阿尔库塔斯Archytas

阿尔库塔斯 是一位数学家、政治家和哲学家，生活在大希腊的 阿尔库塔斯，大希腊是意大利南部的一个地区，在公元前五世纪处于希腊控制之下。毕达哥拉斯学派曾一度在整个大希腊势力强大，但遭到攻击和驱逐，直到只有 阿尔库塔斯 城仍是他们的据点。阿尔库塔斯 在 阿尔库塔斯 领导毕达哥拉斯学派，并试图联合该地区的希腊城镇，结成联盟以对抗他们的非希腊邻居。他担任 阿尔库塔斯 军队的总司令七年，尽管有法律规定任何人担任该职位不得超过一年。柏拉图 成为他的密友，是在大希腊逗留期间与他结识的。托马斯·利特尔·希思 在 [4] 中写道：-

……据说他通过一封信，使 柏拉图 免于死于 Dionysius 之手。

事实上，柏拉图 曾多次前往西西里，正是在公元前 361 年的第三次旅行中，他被 Dionysius II 扣留。柏拉图 写信给 阿尔库塔斯，后者派船去营救他。关于 阿尔库塔斯 与 柏拉图 之间关系的更多细节，请参阅有趣的文章 [8]。

鉴于上述故事以及阿尔库塔斯在苏格拉底之后的结论，将他列入前苏格拉底哲学家的著作中，如[3]所做的那样，可能显得奇怪。然而这样做是因为阿尔库塔斯哲学的风格，而非严格的年代顺序。

阿尔库塔斯 是 菲洛劳斯 的学生，因此是 毕达哥拉斯 哲学的坚定支持者，相信数学提供了理解万物的途径。尽管 阿尔库塔斯 研究许多主题，但由于他是毕达哥拉斯学派成员，数学是他的主要学科，所有其他学科都被视为依赖于数学。他声称数学由四个分支组成，即几何、算术、天文学和音乐。他还相信数学研究在其他方面也很重要，他著作中保存下来的一段残篇表明了这一点（见 [3] 或 [6]）：-

在我看来，数学家似乎具有卓越的洞察力，他们会正确地思考存在着的具体事物，这一点儿也不奇怪；因为既然他们能卓越地洞察宇宙的物理，他们也很可能对存在着的具体事物有卓越的见解。确实，他们传给了我们关于星辰的速度及其升落、关于几何、算术、天文，尤其是音乐的敏锐洞察力。这些似乎是姊妹科学，因为它们关注的是存在的前两种相关形式[数和量]。

这段残篇来自他某部著作的序言，有人称该著作题为On Mathematics，而另一些人则称其题为On Harmonics。当然，在这段引文之后，讨论了音高、频率和一种声音理论。它确实包含一些错误，但仍是一部非凡的著作，并构成了柏拉图著作中声音理论的基础。

阿尔库塔斯研究了调和平均数并给它起了这个名字（在更早的时代它被称为子相反）。他研究这个的原因是他对倍立方问题的兴趣，即找到一个立方体的边，其体积是给定立方体的两倍。希俄斯的希波克拉底将问题简化为找到两个比例中项。阿尔库塔斯用一个非凡的几何解法解决了这个问题（当然不是尺规作图）。

阿尔库塔斯在解决两条线段之间找到两个比例中项的问题时引入的一个有趣创新是将运动引入几何。他的方法使用一个在三维空间中旋转的半圆，以及它切割另一个三维曲面所形成的曲线。

我们通过阿什凯隆的欧托基奥斯的阿什凯隆的欧托基奥斯的著作了解到阿尔库塔斯对倍立方问题的解法。在这些著作中，阿什凯隆的欧托基奥斯声称引用了罗得岛的罗德岛的欧德摩斯在History of geometry中给出的描述，但引文的准确性受到[10]作者的怀疑。

阿尔库塔斯的另一个有趣的数学发现是，不存在一个数，它是两个比率为$(n+1) : n$的数之间的几何中项。他的证明最有趣的地方在于，它接近多年后欧几里得给出的证明，并且它引用了后来出现在欧几里得的Elements第七卷中的已知定理。

刚刚给出的论证使巴特尔·伦德特·范德瓦尔登声称（例如见[5]），Elements第七卷中出现的许多结果早于阿尔库塔斯。他声称，显然有一些著作，在欧几里得撰写Elements之前许多年写成，涵盖了相同的材料。阿尔库塔斯建立在这项早期工作之上，他的发现很大程度上就是欧几里得在Elements第八卷中提出的那些。按照巴特尔·伦德特·范德瓦尔登的这些论证，现在广泛接受的是，欧几里得借用了阿尔库塔斯的工作用于Elements第八卷。

阿尔库塔斯有时被称为力学之父，据说他发明了两种机械装置。一种装置是一只机械鸟[2]：-

这只鸟显然悬挂在一根枢轴杆的末端，整个装置通过蒸汽或压缩空气的喷射而旋转。

另一种机械装置是给儿童的拨浪鼓，用亚里士多德的话说（例如见[4]）是有用的：-

……给孩子们玩，以占据他们的时间，从而防止他们打坏家里的东西（因为年轻人无法保持静止）。

对于公元前400年的一位发明家来说，这确实是一个非凡的现代思想！事实上，这种对应用数学的兴趣与柏拉图的纯数学思想形成对比，而这种对比构成了波兰作家C K Norwid（1821-1883）所写一首诗的基础。这首引人入胜的诗由爱德华·马尔切夫斯基在[9]中讨论并给出了法文翻译。

西里西亚的辛普利修斯在其Physics中引用了阿尔库塔斯的观点，即宇宙是无限的（在托马斯·利特尔·希思的翻译[4]中）：-

如果我处在外面，比如在恒星的天空，我能向外伸出手或手杖吗？假设我不能，那是荒谬的：如果我能伸出去，那么外面的东西必定是物体或空间（正如我们将看到的，是哪一个并无区别）。那么我们同样可以再到那之外，如此类推，每到达一个新的界限就问同样的问题；如果总有一个新的地方可以伸出手杖，这显然涉及无限延伸。如果这样延伸的是物体，命题就得到了证明；但即使它是空间，那么，既然空间是物体所在或可能在的地方，而在永恒事物的情况下，我们必须把潜在存在的东西当作存在，同样可以推出必定有物体和空间无限延伸。

当涉及政治哲学和伦理学时，阿尔库塔斯同样将他的思想建立在数学基础之上。他写道（例如见[3]或[6]）：-

当数学推理被发现时，它制约政治派系并增进和谐，因为它的存在没有不公平的优势，平等占据主导。有了数学推理，我们就能消除彼此交往中的差异。通过它，穷人从强者那里取得，富人给予贫困者，双方都相信它能获得平等的份额……

最后，我们再次引用阿尔库塔斯关于其学习理论的著作。这段残篇出现在[3]或[6]中：-

要想通晓自己不懂的事物，要么向他人学习，要么自己探求。学习来自他人，是外来的；而探求则出于自身、依靠自身。不寻求而有所发现既困难又罕见，但若加以寻求，便易于驾驭、轻松可行，尽管不懂如何寻求的人无法找到。

Archytas of Tarentum was a mathematician, statesman and philosopher who lived in Tarentum in Magna Graecia, an area of southern Italy which was under Greek control in the fifth century BC. The Pythagoreans, who had at one stage been strong throughout Magna Graecia, were attacked and expelled until only the town of Tarentum remained a stronghold for them. Archytas led the Pythagoreans in Tarentum and tried to unite the Greek towns in the area to form an alliance against their non-Greek neighbours. He was commander in chief of the forces in Tarentum for seven years despite there being a law that nobody could hold the post for more than a year. Plato, who became a close friend, made his acquaintance while staying in Magna Graecia. Heath writes in [4]:-

... he is said, by means of a letter, to have saved Plato from death at the hands of Dionysius.

In fact Plato made a number of trips to Sicily and it was on the third of these trips in 361 BC that he was detained by Dionysius II. Plato wrote to Archytas who sent a ship to rescue him. For more details on the relationship between Archytas and Plato consult the interesting article [8].

Given the above story and the conclusion that Archytas came after Socrates, it may seem strange to include him in works on pre-socratic philosophers as is done in [3]. This is done, however, because of the style of Archytas's philosophy rather than the strict chronology.

Archytas was a pupil of Philolaus and so was a firm supporter of the philosophy of Pythagoras believing that mathematics provided the path to the understanding of all things. Although Archytas studied many topics, since he was a Pythagorean, mathematics was his main subject and all other disciplines were seen as dependent on mathematics. He claimed that mathematics was composed of four branches, namely geometry, arithmetic, astronomy and music. He also believed that the study of mathematics was important in other respects as a fragment of his writings that has been preserved shows (see [3] or [6]):-

Mathematicians seem to me to have excellent discernment, and it is not at all strange that they should think correctly about the particulars that are; for inasmuch as they can discern excellently about the physics of the universe, they are also likely to have excellent perspective on the particulars that are. Indeed, they have transmitted to us a keen discernment about the velocities of the stars and their risings and settings, and about geometry, arithmetic, astronomy, and, not least of all, music. These seem to be sister sciences, for they concern themselves with the first two related forms of being [number and magnitude].

This fragment comes from the preface to one of his works which some claim was entitled On Mathematics while others claim that it was entitled On Harmonics. Certainly, coming after this quote, there is a discussion of pitch, frequency and a theory of sound. It does contain some errors but it is still a remarkable piece of work and formed the basis for the theory of sound in the writings of Plato.

Archytas worked on the harmonic mean and gave it that name (it had been called sub-contrary in earlier times). The reason he worked on this was his interest in the problem of duplicating the cube, finding the side of a cube with volume twice that of a given cube. Hippocrates reduced the problem to finding two mean proportionals. Archytas solved the problem with a remarkable geometric solution (not of course a ruler and compass construction).

One interesting innovation which Archytas brought into his solution of finding two mean proportionals between two line segments was to introduce movement into geometry. His method uses a semicircle rotating in three dimensional space and the curve formed by it cutting another three dimensional surface.

We know of Archytas's solution to the problem of duplicating the cube through the writings of Eutocius of Ascalon. In these Eutocius claims to quote the description given in History of geometry by Eudemus of Rhodes but the accuracy of the quotation is doubted by the authors of [10].

Another interesting mathematical discovery due to Archytas is that there can be no number which is a geometric mean between two numbers in the ratio $(n+1) : n$. The most interesting thing about his proof is that it is close to that given by Euclid many years later, and also that it quotes known theorems which would later appear in Euclid's Elements Book VII.

The arguments just given led van der Waerden to claim (see for example [5]) that many of the results which appear in Book VII of the Elements predate Archytas. Clearly, he claims, there were some works, written many years before Euclid wrote the Elements, which covered the same material. Archytas built on this earlier work and his discoveries are then largely those presented by Euclid in the Elements Book VIII. Following these arguments of van der Waerden it is now widely accepted that Euclid borrowed Archytas's work for Book VIII of the Elements.

Archytas is sometimes called the founder of mechanics and he is said to have invented two mechanical devices. One device was a mechanical bird [2]:-

The bird was apparently suspended from the end of a pivoted bar, and the whole apparatus revolved by means of a jet of steam or compressed air.

Another mechanical device was a rattle for children which was useful, in Aristotle's words (see for example [4]):-

... to give to children to occupy them, and so prevent them from breaking things about the house (for the young are incapable of keeping still).

This does seem a remarkably modern thought for an inventor in 400 BC! In fact this interest in applying mathematics is in contrast to the pure mathematical ideas of Plato and this contrast formed the basis for a poem written by the Polish author C K Norwid (1821-1883). This fascinating poem is discussed and given in French translation by Marczewski in [9].

Simplicius, in his Physics, quotes Archytas's view that the universe is infinite (in Heath's translation [4]):-

If I were at the outside, say at the heaven of the fixed stars, could I stretch my hand or my stick outward or not? To suppose that I could not is absurd: and if I can stretch it out, that which is outside must be either body or space (it makes no difference which it is as we shall see). We may then in the same way get to the outside of that again, and so on, asking on arrival at each new limit the same question; and if there is always a new place to which the stick may be held out, this clearly involves extension without limit. If now what so extends is body, the proposition is proved; but even if it is space, then, since space is that in which body is or can be, and in the case of eternal things we must treat that which potentially is as being, it follows equally that there must be body and space extending without limit.

When it came to a philosophy of politics and ethics, again Archytas based his ideas on mathematical foundations. He wrote (see for example [3] or [6]):-

When mathematical reasoning has been found, it checks political faction and increases concord, for there is no unfair advantage in its presence, and equality reigns. With mathematical reasoning we smooth out differences in our dealings with each other. Through it the poor take from the powerful, and the rich give to the needy, both trusting in it to obtain an equal share...

Finally we quote again from the writings of Archytas about his theory of how to learn. The fragment appears in [3] or [6]:-

To become knowledgeable about things one does not know, one must either learn from others or find out for oneself. Now learning derives from someone else and is foreign, whereas finding out is of and by oneself. Finding out without seeking is difficult and rare, but with seeking it is manageable and easy, though someone who does not know how to seek cannot find.