希俄斯的希波克拉底Hippocrates of Chios

希俄斯的希波克拉底 曾在雅典讲学，并研究 squaring the circle 和 倍立方 的古典问题。关于他的生平所知甚少，但据说他是一位出色的几何学家，而在其他方面却愚钝且缺乏见识。有人声称他因天真而被骗走了一大笔钱。扬布利科斯 [4] 写道：

毕达哥拉斯学派的一员 [希俄斯的希波克拉底] 失去了财产，当这一不幸降临时，他获准通过教授几何学来赚钱。

托马斯·利特尔·希思 [6] 讲述了这个故事的两个版本：

故事的一个版本是，[希俄斯的希波克拉底] 曾是一名商人，但因被海盗船俘获而失去了全部财产。随后他来到雅典追诉那些作恶者，并在长期逗留期间听课，最终在几何学上达到了如此高的造诣，以至于他试图化圆为方。

托马斯·利特尔·希思 还讲述了 亚里士多德 所讲的另一个版本的故事：

……他任由 费隆 的海关官员骗走了一大笔钱，从而在 亚里士多德 看来，这证明他虽然是一位优秀的几何学家，但在日常事务中却愚钝且无能。

有人认为，他在雅典的这段“长期停留”大约在公元前450年至公元前430年之间。

在试图化圆为方的过程中，希俄斯的希波克拉底能够利用他的定理——两个圆的面积之比等于它们半径的平方之比——求出lunes即某些月牙形图形的面积。我们将在下面更全面地描述这一令人印象深刻的成就。

希俄斯的希波克拉底还表明，如果能够在一个数与其两倍之间确定两个比例中项，那么一个立方体就可以被加倍。这对倍立方问题的尝试产生了重大影响，此后所有的努力都转向了比例中项问题。

他是第一个撰写Elements of Geometry的人，尽管他的著作现已失传，但其中必定包含了后来欧几里得在Elements第1卷和第2卷中收录的许多内容。普罗克洛是最后一位重要的希腊哲学家，生活于约公元450年，他写道：

希俄斯的希波克拉底，月牙形求积的发现者，……是据记载第一个实际编纂“几何原本”的人。

希俄斯的希波克拉底的书还包含了对二次方程的几何解法，并包含了早期的积分方法。

罗得岛的罗德岛的欧德摩斯是亚里士多德的学生，他写了History of Geometry，其中描述了希俄斯的希波克拉底关于月牙的贡献。这部著作没有流传下来，但西里西亚的辛普利修斯的西里西亚的辛普利修斯，在约530年写作时，能够接触到罗德岛的欧德摩斯的著作，他引用了关于希俄斯的希波克拉底月牙的那段话，“除了少数增补外逐字逐句”，这些增补取自欧几里得的Elements，以使描述更清楚。

我们将首先引用罗德岛的欧德摩斯中关于希俄斯的希波克拉底的月牙形的部分段落，遵循那些将西里西亚的辛普利修斯所添加的内容从欧几里得的Elements中剥离出来的数学史家的做法。关于我们给出的译文以及哪些部分出自罗德岛的欧德摩斯的讨论，参见[6]：

月牙形的求积，由于月牙形与圆的密切关系，曾被认为属于一类不常见的命题，最早由希俄斯的希波克拉底研究，他的论述被认为是正确的；因此我们将详细讨论并描述它们。他从以下命题开始，并将其作为对此目的有用的第一个定理：相似圆弓形彼此之比等于其底上正方形之比。他通过首先证明直径上的正方形与圆有相同的比来证明这一点。

在继续引用之前，我们应当注意，希俄斯的希波克拉底试图“化月牙为方”，他的意思是要构造一个与月牙面积相等的正方形。这正是“化圆为方”问题的含义，即构造一个面积等于圆面积的正方形。再次按照托马斯·利特尔·希思在[6]中的翻译：-

在证明了这一点之后，他接着展示了以何种方式能够化一个外弧为半圆周的月牙为方。他通过在一个等腰直角三角形和一个与两边所截弓形相似的圆弓形外作一个半圆来实现这一点。然后，由于底边上的弓形等于两边上的弓形之和，因此，当把底边弓形上方的三角形部分加到两者上时，月牙将等于三角形。因此，月牙被证明等于三角形后，就可以化为方了。

插图：Lune.gif ↗
要理解希俄斯的希波克拉底在这里的论证，请看图示。

$ABCD$是一个正方形，$O$是它的中心。图中的两个圆是以$O$为中心过$A, B, C$和$D$的圆，以及以$D$为中心过$A$和$C$的圆。

首先注意，$AB$上标有1的弓形在圆心处张成直角（角$AOB$），而$AC$上的弓形2也在圆心处张成直角（角$ADC$）。

因此，$AB$上的弓形1和$AC$上的弓形2相似。现在
$\Large\frac {\text{segment 1}}{\text{segment 2}} \normalsize = \Large\frac {AB^{2}}{AC^{2}} \normalsize = \Large\frac{1}{2}$因为由毕达哥拉斯的定理$AB^{2}+ BC^{2}= AC^{2}$，并且$AB = BC$所以$AC^{2}= 2AB^{2}$。

既然线段2是线段1的两倍，那么线段2就等于标有1的两条线段之和。

然后希俄斯的希波克拉底论证说，从半圆$ABC$中去掉两条线段1后得到的是三角形$ABC$，而三角形是可以化为等面积正方形的（如何构造一个与给定三角形等面积的正方形是众所周知的）。

然而，如果我们从半圆$ABC$中减去弓形2，就得到第二幅图中所示的月牙。因此，希俄斯的希波克拉底证明了月牙可以化为方。

然而，希俄斯的希波克拉底在研究月牙方面走得更远。我们详细考察的这个证明是月牙的外弧为半圆弧的情形。他还研究了外弧小于半圆弧的情形，以及外弧大于半圆弧的情形，在每种情形下都证明了月牙可以化为方。这是一项非凡的成就，也是化圆为方尝试中的一个重大步骤。正如托马斯·利特尔·希思在[6]中所写：-

……他希望表明，如果圆不能用这些方法化为方，那么这些方法可以用来求某些由圆弧围成的图形的面积，即某些月牙，甚至某个圆与某个月牙之和的面积。

还有一项数学史家认为希俄斯的希波克拉底取得的非凡成就，尽管我们没有直接的证明，因为他的著作没有流传下来。在希俄斯的希波克拉底对月牙的研究中，如罗德岛的欧德摩斯所描述的，他使用了圆与圆之比等于其直径上的正方形之比的定理。这个定理由欧几里得在Elements中证明，在那里是通过欧多克索斯的穷竭法来证明的。然而，欧多克索斯出生在希俄斯的希波克拉底去世后几年之内，因此随之而来的是一个引人入胜的问题：希俄斯的希波克拉底是如何证明这个定理的。由于罗德岛的欧德摩斯似乎完全满意希俄斯的希波克拉底确实有一个正确的证明，从这些间接证据来看，我们几乎可以肯定地推断，希俄斯的希波克拉底本人至少发展了穷竭法的一个变体。

Hippocrates of Chios taught in Athens and worked on the classical problems of squaring the circle and duplicating the cube. Little is known of his life but he is reported to have been an excellent geometer who, in other respects, was stupid and lacking in sense. Some claim that he was defrauded of a large sum of money because of his naiveté. Iamblichus [4] writes:-

One of the Pythagoreans [Hippocrates] lost his property, and when this misfortune befell him he was allowed to make money by teaching geometry.

Heath [6] recounts two versions of this story:-

One version of the story is that [Hippocrates] was a merchant, but lost all his property through being captured by a pirate vessel. He then came to Athens to persecute the offenders and, during a long stay, attended lectures, finally attaining such proficiency in geometry that he tried to square the circle.

Heath also recounts a different version of the story as told by Aristotle:-

... he allowed himself to be defrauded of a large sum by custom-house officers at Byzantium, thereby proving, in Aristotle's opinion, that, though a good geometer, he was stupid and incompetent in the business of ordinary life.

The suggestion is that this 'long stay' in Athens was between about 450 BC and 430 BC.

In his attempts to square the circle, Hippocrates was able to find the areas of lunes, certain crescent-shaped figures, using his theorem that the ratio of the areas of two circles is the same as the ratio of the squares of their radii. We describe this impressive achievement more fully below.

Hippocrates also showed that a cube can be doubled if two mean proportionals can be determined between a number and its double. This had a major influence on attempts to duplicate the cube, all efforts after this being directed towards the mean proportionals problem.

He was the first to write an Elements of Geometry and although his work is now lost it must have contained much of what Euclid later included in Books 1 and 2 of the Elements. Proclus, the last major Greek philosopher, who lived around 450 AD wrote:-

Hippocrates of Chios, the discoverer of the quadrature of the lune, ... was the first of whom it is recorded that he actually compiled "Elements".

Hippocrates' book also included geometrical solutions to quadratic equations and included early methods of integration.

Eudemus of Rhodes, who was a pupil of Aristotle, wrote History of Geometry in which he described the contribution of Hippocrates on lunes. This work has not survived but Simplicius of Cilicia, writing in around 530, had access to Eudemus's work and he quoted the passage about the lunes of Hippocrates 'word for word except for a few additions' taken from Euclid's Elements to make the description clearer.

We will first quote part of the passage of Eudemus about the lunes of Hippocrates, following the historians of mathematics who have disentangled the additions from Euclid's Elements which Simplicius added. See [6] both for the translation which we give and for a discussion of which parts are due to Eudemus:-

The quadratures of lunes, which were considered to belong to an uncommon class of propositions on account of the close relation of lunes to the circle, were first investigated by Hippocrates, and his exposition was thought to be correct; we will therefore deal with them at length and describe them. He started with, and laid down as the first of the theorems useful for the purpose, the proposition that similar segments of circles have the same ratio to one another as the squares on their bases. And this he proved by first showing that the squares on the diameters have the same ratio as the circles.

Before continuing with the quote we should note that Hippocrates is trying to 'square a lune' by which he means to construct a square equal in area to the lune. This is precisely what the problem of 'squaring the circle' means, namely to construct a square whose area is equal to the area of the circle. Again following Heath's translation in [6]:-

After proving this, he proceeded to show in what way it was possible to square a lune the outer circumference of which is that of a semicircle. This he affected by circumscribing a semicircle about an isosceles right-angled triangle and a segment of a circle similar to those cut off by the sides. Then, since the segment about the base is equal to the sum of those about the sides, it follows that, when the part of the triangle above the segment about the base is added to both alike, the lune will be equal to the triangle. Therefore the lune, having been proved equal to the triangle, can be squared.

插图：Lune.gif ↗
To follow Hippocrates' argument here, look at the diagram.

$ABCD$ is a square and $O$ is its centre. The two circles in the diagram are the circle with centre $O$ through $A, B, C$ and $D$, and the circle with centre $D$ through $A$ and $C$.

Notice first that the segment marked 1 on $AB$ subtends a right angle at the centre of the circle (the angle $AOB$) while the segment 2 on $AC$ also subtends a right angle at the centre (the angle $ADC$).

Therefore the segment 1 on $AB$ and the segment 2 on $AC$ are similar. Now
$\Large\frac {\text{segment 1}}{\text{segment 2}} \normalsize = \Large\frac {AB^{2}}{AC^{2}} \normalsize = \Large\frac{1}{2}$ since $AB^{2}+ BC^{2}= AC^{2}$ by Pythagoras's theorem, and $AB = BC$ so $AC^{2}= 2AB^{2}$.

Now since segment 2 is twice segment 1, the segment 2 is equal to the sum of the two segments marked 1.

Then Hippocrates argues that the semicircle $ABC$ with the two segments 1 removed is the triangle $ABC$ which can be squared (it was well known how to construct a square equal to a triangle).

However, if we subtract the segment 2 from the semicircle $ABC$ we get the lune shown in the second diagram. Thus Hippocrates has proved that the lune can be squared.

However, Hippocrates went further than this in studying lunes. The proof we have examined in detail is one where the outer circumference of the lune is the arc of a semicircle. He also studied the cases where the outer arc was less than that of a semicircle and also the case where the outer arc was greater than a semicircle, showing in each case that the lune could be squared. This was a remarkable achievement and a major step in attempts to square the circle. As Heath writes in [6]:-

... he wished to show that, if circles could not be squared by these methods, they could be employed to find the area of some figures bounded by arcs of circles, namely certain lunes, and even of the sum of a certain circle and a certain lune.

There is one further remarkable achievement which historians of mathematics believe that Hippocrates achieved, although we do not have a direct proof since his works have not survived. In Hippocrates' study of lunes, as described by Eudemus, he uses the theorem that circles are to one another as the squares on their diameters. This theorem is proved by Euclid in the Elements and it is proved there by the method of exhaustion due to Eudoxus. However, Eudoxus was born within a few years of the death of Hippocrates, and so there follows the intriguing question of how Hippocrates proved this theorem. Since Eudemus seems entirely satisfied that Hippocrates does indeed have a correct proof, it seems almost certain from this circumstantial evidence that we can deduce that Hippocrates himself developed at least a variant of the method of exhaustion.