希庇亚Hippias

希庇亚 是一位政治家和哲学家，他四处游历，靠提供服务赚钱。他讲授诗歌、语法、历史、政治、考古学、数学和天文学。柏拉图 将他描述为一个虚荣的人，既傲慢又自夸，拥有广泛但肤浅的知识。托马斯·利特尔·希思 在 [3] 中写到他的这一性格特点时告诉我们：-

他声称……曾带着所有穿戴都是自己制作的东西去过一次奥林匹亚节，包括戒指和凉鞋（刻有花纹）、油瓶、刮身板、鞋子、衣服，以及一条昂贵类型的波斯腰带；他还带了诗歌、史诗、悲剧、酒神颂和各种散文作品。

至于 希庇亚 的学术成就，托马斯·利特尔·希思 写道：-

他是计算科学、几何学、天文学、‘节奏与和声以及正确书写’的大师。他还有一种奇妙的记忆术系统，使他一旦听到一串五十个名字就能全部记住。

一个相当有趣的故事，与其说是在讲希庇亚，不如说是在讲斯巴达人：据说他在斯巴达讲学没有收到报酬，因为[3]：-

……斯巴达人无法忍受关于天文学、几何学或计算的讲学；他们中只有极少数人甚至能计数；他们喜欢的是历史和考古学。

据说希庇亚讲授考古学，他在斯巴达讲学时似乎选错了主题！

希庇亚对数学的唯一贡献似乎是割圆曲线，他可能曾用它来解决trisecting an angle和squaring the circle。这条曲线可用于将角分成任意数量的等份。也许我们能给予希庇亚的最高赞誉，是报告某些数学史家的论点，他们声称发现割圆曲线的希庇亚不可能是希庇亚，因为当时的几何学尚未发达到足以让他做出这些发现的程度。然而，他们的论点并未被普遍接受，且有充分证据将割圆曲线的发现归功于希庇亚。

托马斯·利特尔·希思 [3] 写道：-

大概在公元前420年左右，希庇亚为了三等分任意角的目的发明了称为割圆曲线的曲线。

然而这远不确定，并且有一些证据表明，公元前一世纪写作的杰米纽斯拥有一部希庇亚关于割圆曲线的论著，其中指出了如何用它化圆为方。如果情况确实如此，那么希庇亚的论著必定在这时与公元三世纪的Sporus之间丢失了。

帕普斯于340年写了他的几何学主要著作Synagoge。这是一部八卷本的数学著作集。第四卷包含对希庇亚的割圆曲线的描述。
插图：Quadratrix.gif ↗
看割圆曲线的图。

$ABCD$是正方形，$BED$是圆的一部分，圆心为$A$，半径为$AB$。当半径$AB$绕$A$旋转到位置$AD$时，直线$BC$以相同速率平行于自身移动，最终到达$AD$。那么，旋转半径$AB$与移动直线$BC$的交点$F$的轨迹就是割圆曲线。因此

$\Large\frac{\text{angle } EAD}{\text{angle } BAD}\normalsize = \Large\frac{\text{arc } ED}{\text{arc } BED}\normalsize =\Large\frac{FH}{AB}$，

所以，取$AB = 1$，

角$EAD$ = 弧$ED$ = $FH \times \Large\frac{\pi}{2}$。

要将角$FAD$按给定比，比如$p : q$，进行分割，然后在直线$FH$上取一点$P$，使其按比$p : q$分割该直线。
过$P$作平行于$AD$的直线，与割圆曲线交于$Q$。于是$AQ$按比$p : q$分割角$FAD$。

帕普斯还给出了化圆为方所需的更为复杂的作图版本。然而，帕普斯报告说，Sporus对希庇亚的方法有两点批评，他同意这些批评。第二点具体涉及我们尚未描述的化圆为方所需的作图。但第一点涉及割圆曲线本身的作图。帕普斯报告说，Sporus写道（见[3]）：-

作图被认为所要达到的目的本身，实际上已在假设中被假定了。因为，若从B出发的两个点，一个沿直线运动到A，另一个沿圆周运动到D，且所用时间相等，除非你先知道直线AB与圆周BED之比，否则这怎么可能呢？事实上，这个比也必定是运动速度之比。因为，如果你采用并未明确调整到这个比的速度，那么除非纯属偶然，否则你怎么能使这两个运动在同一时刻结束呢？这样看来，这件事岂不是荒谬的吗？

这里的要点似乎在于希庇亚究竟想用他的割圆曲线表明什么。当然，他完全清楚自己并没有为化圆为方提供一种尺规作图。正如帕普斯和Sporus所暗示的，他关于化圆为方究竟证明了什么，远不清楚。

Hippias of Elis was a statesman and philosopher who travelled from place to place taking money for his services. He lectured on poetry, grammar, history, politics, archaeology, mathematics and astronomy. Plato describes him as a vain man being both arrogant and boastful, having a wide but superficial knowledge. Heath tells us something of this character when he writes in [3]:-

He claimed ... to have gone once to the Olympian festival with everything that he wore made by himself, ring and sandal (engraved), oil-bottle, scraper, shoes, clothes, and a Persian girdle of expensive type; he also took poems, epics, tragedies, dithyrambs, and all sorts of prose works.

As to Hippias's academic achievements, Heath writes:-

He was a master of the science of calculation, geometry, astronomy, 'rhythms and harmonies and correct writing'. He also had a wonderful system of mnemonics enabling him, if he once heard a string of fifty names to remember them all.

A rather nice story, which says more of the Spartans than it does of Hippias, is that it was reported that he received no payment for the lectures he gave in Sparta since [3]:-

... the Spartans could not endure lectures on astronomy or geometry or calculation; it was only a small minority of them who could even count; what they liked was history and archaeology.

Since Hippias was reported to give lectures on archaeology, he seems to have chosen the wrong topics when he lectured in Sparta!

Hippias's only contribution to mathematics seems to be the quadratrix which may have been used by him for trisecting an angle and squaring the circle. The curve may be used for dividing an angle into any number of equal parts. Perhaps the highest compliment that we can pay to Hippias is to report on the arguments of certain historians of mathematics who have claimed that the Hippias who discovered the quadratrix cannot be Hippias of Elis since geometry was not far enough advanced at this time to have allowed him to make these discoveries. However, their arguments are not generally accepted and there is ample evidence to attribute the discovery of the quadratrix to Hippias of Elis.

Heath [3] writes:-

It was probably about 420 BC that Hippias of Elis invented the curve known as the quadratrix for the purpose of trisecting any angle.

However this is far from certain and there is some evidence to suggest that Geminus, writing in the first century BC, had in his possession a treatise by Hippias of Elis on the quadratrix which indicated how it could be used to square the circle. If this is indeed the case then the treatise by Hippias must have been lost between this time and that of Sporus in the third century AD.

Pappus wrote his major work on geometry Synagoge in 340. It is a collection of mathematical writings in eight books. Book IV contains a description of the quadratrix of Hippias.
插图：Quadratrix.gif ↗
Look at the diagram of the quadratrix.

$ABCD$ is a square and $BED$ is part of a circle, centre $A$ radius $AB$. As the radius $AB$ rotates about $A$ to move to the position $AD$ then the line $BC$ moves at the same rate parallel to itself to end at $AD$. Then the locus of the point of intersection $F$ of the rotating radius $AB$ and the moving line $BC$ is the quadratrix. Hence

$\Large\frac{\text{angle } EAD}{\text{angle } BAD}\normalsize = \Large\frac{\text{arc } ED}{\text{arc } BED}\normalsize =\Large\frac{FH}{AB}$,

so, taking $AB = 1$,

angle $EAD$ = arc $ED$ = $FH \times \Large\frac{\pi}{2}$.

To divide the angle $FAD$ in a given ratio, say $p : q$, then draw a point $P$ on the line $FH$ dividing it in the ratio $p : q$.
Draw a line through $P$ parallel to $AD$ to meet the quadratrix at $Q$. Then $AQ$ divides angle $FAD$ in the ratio $p : q$.

Pappus also gives the rather more complicated version of the construction necessary to square the circle. However, Pappus reports that Sporus had two criticisms of Hippias's method with which he agrees. The second is specifically related to the construction necessary for squaring the circle which we have not described. The first however relates to the construction of the quadratrix itself. Pappus reports that Sporus writes (see [3]):-

The very thing for which the construction is thought to serve is actually assumed in the hypothesis. For how is it possible, with two points starting from B, to make one of them move along a straight line to A and the other along a circumference to D in an equal time, unless you first know the ratio of the straight line AB to the circumference BED? In fact this ratio must also be that of the speeds of motion. For, if you employ speeds not definitely adjusted to this ratio, how can you make the motions end at the same moment, unless this should sometime happen by pure chance? Is not the thing thus shown to be absurd?

The point here seems to be a question of what exactly Hippias is trying to show with his quadratrix. Certainly he knew perfectly well that he was not providing a ruler and compass construction for squaring the circle. Exactly what he has proved concerning squaring the circle is, as Pappus and Sporus suggest, far from clear.