟》中所述，这似乎与日晷科学有相似之处。

海伦的大量著作流传下来，尽管其中一些的作者身份存在争议。我们将在下面海伦著作列表中讨论一些分歧。这些著作分为几类：技术著作、机械著作和数学著作。流传下来的著作有：

1. On the dioptra涉及theodolites和测量。它包含一章关于天文学的内容，给出了一种利用在每座城市观测到月食的当地时刻之差来求海伦与罗马之间距离的方法。克劳狄乌斯·托勒密似乎不知道这种方法，这一事实使历史学家们错误地认为海伦生活在克劳狄乌斯·托勒密之后；
2. The pneumatica 分两卷，研究由空气、蒸汽或水压驱动的机械装置。下面将更详细地描述；
3. The automaton theatre 描述了一个由绳索、鼓和重物驱动的木偶剧院；
4. Belopoeica 描述了如何建造战争器械。它与 费隆 的著作以及 维特鲁威 的著作有一些相似之处，后者是生活在公元前 1 世纪的罗马建筑师和工程师；
5. 关于弩炮的The cheirobalistra被认为是弩炮词典的一部分，但几乎可以肯定不是海伦所写；
6. Mechanica 分为三卷，为建筑师而写，下文有更详细的描述；
7. Metrica 给出了测量方法。我们在下文给出更多细节；
8. Definitiones 包含 133 个几何术语的定义，从点、线等开始。在 [15] 中，威尔伯·诺尔 令人信服地论证了这部著作实际上出自 丢番图 之手；
9. Geometria似乎是Metrica第一章的一个不同版本，完全基于例子。虽然基于海伦的工作，但人们认为它不是他写的；
10. Stereometrica 测量三维物体，至少部分基于 Metrica 的第二章，同样基于例子。它同样被认为基于 海伦 的工作，但被许多后来的编者大大修改；
11. Mensurae 测量了各种各样的不同对象，并与 Stereometrica 和 Metrica 的部分内容相关，尽管它主要必定是较晚作者的工作；
12. Catoptrica讨论镜子，一些历史学家将其归于克劳狄乌斯·托勒密，尽管现在大多数人似乎相信这是海伦的真作。在这部著作中，海伦指出视觉源于眼睛发出的光线。他认为这些光线以无限速度传播。

让我们更深入地考察海伦的一些工作。他的论著Metrica的第一卷讨论了三角形、四边形、3至12边的正多边形、圆锥、圆柱、棱柱、棱锥、球体等的面积。还给出了一种方法，用于近似一个数的平方根，这种方法在2000年前就为巴比伦人所知。海伦以如下形式给出（例如见[5]）：-

由于720没有其边有理的，我们可以如下获得其边，差异非常小。由于下一个后继的平方数是729，其边为27，将720除以27。这得到$26\large\frac{2}{3}\normalsize$。将27加到这上，得到$53\large\frac{2}{3}\normalsize$，取一半即$26\large\frac{5}{6}\normalsize$。因此720的边将非常接近$26\large\frac{5}{6}\normalsize$。事实上，如果我们将$26\large\frac{5}{6}\normalsize$乘以自身，乘积是$720\large\frac{1}{36}\normalsize$，所以平方中的差异是$\large\frac{1}{36}\normalsize$。如果我们希望使差异比$\large\frac{1}{36}\normalsize$更小，我们将取$720\large\frac{1}{36}\normalsize$而不是729（或者更确切地说，我们应该取$26\large\frac{5}{6}\normalsize$而不是27），并以同样的方式进行，我们将发现得到的差异远小于$\large\frac{1}{36}\normalsize$。

海伦 还在 Metrica 第一卷中证明了他著名的公式：-

如果 A 是边长分别为 a、b、c 的三角形的面积，且 $s = \large\frac{1}{2}\normalsize (a + b + c)$，则
$A^{2} = s (s - a)(s - b)(s - c)$。

在 Metrica 第二卷中，海伦 考虑了球、圆柱、圆锥、棱柱、棱锥等各种三维图形的体积测量。他的序言很有意思，部分原因是关于 阿基米德 工作的知识似乎并不像人们可能预期的那样广为人知（例如见 [5]）：-

在测量了曲面——无论是直线形的还是非直线形的——之后，接下来就应当处理立体，这些立体的表面我们已在上一卷中测量过，包括平面和球面、锥面和柱面，以及不规则曲面。处理这些立体的方法，鉴于其令人惊讶的性质，被某些记述其起源的传统说法的作者归之于 阿基米德。但无论它们属于 阿基米德 还是另一个人，都有必要对这些结果作一概述。

Metrica 第三卷讨论按给定比例分割面积和体积。这是 欧几里得 在其著作 On divisions of figures 中研究过的问题，而 海伦 的第三卷与 欧几里得 的工作有许多共同之处。同样在第三卷中，海伦 给出了一种求一个数的立方根的方法。特别是 海伦 求出了 100 的立方根，而 [9] 的作者们给出了 $N$ 的立方根的一般公式，海伦 似乎在他的计算中使用了这个公式：

$a + \Large \frac {b d}{b d + aD} \normalsize(b - a)$，其中$a^{3} < N < b^{3}, d = N - a^{3}, D = b^{3} - N$。

在9中指出，这是一个非常精确的公式，但是，除非将错误归咎于一位拜占庭抄写员，他们得出结论，海伦可能借用了这个精确的公式，却不理解如何一般地使用它。

Pneumatica是一部奇特的作品，分为两卷，第一卷有43章，第二卷有37章。海伦从流体内压力的理论考量开始。其中一些理论是正确的，但毫不奇怪，有些则相当错误。接着描述了一整套或许最好被描述为儿童机械玩具的东西[1]：-

能分别或按固定比例倒出酒或水的诡计罐，会唱歌的鸟和发声的喇叭，在祭坛上点火时就会移动的木偶，被提供水时就会喝水的动物……

尽管这一切对一位科学家来说似乎非常琐碎，但看来海伦是在利用这些玩具作为向学生教授物理学的载体。这似乎是一种尝试，使科学理论与当时学生所熟悉的日常物品相关联。

相当引人注目的是，书中描述了100多种机器，例如灭火机、风琴、投币式机器，以及一种称为汽转球的蒸汽动力发动机。海伦的汽转球与喷气发动机有许多共同之处，在2中描述如下：-

汽转球是一个空心球体，其安装方式使其能够在一对空心管上转动，这些空心管从一个大锅中向球体提供蒸汽。蒸汽从球体上从其赤道处突出的一个或多个弯曲管中逸出，使球体旋转。汽转球是已知的第一个将蒸汽转化为旋转运动的装置。

海伦写了许多重要的力学论著。它们给出了提升重物的方法，并描述了简单的机械装置。特别是Mechanica相当紧密地基于阿基米德的思想。第一卷考察如何按照给定形状的给定比例构造三维形状。它还考察了运动理论、某些静力学问题以及天平理论。

在第二卷中海伦讨论了用杠杆、滑轮、楔子或螺旋提升重物。其中讨论了平面图形的重心。第三卷考察了用雪橇等手段运输物体的方法、起重机的使用，并探讨了榨酒机。

另有一些著作被归于海伦名下，其中一些我们存有残篇，另一些则仅有提及。存有残篇的著作包括一部关于Water clocks的四卷本著作，以及Commentary on Euclid's Elements，后者必定至少涵盖了Elements的前八卷。被提及但无任何痕迹存留的海伦著作，包括阿什凯隆的欧托基奥斯提到的Camarica或On vaultings，以及帕普斯提到的Zygia或On balancing。此外，在Fihrist这部十世纪伊斯兰文化概览中，还提到了海伦的一部关于如何使用星盘的著作。

最后，考察各位作者对海伦的质量和重要性所表达的看法是很有意思的。奥托·纽格包尔写道[7]：-

对数学楔形文字文本的解读清楚地表明，希腊数学中大量“赫伦式”类型的内容不过是巴比伦数学传统延续1800年后的最后阶段。

有些人认为海伦是一个无知的手工匠人，他抄袭了自己书中的内容却不理解自己所写的东西。这种指责尤其针对Pneumatica，但Drachmann在[1]中写道：-

……在我看来，这种流畅而相当散漫的风格表明，此人精通自己的主题，正在向一群对此了解甚多、或可能被期望了解甚多的听众作快速概述。

一些学者认可海伦作为测量员的实践技能，但声称他的科学知识微不足道。然而，Mahony在[1]中写道：-

根据最近的学术研究，他现在看来是一位受过良好教育且常常富有独创性的应用数学家，同时也是从巴比伦人、经阿拉伯人、到文艺复兴时期欧洲的连续实用数学传统中的一个重要环节。

最后托马斯·利特尔·希思在[5]中写道：-

海伦的手册实用性如此之大，它们自然应该大受欢迎，同样自然的是，其中最受欢迎的至少应该被后来的作者重新编辑、修改和增补；这对于像欧几里得的《几何原本》那样，在希腊、拜占庭、罗马和阿拉伯教育中几个世纪以来一直常规使用的书籍来说是不可避免的。

Sometimes called Hero, Heron of Alexandria was an important geometer and worker in mechanics. Perhaps the first comment worth making is how common the name Heron was around this time and it is a difficult problem in the history of mathematics to identify which references to Heron are to the mathematician described in this article and which are to others of the same name. There are additional problems of identification which we discuss below.

A major difficulty regarding Heron was to establish the date at which he lived. There were two main schools of thought on this, one believing that he lived around 150 BC and the second believing that he lived around 250 AD. The first of these was based mainly on the fact that Heron does not quote from any work later than Archimedes. The second was based on an argument which purported to show that he lived later that Ptolemy, and, since Pappus refers to Heron, before Pappus.

Both of these arguments have been shown to be wrong. There was a third date proposed which was based on the belief that Heron was a contemporary of Columella. Columella was a Roman soldier and farmer who wrote extensively on agriculture and similar subjects, hoping to foster in people a love for farming and a liking for the simple life. Columella, in a text written in about 62 AD [5]:-

... gave measurements of plane figures which agree with the formulas used by Heron, notably those for the equilateral triangle, the regular hexagon (in this case not only the formula but the actual figures agree with Heron's) and the segment of a circle which is less than a semicircle ...

However, most historians believed that both Columella and Heron were using an earlier source and claimed that the similarity did not prove any dependence. We now know that those who believed that Heron lived around the time of Columella were in fact correct, for Neugebauer in 1938 discovered that Heron referred to a recent eclipse in one of his works which, from the information given by Heron, he was able to identify with one which took place in Alexandria at 23.00 hours on 13 March 62.

From Heron's writings it is reasonable to deduce that he taught at the Museum in Alexandria. His works look like lecture notes from courses he must have given there on mathematics, physics, pneumatics, and mechanics. Some are clearly textbooks while others are perhaps drafts of lecture notes not yet worked into final form for a student textbook.

Pappus describes the contribution of Heron in Book VIII of his Mathematical Collection. Pappus writes (see for example [8]):-

The mechanicians of Heron's school say that mechanics can be divided into a theoretical and a manual part; the theoretical part is composed of geometry, arithmetic, astronomy and physics, the manual of work in metals, architecture, carpentering and painting and anything involving skill with the hands.

... the ancients also describe as mechanicians the wonder-workers, of whom some work by means of pneumatics, as Heron in his Pneumatica, some by using strings and ropes, thinking to imitate the movements of living things, as Heron in his Automata and Balancings, ... or by using water to tell the time, as Heron in his Hydria, which appears to have affinities with the science of sundials.

A large number of works by Heron have survived, although the authorship of some is disputed. We will discuss some of the disagreements in our list of Heron's works below. The works fall into several categories, technical works, mechanical works and mathematical works. The surviving works are:

1. On the dioptra dealing with theodolites and surveying. It contains a chapter on astronomy giving a method to find the distance between Alexandria and Rome using the difference between local times at which an eclipse of the moon is observed at each cities. The fact that Ptolemy does not appear to have known of this method led historians to mistakenly believe Heron lived after Ptolemy;
2. The pneumatica in two books studying mechanical devices worked by air, steam or water pressure. It is described in more detail below;
3. The automaton theatre describing a puppet theatre worked by strings, drums and weights;
4. Belopoeica describing how to construct engines of war. It has some similarities with work by Philon and also work by Vitruvius who was a Roman architect and engineer who lived in the 1st century BC;
5. The cheirobalistra about catapults is thought to be part of a dictionary of catapults but was almost certainly not written by Heron;
6. Mechanica in three books written for architects and described in more detail below;
7. Metrica which gives methods of measurement. We give more details below;
8. Definitiones contains 133 definitions of geometrical terms beginning with points, lines etc. In [15] Knorr argues convincingly that this work is in fact due to Diophantus;
9. Geometria seems to be a different version of the first chapter of the Metrica based entirely on examples. Although based on Heron's work it is not thought to be written by him;
10. Stereometrica measures three-dimensional objects and is at least in part based on the second chapter of the Metrica again based on examples. Again it is though to be based on Heron's work but greatly changed by many later editors;
11. Mensurae measures a whole variety of different objects and is connected with parts of Stereometrica and Metrica although it must be mainly the work of a later author;
12. Catoptrica deals with mirrors and is attributed by some historians to Ptolemy although most now seem to believe that this is a genuine work of Heron. In this work, Heron states that vision results from light rays emitted by the eyes. He believes that these rays travel with infinite velocity.

Let us examine some of Heron's work in a little more depth. Book I of his treatise Metrica deals with areas of triangles, quadrilaterals, regular polygons of between 3 and 12 sides, surfaces of cones, cylinders, prisms, pyramids, spheres etc. A method, known to the Babylonians 2000 years before, is also given for approximating the square root of a number. Heron gives this in the following form (see for example [5]):-

Since 720 has not its side rational, we can obtain its side within a very small difference as follows. Since the next succeeding square number is 729, which has 27 for its side, divide 720 by 27. This gives $26\large\frac{2}{3}\normalsize$. Add 27 to this, making $53\large\frac{2}{3}\normalsize$, and take half this or $26\large\frac{5}{6}\normalsize$. The side of 720 will therefore be very nearly $26\large\frac{5}{6}\normalsize$. In fact, if we multiply $26\large\frac{5}{6}\normalsize$ by itself, the product is $720\large\frac{1}{36}\normalsize$, so the difference in the square is $\large\frac{1}{36}\normalsize$. If we desire to make the difference smaller still than $\large\frac{1}{36}\normalsize$, we shall take $720\large\frac{1}{36}\normalsize$ instead of 729 (or rather we should take $26\large\frac{5}{6}\normalsize$ instead of 27), and by proceeding in the same way we shall find the resulting difference much less than $\large\frac{1}{36}\normalsize$.

Heron also proves his famous formula in Book I of the Metrica :-

if A is the area of a triangle with sides a, b and c and $s = \large\frac{1}{2}\normalsize (a + b + c)$ then
     $A^{2} = s (s - a)(s - b)(s - c)$.

In Book II of Metrica, Heron considers the measurement of volumes of various three dimensional figures such as spheres, cylinders, cones, prisms, pyramids etc. His preface is interesting, partly because knowledge of the work of Archimedes does not seem to be as widely known as one might expect (see for example [5]):-

After the measurement of surfaces, rectilinear or not, it is proper to proceed to solid bodies, the surfaces of which we have already measured in the preceding book, surfaces plane and spherical, conical and cylindrical, and irregular surfaces as well. The methods of dealing with these solids are, in view of their surprising character, referred to Archimedes by certain writers who give the traditional account of their origin. But whether they belong to Archimedes or another, it is necessary to give a sketch of these results as well.

Book III of Metrica deals with dividing areas and volumes according to a given ratio. This was a problem which Euclid investigated in his work On divisions of figures and Heron's Book III has a lot in common with the work of Euclid. Also in Book III, Heron gives a method to find the cube root of a number. In particular Heron finds the cube root of 100 and the authors of [9] give a general formula for the cube root of $N$ which Heron seems to have used in his calculation:

$a + \Large \frac {b d}{b d + aD} \normalsize(b - a)$, where $a^{3} < N < b^{3}, d = N - a^{3}, D = b^{3} - N$.

In [9] it is remarked that this is a very accurate formula, but, unless a Byzantine copyist is to be blamed for an error, they conclude that Heron might have borrowed this accurate formula without understanding how to use it in general.

The Pneumatica is a strange work which is written in two books, the first with 43 chapters and the second with 37 chapters. Heron begins with a theoretical consideration of pressure in fluids. Some of this theory is right but, not surprisingly, some is quite wrong. Then there follows a description of a whole collection of what might best be described as mechanical toys for children [1]:-

Trick jars that give out wine or water separately or in constant proportions, singing birds and sounding trumpets, puppets that move when a fire is lit on an altar, animals that drink when they are offered water ...

Although all this seems very trivial for a scientist to be involved with, it would appear that Heron is using these toys as a vehicle for teaching physics to his students. It seems to be an attempt to make scientific theories relevant to everyday items that students of the time would be familiar with.

There is, rather remarkably, descriptions of over 100 machines such as a fire engine, a wind organ, a coin-operated machine, and a steam-powered engine called an aeolipile. Heron's aeolipile, which has much in common with a jet engine, is described in [2] as follows:-

The aeolipile was a hollow sphere mounted so that it could turn on a pair of hollow tubes that provided steam to the sphere from a cauldron. The steam escaped from the sphere from one or more bent tubes projecting from its equator, causing the sphere to revolve. The aeolipile is the first known device to transform steam into rotary motion.

Heron wrote a number of important treatises on mechanics. They give methods of lifting heavy weights and describe simple mechanical machines. In particular the Mechanica is based quite closely on ideas due to Archimedes. Book I examines how to construct three dimensional shapes in a given proportion to a given shape. It also examines the theory of motion, certain statics problems, and the theory of the balance.

In Book II Heron discusses lifting heavy objects with a lever, a pulley, a wedge, or a screw. There is a discussion on centres of gravity of plane figures. Book III examines methods of transporting objects by such means as sledges, the use of cranes, and looks at wine presses.

Other works have been attributed to Heron, and for some of these we have fragments, for others there are only references. The works for which fragments survive include one on Water clocks in four books, and Commentary on Euclid's Elements which must have covered at least the first eight books of the Elements. Works by Heron which are referred to, but no trace survives, include Camarica or On vaultings which is mentioned by Eutocius and Zygia or On balancing mentioned by Pappus. Also in the Fihrist, a tenth century survey of Islamic culture, a work by Heron on how to use an astrolabe is mentioned.

Finally it is interesting to look at the opinions that various writers have expressed as to the quality and importance of Heron. Neugebauer writes [7]:-

The decipherment of the mathematical cuneiform texts made it clear that much of the "Heronic" type of Greek mathematics is simply the last phase of the Babylonian mathematical tradition which extends over 1800 years.

Some have considered Heron to be an ignorant artisan who copied the contents of his books without understanding what he wrote. This in particular has been levelled against the Pneumatica but Drachmann, writing in [1], says:-

... to me the free flowing, rather discursive style suggests a man well versed in his subject who is giving a quick summary to an audience that knows, or who might be expected to know, a good deal about it.

Some scholars have approved of Heron's practical skills as a surveyor but claimed that his knowledge of science was negligible. However, Mahony writes in [1]:-

In the light of recent scholarship, he now appears as a well-educated and often ingenious applied mathematician, as well as a vital link in a continuous tradition of practical mathematics from the Babylonians, through the Arabs, to Renaissance Europe.

Finally Heath writes in [5]:-

The practical utility of Heron's manuals being so great, it was natural that they should have great vogue, and equally natural that the most popular of them at any rate should be re-edited, altered and added to by later writers; this was inevitable with books which, like the "Elements" of Euclid, were in regular use in Greek, Byzantine, Roman, and Arabian education for centuries.