西奥多罗斯Theodoros of Cyrene

西奥多罗斯是普罗泰戈拉的学生，他自己是柏拉图的导师，教他数学，也是泰阿泰德的导师。柏拉图往返于埃及，在这样的场合他与西奥多罗斯在西奥多罗斯共度时光。然而，西奥多罗斯并没有在西奥多罗斯度过他的一生，因为他肯定在苏格拉底活着的时候在雅典。

除了数学方面的工作，西奥多罗斯 还[5]：-

……在天文学、算术、音乐以及所有教育科目方面……杰出。

作为 毕达哥拉斯 学派的成员，西奥多罗斯 是昔勒尼道德哲学学派的主要哲学家之一。他认为快乐和痛苦既非善也非恶。他相信，愉快和智慧足以带来幸福。

我们对 西奥多罗斯 的了解来自 柏拉图，他在其著作 Theaetetus 中写到了他。西奥多罗斯 因其对 无理数 数的发展所作的贡献而为数学家所铭记，而 柏拉图 所描述的正是他工作的这一方面（例如见[5]）：-

[西奥多罗斯]向我们证明关于平方根的某个事实，我指的是面积为三个平方单位和五个平方单位的正方形的边（即根），这些根的长度与单位长度不可可公度的，他就这样继续下去，把所有个别情形一直做到十七个平方单位的根，到了这一点，不知什么缘故，他停住了。

我们对西奥多罗斯数学成就的全部了解都来自柏拉图中的这段话。然而，立刻就有一些值得注意的地方。第一点是，柏拉图并没有把证明二的平方根是无理数归功于西奥多罗斯。这一定是因为√2在西奥多罗斯研究这个问题之前就已经被证明是无理数了，有人声称这是由毕达哥拉斯本人证明的。

毫无疑问，西奥多罗斯会利用毕达哥拉斯的定理构造出长度为√3、√5等的线段。同样清楚的是，西奥多罗斯在这里没有一般性的结果，因为柏拉图接着描述了西奥多罗斯的结果如何启发泰阿泰德和苏格拉底去考察推广：-

我们两人（泰阿泰德和苏格拉底）想到，既然这些平方根在数量上似乎是无限的，那就试着找到一个集合名词，用来指称所有这些根……

所以接下来自然要问的问题是，西奥多罗斯是如何证明√3、√5、…、√17是无理数的，却没有给出一个能清楚地证明任何非平方数都是无理数的证明。证明√2是无理数的通常证明，即假设$√2 = \large\frac{p}{q}\normalsize$，其中$\large\frac{p}{q}\normalsize$是一个最简形式的有理的，并通过表明$p$和$q$都是偶数来导出矛盾，这个证明西奥多罗斯应该是知道的。这个证明很容易推广（对于以数而非长度来思考的现代数学家而言），以表明对于任何非平方数$n$，$√n$都是无理数。几乎无法想象西奥多罗斯会对√3、√5、…、√17中的每一个都使用这个证明，而在做到17之前还没有得到一个一般定理。

Zeuthen在1915年提出了一个有趣的建议。他提出，西奥多罗斯可能使用了后来出现在欧几里得的Elements中的结果，即：-

如果两个不等量中较小的一个不断从较大的一个中减去，而余下的量永远不能量尽它前面的量，那么这两个量就是不可公度的。

托马斯·利特尔·希思 [5] 说明了如何利用这一结果来证明 √5 是无理数。从 1 和 √5 开始。

$\large\frac{√5}{1}\normalsize = 2 +(√5-2)$
$\large\frac{1}{√5-2}\normalsize = 4 + (√5-2)^{2}$
$\large\frac{√5-2}{(√5-2)^{2}\normalsize} = \large\frac{1}{√5-2}\normalsize = 4 + (√5-2)^{2}$
.......

现在这个过程显然不会终止，因为比值 $1 : (√5-2)$ 与 $(√5-2) : (√5-2)^{2}$ 相同。托马斯·利特尔·希思 [5] 给出了这一过程的一个几何版本，从边长为 1、2 和 √5 的直角三角形开始，这可能接近 西奥多罗斯 所使用的方法。然而，对于 西奥多罗斯 的方法，除了猜测之外几乎无能为力。

Theodorus of Cyrene was a pupil of Protagoras and himself the tutor of Plato, teaching him mathematics, and also the tutor of Theaetetus. Plato travelled to and from Egypt and on such occasions he spent time with Theodorus in Cyrene. Theodorus, however, did not spend his whole life in Cyrene for he was certainly in Athens at a time when Socrates was alive.

Theodorus, in addition to his work in mathematics, was [5]:-

... distinguished ... in astronomy, arithmetic, music and all educational subjects.

A member of the society of Pythagoras, Theodorus was one of the main philosophers in the Cyrenaic school of moral philosophy. He believed that pleasures and pains are neither good nor bad. Cheerfulness and wisdom, he believed, were sufficient for happiness.

Our knowledge of Theodorus comes through Plato who wrote about him in his work Theaetetus. Theodorus is remembered by mathematicians for his contribution to the development of irrational numbers and it is this aspect of his work which Plato describes (see for example [5]):-

[Theodorus] was proving to us a certain thing about square roots, I mean the side (i.e. root) of a square of three square units and of five square units, that these roots are not commensurable in length with the unit length, and he went on in this way, taking all the separate cases up to the root of seventeen square units, at which point, for some reason, he stopped.

Our whole knowledge of Theodorus's mathematical achievements are given by this passage from Plato. Yet there are points of interest which immediately arise. The first point is that Plato does not credit Theodorus with a proof that the square root of two was irrational. This must be because √2 was proved irrational before Theodorus worked on the problem, some claim this was proved by Pythagoras himself.

There is no doubt that Theodorus would have constructed lines of length √3, √5 etc. using Pythagoras's theorem. It is also clear that Theodorus had no general result here, for Plato goes on to describe how Theodorus's results inspired Theaetetus and Socrates to look at generalisations:-

The idea occurred to the two of us (Theaetetus and 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....

So the question which naturally comes next is how did Theodorus prove that √3, √5, ..., √17 were irrational without giving a proof which would clearly prove that any non-square number was irrational. The usual proof that √2 is irrational, namely the one which supposes that $√2 = \large\frac{p}{q}\normalsize$ where $\large\frac{p}{q}\normalsize$ is a rational in its lowest terms and derives a contradiction by showing that $p$ and $q$ are both even, would have been known to Theodorus. This proof generalises easily (for a modern mathematicians thinking in terms of numbers rather than lengths) to show $√n$ is irrational for any non-square $n$. It is almost impossible to conceive that Theodorus would have used this proof on each of √3, √5, ..., √17 without obtaining a general theorem long before he got to 17.

An interesting proposal was made by Zeuthen in 1915. He suggested that Theodorus may have used the result which would later appear in Euclid's Elements namely:-

If, when the lesser of two unequal magnitudes is continually subtracted in turn from the greater, that which is left never measures the one before it, the magnitudes will be incommensurable.

Heath [5] illustrates the use of this result to show that √5 is irrational. Start with 1 and √5.

$\large\frac{√5}{1}\normalsize = 2 +(√5-2)$
$\large\frac{1}{√5-2}\normalsize = 4 + (√5-2)^{2}$
$\large\frac{√5-2}{(√5-2)^{2}\normalsize} = \large\frac{1}{√5-2}\normalsize = 4 + (√5-2)^{2}$
.......

The process now clearly fails to terminate since the ratio $1 : (√5-2)$ is the same as $(√5-2) : (√5-2)^{2}$. Heath [5] gives a geometric version of this, starting with a right-angled triangle with sides 1, 2 and √5 which may be close to the method that Theodorus used. However there is little chance to do more than guess at Theodorus's method.