泰阿泰德Theaetetus

我们对泰阿泰德生平的了解大多来自柏拉图的著作。很明显，柏拉图对泰阿泰德极为尊重，他写了两篇对话，其中泰阿泰德是主要人物，其中一篇对话是Theaetetus，另一篇是智者。

在Theaetetus中，记录了苏格拉底、泰阿泰德和他的老师西奥多罗斯（来自西奥多罗斯）之间的讨论。这次对话发生在公元前399年，当时泰阿泰德被描述为一个年轻人。这使我们能够相当准确地确定泰阿泰德的出生日期（尽管有些人声称希腊词可以描述一个最多21岁的男子）。再次从柏拉图中我们得知，泰阿泰德的父亲，苏尼翁的Euphronius，是一个富有的人，留下了大笔财富。然而，这笔钱被遗嘱的受托人挥霍了，但尽管如此，泰阿泰德对周围的人都很慷慨。

从外表看，泰阿泰德长着朝天鼻和凸出的眼睛，但柏拉图形容他拥有美丽的心灵，还称他是完美的绅士。西奥多罗斯谈到他所有的学生时说[1]：-

……他从未发现有一个如此天赋异禀。

在Suda Lexicon（一部10世纪希腊词典编纂者的著作）中有两处提到“泰阿泰德”。第一处写道（例如见[1]）：-

泰阿泰德，泰阿泰德，天文学家、哲学家、苏格拉底的弟子，在赫拉克利亚任教。他是第一个构造所谓五种立体的人。他生活在伯罗奔尼撒战争之后。

伯罗奔尼撒战争是泰阿泰德与斯巴达之间从公元前431年到公元前404年进行的战争，所以这里的年代是一致的，因为泰阿泰德在战争结束时将是13岁，因此说他“生活在伯罗奔尼撒战争之后”是合理的。

Suda Lexicon中的第二处提到（例如见[1]）写道：-

泰阿泰德，生于本都的赫拉克利亚，哲学家，柏拉图的学生。

当然，尚不清楚这些是指同一个人还是两个不同的人。许多数学史家认为这些指的是同一个人。然而，1中的Bulmer-Thomas认为5中Allman的解释最有可能。根据这一理论，第二个泰阿泰德是第一个的儿子。如果是这样，那么他出生时泰阿泰德正在赫拉克利亚教书，并且会被他的父亲送到泰阿泰德，在那里跟随柏拉图在科学院接受教育。

泰阿泰德 于公元前369年参加了 泰阿泰德 与科林斯之间的战役。他在战斗中表现出色，但负伤并被送回 泰阿泰德。由于在战斗中受的伤，泰阿泰德 染上了痢疾，并在 泰阿泰德 去世。

泰阿泰德对数学做出了非常重要的贡献，尽管他的著作无一留存，但我们确实对他的贡献了解很多。欧几里得的Elements的第X卷和第XIII卷几乎可以肯定是对泰阿泰德工作的描述。这意味着，第X卷中描述的是泰阿泰德关于无理数长度的研究，许多人认为这是Elements中最出色的工作。帕普斯在为其对欧几里得的Elements第X卷所作的评注撰写的导言中写道（例如见[1]）：-

欧几里得关于“Elements”的论著第X卷的目的，是研究可公度的与不可公度量，即有理的与无理连续量。这门科学起源于毕达哥拉斯的学派，但在雅典人泰阿泰德手中经历了重要的发展，他由于在这一数学分支以及其他分支中的天赋而受到公正的钦佩。作为最有天赋的人之一，他耐心地探究这些科学分支中所包含的真理……并且在我看来，是就上述量建立精确区分和不可反驳证明的主要推动者。

因此，帕普斯告诉我们，泰阿泰德受到西奥多罗斯工作的启发而研究不可公度量，并且他对该理论做出了重大贡献。在托马斯·利特尔·希思的译本中，例如见[3]（我们以略有不同的形式重复帕普斯上述引文的一部分），无理数理论：-

……由雅典人泰阿泰德得到了相当大的发展，他在数学的这一部分以及其他部分中，证明了理应受到钦佩的能力。……至于上述量的精确区分，以及这一理论所产生的命题的严格证明，我相信它们主要是由这位数学家建立的。因为泰阿泰德已经区分了长度可公度的平方根与不可公度的平方根，并且按照不同的中项划分了更为一般地已知的无理线段，把中间的线归于几何，把二项式归于算术，把余理数归于和声，如罗德岛的欧德摩斯所述……

B L 巴特尔·伦德特·范德瓦尔登 在 [4] 中论证 Elements 第十卷完全是 泰阿泰德 的工作。他写道：-

研究过中项、二项和余项的同一位 泰阿泰德，是否也定义并研究了另外十种无理量，还是它们后来才被引入？在我看来，这一切都是一位数学家的成果。因为对13种无理量的研究是一个整体。贯穿全书的是同一个基本思想，所有情形都应用了相同的证明方法。……因此，整卷书都是 泰阿泰德 的工作。

然而，正如 Bulmer-Thomas 在 [1] 中指出的，巴特尔·伦德特·范德瓦尔登 的论证只有在我们假设 欧几里得 没有做大量工作来统一方法并给第十卷的工作提供一致进路时才成立。Bulmer-Thomas 更倾向于这样的猜测：尽管第十卷基于 泰阿泰德 的工作，但其中也有许多应归于 欧几里得。

柏拉图 在其著作 Theaetetus 中写道，泰阿泰德 描述了他如何推广 西奥多罗斯 关于 √3、√5、……、√17 是无理数的证明（例如见 [3]）：-

我们两人[泰阿泰德和年轻的苏格拉底]想到，既然这些平方根在数量上似乎是无限的，就试图找到一个集合名词来指称所有这些根。……我们把一般的数分成两类。可以表示为相等乘以相等的数，我们把它比作正方形，称之为平方数……中间的数，如三、五以及任何不能表示为相等乘以相等的数……我们把它比作长方形，称之为 矩形数……

然而，在 [10] 中，Paiow 论证说 西奥多罗斯 有一个一般方法，只是出于教学理由才呈现了特殊情形。如果他的论证成立，那么当然，泰阿泰德 就不会是第一个证明一般结果的人。

泰阿泰德 也被认为是 欧多克索斯 著作中出现的比例论的作者。

泰阿泰德 是第一个研究 八面体 和 二十面体 的人，据信 欧几里得 的 Elements 第十三卷是基于他的工作。一条注释（被认为是 杰米纽斯 所作）指出 [3]：-

……所谓的五个柏拉图立体，然而它们并不属于 柏拉图，其中三个属于毕达哥拉斯学派，即立方体、金字塔和 正十二面体，而八面体和二十面体则属于 泰阿泰德。

我们在上面引用了 帕普斯 中他描述 泰阿泰德 关于中项、二项和余项的工作。给定两个量 $a, b$，中项是 $ab$，二项是 $a + b$，余项是 $a - b$。容易看出，中项和二项分别与几何平均和算术平均密切相关。不太清楚的是，余项 $a - b$ 与现代调和平均有何关系。

Most of what we know of Theaetetus's life comes from the writing of Plato. It is clear that Plato held Theaetetus in the highest regard and he wrote two dialogues which had Theaetetus as the principal character, one of the dialogues being Theaetetus while the other is the Sophist.

In Theaetetus a discussion between Socrates, Theaetetus and his teacher Theodorus of Cyrene is recorded. This conversation took place in 399 BC and Theaetetus is described as a youth at the time. This allows us to give a fairly accurate date for Theaetetus's birth (although some have claimed that the Greek word could describe a man of up to 21 years old). Again from Plato we learn that Theaetetus's father, Euphronius of Sunium, was a wealthy man and left a large fortune. However, the money was squandered by the trustees of the will but despite this Theaetetus was generous to all around him.

In appearance Theaetetus had a snub nose and protruding eyes but he is described by Plato as having a beautiful mind and he is also described as being the perfect gentleman. Theodorus said that of all his pupils [1]:-

... he had never found one so marvellously gifted.

There are two references to a 'Theaetetus' in the Suda Lexicon (a work of a 10th century Greek lexicographer). The first states (see for example [1]):-

Theaetetus, of Athens, astronomer, philosopher, disciple of Socrates, taught at Heraclea. He was the first to construct the so-called five solids. He lived after the Peloponnesian war.

The Peloponnesian War was fought between Athens and Sparta from 431 BC to 404 BC so the dates here are consistent since Theaetetus would be 13 years old when the War ended so saying the he 'lived after the Peloponnesian war' is reasonable.

The second reference in the Suda Lexicon states (see for example [1]):-

Theaetetus, of Heraclea in Pontus, philosopher and pupil of Plato.

Of course it is unclear whether these refer to the same person or to two different people. There are many historians of mathematics who believe that these refer to the same person. Bulmer-Thomas in [1], however, thinks that Allman's explanation in [5] is the most likely. According to this theory the second Theaetetus was the son of the first. If this is so then he would have been born when Theaetetus of Athens was teaching in Heraclea and would have been sent by his father to Athens to be educated at the Academy there under Plato.

Theaetetus took part in the battle between Athens and Corinth in 369 BC. After acquitting himself with distinction in the battle, he was wounded and carried back to Athens. As a result of the wounds that he received in the battle, Theaetetus contracted dysentery and died in Athens.

Theaetetus made very important contributions to mathematics and despite none of his writing having survived we do know a great deal about his contribution. Book X and Book XIII of Euclid's Elements are almost certainly a description of Theaetetus's work. This means that it was Theaetetus's work on irrational lengths which is described in the Book X, thought by many to be the finest work of the Elements. Pappus wrote in the introduction to his commentary to Book X of Euclid's Elements (see for example [1]):-

The aim of Book X of Euclid's treatise on the "Elements" is to investigate the commensurable and the incommensurable, the rational and irrational continuous quantities. This science has its origin in the school of Pythagoras, but underwent an important development in the hands of the Athenian, Theaetetus, who is justly admired for his natural aptitude in this as in other branches of mathematics. One of the most gifted of men, he patiently pursued the investigation of truth contained in these branches of science ... and was in my opinion the chief means of establishing exact distinctions and irrefutable proofs with respect to the above mentioned quantities.

Pappus tells us, therefore, that Theaetetus was inspired by the work of Theodorus to work on incommensurables and that he made major contributions to the theory. In Heath's translation, see for example [3], (we repeat in a slightly different form part of the above quotation by Pappus) the theory of irrationals:-

... was considerably developed by Theaetetus the Athenian, who gave proof, in this part of mathematics as in others, of ability which has been justly admired. ... As for the exact distinctions of the above-named magnitudes and the rigorous demonstrations of the propositions to which this theory gives rise, I believe that they were chiefly established by this mathematician. For Theaetetus had distinguished square roots commensurable in length from those which are incommensurable, and who divided the more generally known irrational lines according to the different means, assigning the medial line to geometry, the binomial to arithmetic and the apotome to harmony, as stated by Eudemus...

B L van der Waerden argues in [4] that Book X of the Elements is entirely the work of Theaetetus. He writes:-

Has the same Theaetetus who studied the medial, the binomial, and the apotome, also defined and investigated the ten other irrationalities, or were they introduced later on? It seems to me that all this is the work of one mathematician. For the study of the 13 irrationalities is a unit. The same fundamental idea prevails throughout the book, the same methods of proof are applied in all cases. ... Hence the entire book is the work of Theaetetus.

However, as Bulmer-Thomas points out in [1], van der Waerden's argument only holds up if we assume that Euclid has not done a lots of work in unifying the methods and giving a consistent approach to the work of Book X. Bulmer-Thomas prefers the conjecture that although Book X is based on Theaetetus's work there is much due to Euclid presented there too.

Plato, writing in his work Theaetetus, has Theaetetus describe how he came to generalise Theodorus's proof that √3, √5, ..., √17 were irrational (see for example [3]):-

The idea occurred to the two of us [Theaetetus and the younger Socrates], seeing that these square roots appeared to be unlimited in multitude, to try to arrive at one collective term by which we could designate all these roots. ... We divided number in general into two classes. The number which can be expressed as equal multiplied by equal we likened to a square in form, and we called it square ... The intermediate number, such as three, five, and any number which cannot be expressed as equal multiplied by equal ... we likened to an oblong figure and called it an oblong number...

In [10], however, Paiow argues that Theodorus had a general method and only presented the particular cases for pedagogical reasons. If his arguments are valid then, of course, Theaetetus would not be the first to prove the general result.

Theaetetus is also thought to be the author of the theory of proportion which appears in Eudoxus's work.

Theaetetus was the first to study the octahedron and the icosahedron and it is believed that Book XIII of Euclid's Elements is based on his work. A comment (thought to be due to Geminus) states [3]:-

... the five so-called Platonic figures which, however, do not belong to Plato, three of the five being due to the Pythagoreans, namely the cube, the pyramid, and the dodecahedron, while the octahedron and the icosahedron are due to Theaetetus.

We quoted from Pappus above where he described Theaetetus's work on the medial, the binomial, and the apotome. Given two magnitudes $a, b$ the medial is $ab$, the binomial is $a + b$, and the apotome is $a - b$. It is easy to see that the medial and the binomial are closely related to the geometric mean and the arithmetic mean respectively. What is much less clear is how the apotome is $a - b$ is related to the modern harmonic mean.