阿基米德Archimedes

阿基米德的父亲是Phidias，一位天文学家。除了这一事实，我们对Phidias一无所知，而我们之所以知道这一点，仅仅是因为阿基米德在他的一部著作The Sandreckoner中给了我们这一信息。阿基米德的一位名叫Heracleides的朋友写了一部他的传记，但遗憾的是这部作品已经遗失。如果这部遗失的作品能被找到，哪怕只是在他人著作中找到摘录，我们对阿基米德的认识将会发生怎样的改变啊。

阿基米德是阿基米德，西西里岛人。一些作者报道说，他曾访问埃及，并在那里发明了一种现在称为阿基米德螺旋的装置。这是一种泵，世界许多地方仍在使用。极有可能的是，当他年轻时，阿基米德曾与亚历山大里亚的欧几里得的继承者们一起学习。当然，他完全熟悉那里发展起来的数学，但使这一推测更加确定的是，他亲自认识在那里工作的数学家，并将自己的结果连同私人信息送到亚历山大里亚。他非常看重科农，亚历山大里亚的数学家之一，既高度评价他作为数学家的能力，也把他视为密友。

在On spirals的序言中，阿基米德讲述了一个关于他在Alexandria的朋友们的趣事。他告诉我们，他习惯向他们发送他最新定理的陈述，但不给出证明。显然那里的一些数学家曾声称这些结果是他们自己的，所以阿基米德说，在他最后一次向他们发送定理时，他包括两个错误的[3]：-

……这样，那些声称发现了一切却拿不出相应证明的人，就可以被驳斥为假装发现了不可能之事。

除了他著作的序言之外，关于阿基米德的信息来自若干来源，例如普鲁塔克、李维等书中的故事。Plutarch告诉我们，阿基米德与国王Hieron II 阿基米德有亲缘关系（例如见[3]）：-

阿基米德……在写给国王Hiero的信中，他是Hiero的朋友和近亲……

同样，至少他与国王Hieron II家族的友谊的证据来自这样一个事实：The Sandreckoner被献给了国王Hieron之子Gelon。

事实上，当时的著作中相当多地提到阿基米德，因为他在自己的时代就赢得了声誉，这是这一时期其他数学家很少能达到的。其原因并不是对新数学思想的广泛兴趣，而是阿基米德发明了许多被用作战争机器的机械。当阿基米德遭到由马塞卢斯指挥的罗马人进攻时，这些机械在防御中特别有效。

Plutarch在其关于罗马统帅Marcellus的著作中写道，在公元前212年的围城战中，阿基米德的战争机械是如何被用来对付罗马人的：-

……当阿基米德开始操作他的机械时，他立刻向陆上部队发射各种投射武器，以及巨大的石块，它们以令人难以置信的声响和猛烈之势落下；无人能够抵挡；因为它们把被击中的人成堆地打倒，打乱了他们所有的队列和行列。与此同时，从城墙上方伸出的巨大长杆越过船只，用从高处放下的重物把一些船击沉；另一些船则被一只铁手或像鹤嘴一样的铁喙举到空中，当他们把船拖到船首，并使其在船尾竖立起来后，就把它们沉入海底；或者船只被内部的机械牵引、旋转，撞向城墙下突出的陡峭岩石，船上的士兵遭到巨大伤亡。一艘船常常被举到空中的很高处（目睹此景令人恐惧），并被来回翻滚、不断摇摆，直到水手全都被抛出去，最后它才撞上岩石，或被摔落下来。

阿基米德曾被他的朋友兼亲戚希伦国王说服去建造这样的机械：-

这些机械[阿基米德]的设计和发明，并非作为任何重要的事物，而仅仅是几何学中的消遣；这是为了遵从希伦国王不久前的愿望和要求，即让他将自己令人钦佩的科学思辨的某些部分付诸实践，并通过使理论真理适应感觉和日常使用，使其更易于为一般民众所欣赏。

或许令人悲哀的是，这个时代的人们对战争机械的欣赏程度远胜于理论数学，但人们不得不指出，在公元第二个千年结束时，世界并没有太大的不同。阿基米德的其他发明，如复合滑轮，也在他的同时代人中为他带来了巨大的声誉。我们再次引用普鲁塔克的话：-

[阿基米德]曾[在给希伦国王的一封信中]声称，给定力，任何给定的重量都可以被移动，甚至夸口说——我们被告知，他依靠论证的力量——如果存在另一个地球，他进入其中就能移动这个地球。希伦对此感到惊讶，并恳求他通过实际实验来证实这个问题，展示某个巨大的重量被一个小型机械移动，于是他相应地从国王的军械库中挑选了一艘货船，这艘船没有巨大的劳力和许多人无法从船坞中拖出；他让许多乘客和满载的货物上船，自己则坐在远处，没有费很大的力气，只是手握滑轮的头部并逐渐拉动绳索，他将船沿直线拉动，平稳而均匀，仿佛它在海中一样。

阿基米德虽然因他的机械发明而声名鹊起，但他相信纯数学是唯一值得追求的事业。普鲁塔克再次优美地描述了阿基米德的态度，然而我们稍后将看到阿基米德实际上确实使用了一些非常实用的方法从纯几何学中发现结果：-

阿基米德拥有如此高尚的精神、如此深邃的灵魂和如此丰富的科学知识宝藏，以至于尽管这些发明现在为他赢得了超越人类智慧的声誉，他仍然不屑于留下任何关于此类主题的评论或著作；相反，他摒弃了整个工程行业以及每一种仅仅用于使用和利润的艺术，认为它们是卑鄙和低下的，他将全部的热爱和抱负置于那些更纯粹的思辨中，那里不可能涉及生活的庸俗需求；这些研究，其优越性相对于所有其他研究是无可置疑的，其中唯一的疑问是，所考察主题的美丽和宏伟，以及证明方法和手段的精确性和说服力，是否最值得我们钦佩。

他对几何学的痴迷被普鲁塔克优美地描述为：-

阿基米德的仆人不顾他的意愿带他去浴场，为他清洗和涂油，然而在那里，他仍然会从几何图形中画出来，甚至在烟囱的余烬中也是如此。当他们用油和香料为他涂油时，他用手指在赤裸的身体上画线，以至于他如此忘我，沉浸在几何学研究的喜悦中，进入狂喜或恍惚状态。

阿基米德的成就非常杰出。大多数数学史家认为他是有史以来最伟大的数学家之一。他完善了一种积分方法，使他能够求出许多物体的面积、体积和表面积。米歇尔·沙勒说阿基米德关于积分的工作（见[7]）：-

……孕育了无穷的微积分，由约翰内斯·开普勒、博纳文图拉·卡瓦列里、皮埃尔·德·费马、哥特弗里德·威廉·莱布尼茨和艾萨克·牛顿构想并完善。

阿基米德能够应用穷竭法，这是积分的早期形式，来获得一系列重要结果，我们在下面描述他的作品时提到其中的一些。阿基米德还给出了π的精确近似值，并表明他能够精确地近似平方根。他发明了一种表示大数的系统。在力学中，阿基米德发现了关于平面图形和立体重心的基本定理。他最著名的定理给出了浸入液体中的物体的重量，称为阿基米德原理。

阿基米德留存下来的著作如下。On plane equilibriums（两卷）、求积 of the 抛物线、On the sphere and cylinder（两卷）、On spirals、On conoids and spheroids、On floating bodies（两卷）、Measurement of a circle和The Sandreckoner。1906年夏，哥本哈根大学古典语文学教授J L Heiberg发现了一份10世纪的手稿，其中包含阿基米德的著作The method。这为我们了解阿基米德如何发现他的许多结果提供了极为宝贵的线索，在进一步详述现存各卷的内容之后，我们将在下文讨论这一点。

阿基米德撰写其著作的先后顺序并不确定。在上文列出这些著作时，我们采用了托马斯·利特尔·希思在[7]中提出的年代顺序，但The Method除外，托马斯·利特尔·希思将其置于On the sphere and cylinder之前。论文[47]探讨了关于阿基米德著作另一种年代顺序的论据。

论著On plane equilibriums运用几何方法阐述了力学的基本原理。阿基米德发现了关于平面图形重心的基本定理，这些定理载于该著作中。他尤其在第1卷中求出了平行四边形、三角形和梯形的重心。第2卷则完全致力于求抛物线弓形的重心。在Quadrature of the parabola中，阿基米德求出了被任意弦截出的抛物线弓形的面积。

在On the sphere and cylinder第1卷中，阿基米德证明了球面面积是大圆的四倍，求出了球面上任意弓形的面积，证明了球的体积是外接的圆柱体积的三分之二，并且球面面积（包括底面在内）是外切圆柱表面积的三分之二。关于阿基米德可能如何借助无穷小得出其中某些结果，[14]中有很好的讨论。在该著作第2卷中，阿基米德最重要的结果是说明如何用一个平面去截给定的球，使得两弓形体积之比为预先给定的比值。

在On spirals中，阿基米德定义了螺线，给出了联系矢径长度与其所转过的角度的基本性质。他给出了关于螺线的切线的结果，并求出了螺线各部分的面积。在著作On conoids and spheroids中，阿基米德考察了旋转抛物面、旋转双曲面，以及将椭圆绕其长轴或短轴旋转所得的椭球面。该著作的主要目的是研究这三类三维图形弓形的体积。有人声称该著作的某些结果缺乏严格性，但[43]中饶有兴味的讨论将此归因于现代的重构。

On floating bodies是阿基米德阐述流体静力学基本原理的著作。他最著名的定理——给出浸入液体中物体的重量，称为Archimedes' principle——就包含在这部著作中。他还研究了各种形状和不同比重的浮体的稳定性。在Measurement of the Circle中，阿基米德证明了π的精确值介于$3\large\frac{10}{71}\normalsize$和$3\large\frac{1}{7}\normalsize$之间。他是通过用具有96条边的正多边形外切和内接一个圆而得到这一结果的。

The Sandreckoner是一部非凡的著作，其中阿基米德提出了一种数系，能够用现代符号表示高达$8 \times 10^{63}$的数字。他在这部著作中论证说，这个数字足够大，可以计算能够装入宇宙的沙粒数量。这部著作中还有重要的历史评论，因为阿基米德必须给出宇宙的尺寸，才能计算它可能包含的沙粒数量。他指出阿里斯塔克斯提出了一个以太阳为中心、包括地球在内的行星围绕它旋转的系统。在引用关于尺寸的结果时，他陈述了归功于欧多克索斯、菲迪亚斯（他的父亲）和阿里斯塔克斯的结果。还有其他来源提到阿基米德关于到天体距离的工作。例如，在[59]中，奥斯本重建并讨论了：-

……一种关于天体距离的理论，被归于阿基米德，但唯一存世的手稿［出自罗马的Hippolytus，约公元220年］中数字的讹误状态意味着这些材料难以处理。

在Method中，阿基米德描述了他发现许多几何结果的方式（见[7]）：-

……某些事情最初通过一种机械方法对我变得清楚，尽管它们后来必须用几何学来证明，因为用上述方法对它们的研究并未提供真正的证明。但当然，当我们事先通过该方法对问题有所了解时，提供证明要比在没有任何先前知识的情况下找到它更容易。

也许阿基米德几何结果的辉煌最好由Plutarch来总结，他写道：-

在整个几何学中不可能找到更困难、更复杂的问题，或更简单、更清晰的解释。有些人将此归因于他的天赋；而另一些人则认为，难以置信的努力和辛劳产生了这些表面上轻松而不费力的结果。你无论进行多少研究都无法成功获得证明，然而，一旦看到，你立刻相信你本会自己发现它；他沿着如此平滑、如此迅捷的路径引导你达到所需的结论。

托马斯·利特尔·希思补充了他对阿基米德工作质量的看法[7]：-

这些论著无一例外都是数学表述的丰碑；对攻击计划的逐步揭示、命题的巧妙排序、对一切与目的并非直接相关之物的严格剔除、整体的完美收束，其完美程度如此令人印象深刻，以致在读者心中唤起一种近乎敬畏的情感。

书中提到了阿基米德现已失传的其他著作。帕普斯提到了一部阿基米德关于半正多面体的著作，阿基米德本人提到了一部他在Sandreckoner中提出的关于数系的著作，帕普斯提到了一部On balances and levers，而Theon提到了一部阿基米德关于镜子的著作。关于更多失传著作的证据在[67]中有所讨论，但这些证据并不完全令人信服。

阿基米德于公元前212年，在第二次布匿战争中罗马人攻占阿基米德时被杀，此前他用战争机械抵御罗马人的一切努力均已失败。Plutarch叙述了流传到他那里的关于他被杀故事的三个版本。第一个版本：

阿基米德……命中注定般地正专注于用图形解决某个问题，他的心思和目光都固定在他所思考的对象上，他全然没有注意到罗马人的入侵，也没有注意到城池已被攻陷。在这种全神贯注的研究与沉思中，一名士兵出其不意地走到他面前，命令他跟随自己去见Marcellus；他不愿在把问题论证清楚之前就去，士兵大怒，拔出剑来将他刺穿。

第二个版本：

……一名罗马士兵拔剑向他冲来，要杀死他；而阿基米德回过头来，恳切地请求他稍等片刻，以免他当时正在研究的东西留下未决和不完整；但士兵丝毫不为他的恳求所动，当即杀死了他。

最后，普鲁塔克听说过的第三种版本：-

……当阿基米德正把数学仪器、日晷、球体和角尺运送给马塞卢斯，用这些东西可以目测太阳的大小，一些士兵看见他，以为他携带的容器里装着金子，就杀了他。

阿基米德认为他最重大的成就是关于一个圆柱外接一个球体的那些成果，他要求把这一图形连同他关于两者之比的结果刻在他的墓碑上。西塞罗于公元前75年在西西里，他写道他如何寻找阿基米德的坟墓（例如见[1]）：-

……发现它四周被围住，覆盖着荆棘和灌木丛；因为我记得，正如我所听说的，墓碑上刻着某些打油诗句，其中说一个球体连同圆柱被放在他的坟墓顶上。于是，在向四周仔细察看之后……我注意到一根小柱子稍稍高出灌木丛，上面有一个球体和圆柱的图形……。奴隶们被派进去带着镰刀……当通往那个地方的通道被打开后，我们走近我们面前的基座；铭文可以辨认出大约一半的诗行，因为后面部分已经磨损了。

也许令人惊讶的是，阿基米德的数学著作在他去世后不久相对鲜为人知。正如克拉格特在[1]中所写：-

与欧几里得的《原本》不同，阿基米德的著作在古代并不广为人知。……确实……阿基米德的个别著作显然在亚历山大里亚被研究过，因为阿基米德经常被亚历山大里亚的三位杰出数学家引用：海伦、帕普斯和亚历山大的席恩。

直到公元六世纪阿什凯隆的欧托基奥斯出版了一些阿基米德著作的版本并附有评注之后，这些非凡的论著才得以更广泛地流传。最后，值得一提的是，今天用来判断阿基米德各种论著版本与原文接近程度的标准，是确定它们是否保留了阿基米德的多利亚方言。

Archimedes' father was Phidias, an astronomer. We know nothing else about Phidias other than this one fact and we only know this since Archimedes gives us this information in one of his works, The Sandreckoner. A friend of Archimedes called Heracleides wrote a biography of him but sadly this work is lost. How our knowledge of Archimedes would be transformed if this lost work were ever found, or even extracts found in the writing of others.

Archimedes was a native of Syracuse, Sicily. It is reported by some authors that he visited Egypt and there invented a device now known as Archimedes' screw. This is a pump, still used in many parts of the world. It is highly likely that, when he was a young man, Archimedes studied with the successors of Euclid in Alexandria. Certainly he was completely familiar with the mathematics developed there, but what makes this conjecture much more certain, he knew personally the mathematicians working there and he sent his results to Alexandria with personal messages. He regarded Conon of Samos, one of the mathematicians at Alexandria, both very highly for his abilities as a mathematician and he also regarded him as a close friend.

In the preface to On spirals Archimedes relates an amusing story regarding his friends in Alexandria. He tells us that he was in the habit of sending them statements of his latest theorems, but without giving proofs. Apparently some of the mathematicians there had claimed the results as their own so Archimedes says that on the last occasion when he sent them theorems he included two which were false [3]:-

... so that those who claim to discover everything, but produce no proofs of the same, may be confuted as having pretended to discover the impossible.

Other than in the prefaces to his works, information about Archimedes comes to us from a number of sources such as in stories from Plutarch, Livy, and others. Plutarch tells us that Archimedes was related to King Hieron II of Syracuse (see for example [3]):-

Archimedes ... in writing to King Hiero, whose friend and near relation he was....

Again evidence of at least his friendship with the family of King Hieron II comes from the fact that The Sandreckoner was dedicated to Gelon, the son of King Hieron.

There are, in fact, quite a number of references to Archimedes in the writings of the time for he had gained a reputation in his own time which few other mathematicians of this period achieved. The reason for this was not a widespread interest in new mathematical ideas but rather that Archimedes had invented many machines which were used as engines of war. These were particularly effective in the defence of Syracuse when it was attacked by the Romans under the command of Marcellus.

Plutarch writes in his work on Marcellus, the Roman commander, about how Archimedes' engines of war were used against the Romans in the siege of 212 BC:-

... when Archimedes began to ply his engines, he at once shot against the land forces all sorts of missile weapons, and immense masses of stone that came down with incredible noise and violence; against which no man could stand; for they knocked down those upon whom they fell in heaps, breaking all their ranks and files. In the meantime huge poles thrust out from the walls over the ships and sunk some by great weights which they let down from on high upon them; others they lifted up into the air by an iron hand or beak like a crane's beak and, when they had drawn them up by the prow, and set them on end upon the poop, they plunged them to the bottom of the sea; or else the ships, drawn by engines within, and whirled about, were dashed against steep rocks that stood jutting out under the walls, with great destruction of the soldiers that were aboard them. A ship was frequently lifted up to a great height in the air (a dreadful thing to behold), and was rolled to and fro, and kept swinging, until the mariners were all thrown out, when at length it was dashed against the rocks, or let fall.

Archimedes had been persuaded by his friend and relation King Hieron to build such machines:-

These machines [Archimedes] had designed and contrived, not as matters of any importance, but as mere amusements in geometry; in compliance with King Hiero's desire and request, some little time before, that he should reduce to practice some part of his admirable speculation in science, and by accommodating the theoretic truth to sensation and ordinary use, bring it more within the appreciation of the people in general.

Perhaps it is sad that engines of war were appreciated by the people of this time in a way that theoretical mathematics was not, but one would have to remark that the world is not a very different place at the end of the second millenium AD. Other inventions of Archimedes such as the compound pulley also brought him great fame among his contemporaries. Again we quote Plutarch:-

[Archimedes] had stated [in a letter to King Hieron] that given the force, any given weight might be moved, and even boasted, we are told, relying on the strength of demonstration, that if there were another earth, by going into it he could remove this. Hiero being struck with amazement at this, and entreating him to make good this problem by actual experiment, and show some great weight moved by a small engine, he fixed accordingly upon a ship of burden out of the king's arsenal, which could not be drawn out of the dock without great labour and many men; and, loading her with many passengers and a full freight, sitting himself the while far off, with no great endeavour, but only holding the head of the pulley in his hand and drawing the cords by degrees, he drew the ship in a straight line, as smoothly and evenly as if she had been in the sea.

Yet Archimedes, although he achieved fame by his mechanical inventions, believed that pure mathematics was the only worthy pursuit. Again Plutarch describes beautifully Archimedes attitude, yet we shall see later that Archimedes did in fact use some very practical methods to discover results from pure geometry:-

Archimedes possessed so high a spirit, so profound a soul, and such treasures of scientific knowledge, that though these inventions had now obtained him the renown of more than human sagacity, he yet would not deign to leave behind him any commentary or writing on such subjects; but, repudiating as sordid and ignoble the whole trade of engineering, and every sort of art that lends itself to mere use and profit, he placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life; studies, the superiority of which to all others is unquestioned, and in which the only doubt can be whether the beauty and grandeur of the subjects examined, of the precision and cogency of the methods and means of proof, most deserve our admiration.

His fascination with geometry is beautifully described by Plutarch:-

Oftimes Archimedes' servants got him against his will to the baths, to wash and anoint him, and yet being there, he would ever be drawing out of the geometrical figures, even in the very embers of the chimney. And while they were anointing of him with oils and sweet savours, with his fingers he drew lines upon his naked body, so far was he taken from himself, and brought into ecstasy or trance, with the delight he had in the study of geometry.

The achievements of Archimedes are quite outstanding. He is considered by most historians of mathematics as one of the greatest mathematicians of all time. He perfected a method of integration which allowed him to find areas, volumes and surface areas of many bodies. Chasles said that Archimedes' work on integration (see [7]):-

... gave birth to the calculus of the infinite conceived and brought to perfection by Kepler, Cavalieri, Fermat, Leibniz and Newton.

Archimedes was able to apply the method of exhaustion, which is the early form of integration, to obtain a whole range of important results and we mention some of these in the descriptions of his works below. Archimedes also gave an accurate approximation to π and showed that he could approximate square roots accurately. He invented a system for expressing large numbers. In mechanics Archimedes discovered fundamental theorems concerning the centre of gravity of plane figures and solids. His most famous theorem gives the weight of a body immersed in a liquid, called Archimedes' principle.

The works of Archimedes which have survived are as follows. On plane equilibriums (two books), Quadrature of the parabola, On the sphere and cylinder (two books), On spirals, On conoids and spheroids, On floating bodies (two books), Measurement of a circle, and The Sandreckoner. In the summer of 1906, J L Heiberg, professor of classical philology at the University of Copenhagen, discovered a 10th century manuscript which included Archimedes' work The method. This provides a remarkable insight into how Archimedes discovered many of his results and we will discuss this below once we have given further details of what is in the surviving books.

The order in which Archimedes wrote his works is not known for certain. We have used the chronological order suggested by Heath in [7] in listing these works above, except for The Method which Heath has placed immediately before On the sphere and cylinder. The paper [47] looks at arguments for a different chronological order of Archimedes' works.

The treatise On plane equilibriums sets out the fundamental principles of mechanics, using the methods of geometry. Archimedes discovered fundamental theorems concerning the centre of gravity of plane figures and these are given in this work. In particular he finds, in book 1, the centre of gravity of a parallelogram, a triangle, and a trapezium. Book two is devoted entirely to finding the centre of gravity of a segment of a parabola. In the Quadrature of the parabola Archimedes finds the area of a segment of a parabola cut off by any chord.

In the first book of On the sphere and cylinder Archimedes shows that the surface of a sphere is four times that of a great circle, he finds the area of any segment of a sphere, he shows that the volume of a sphere is two-thirds the volume of a circumscribed cylinder, and that the surface of a sphere is two-thirds the surface of a circumscribed cylinder including its bases. A good discussion of how Archimedes may have been led to some of these results using infinitesimals is given in [14]. In the second book of this work Archimedes' most important result is to show how to cut a given sphere by a plane so that the ratio of the volumes of the two segments has a prescribed ratio.

In On spirals Archimedes defines a spiral, he gives fundamental properties connecting the length of the radius vector with the angles through which it has revolved. He gives results on tangents to the spiral as well as finding the area of portions of the spiral. In the work On conoids and spheroids Archimedes examines paraboloids of revolution, hyperboloids of revolution, and spheroids obtained by rotating an ellipse either about its major axis or about its minor axis. The main purpose of the work is to investigate the volume of segments of these three-dimensional figures. Some claim there is a lack of rigour in certain of the results of this work but the interesting discussion in [43] attributes this to a modern day reconstruction.

On floating bodies is a work in which Archimedes lays down the basic principles of hydrostatics. His most famous theorem which gives the weight of a body immersed in a liquid, called Archimedes' principle, is contained in this work. He also studied the stability of various floating bodies of different shapes and different specific gravities. In Measurement of the Circle Archimedes shows that the exact value of π lies between the values $3\large\frac{10}{71}\normalsize$ and $3\large\frac{1}{7}\normalsize$. This he obtained by circumscribing and inscribing a circle with regular polygons having 96 sides.

The Sandreckoner is a remarkable work in which Archimedes proposes a number system capable of expressing numbers up to $8 \times 10^{63}$ in modern notation. He argues in this work that this number is large enough to count the number of grains of sand which could be fitted into the universe. There are also important historical remarks in this work, for Archimedes has to give the dimensions of the universe to be able to count the number of grains of sand which it could contain. He states that Aristarchus has proposed a system with the sun at the centre and the planets, including the Earth, revolving round it. In quoting results on the dimensions he states results due to Eudoxus, Phidias (his father), and to Aristarchus. There are other sources which mention Archimedes' work on distances to the heavenly bodies. For example in [59] Osborne reconstructs and discusses:-

...a theory of the distances of the heavenly bodies ascribed to Archimedes, but the corrupt state of the numerals in the sole surviving manuscript [due to Hippolytus of Rome, about 220 AD] means that the material is difficult to handle.

In the Method, Archimedes described the way in which he discovered many of his geometrical results (see [7]):-

... certain things first became clear to me by a mechanical method, although they had to be proved by geometry afterwards because their investigation by the said method did not furnish an actual proof. But it is of course easier, when we have previously acquired, by the method, some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge.

Perhaps the brilliance of Archimedes' geometrical results is best summed up by Plutarch, who writes:-

It is not possible to find in all geometry more difficult and intricate questions, or more simple and lucid explanations. Some ascribe this to his natural genius; while others think that incredible effort and toil produced these, to all appearances, easy and unlaboured results. No amount of investigation of yours would succeed in attaining the proof, and yet, once seen, you immediately believe you would have discovered it; by so smooth and so rapid a path he leads you to the conclusion required.

Heath adds his opinion of the quality of Archimedes' work [7]:-

The treatises are, without exception, monuments of mathematical exposition; the gradual revelation of the plan of attack, the masterly ordering of the propositions, the stern elimination of everything not immediately relevant to the purpose, the finish of the whole, are so impressive in their perfection as to create a feeling akin to awe in the mind of the reader.

There are references to other works of Archimedes which are now lost. Pappus refers to a work by Archimedes on semi-regular polyhedra, Archimedes himself refers to a work on the number system which he proposed in the Sandreckoner, Pappus mentions a treatise On balances and levers, and Theon mentions a treatise by Archimedes about mirrors. Evidence for further lost works are discussed in [67] but the evidence is not totally convincing.

Archimedes was killed in 212 BC during the capture of Syracuse by the Romans in the Second Punic War after all his efforts to keep the Romans at bay with his machines of war had failed. Plutarch recounts three versions of the story of his killing which had come down to him. The first version:-

Archimedes ... was ..., as fate would have it, intent upon working out some problem by a diagram, and having fixed his mind alike and his eyes upon the subject of his speculation, he never noticed the incursion of the Romans, nor that the city was taken. In this transport of study and contemplation, a soldier, unexpectedly coming up to him, commanded him to follow to Marcellus; which he declining to do before he had worked out his problem to a demonstration, the soldier, enraged, drew his sword and ran him through.

The second version:-

... a Roman soldier, running upon him with a drawn sword, offered to kill him; and that Archimedes, looking back, earnestly besought him to hold his hand a little while, that he might not leave what he was then at work upon inconclusive and imperfect; but the soldier, nothing moved by his entreaty, instantly killed him.

Finally, the third version that Plutarch had heard:-

... as Archimedes was carrying to Marcellus mathematical instruments, dials, spheres, and angles, by which the magnitude of the sun might be measured to the sight, some soldiers seeing him, and thinking that he carried gold in a vessel, slew him.

Archimedes considered his most significant accomplishments were those concerning a cylinder circumscribing a sphere, and he asked for a representation of this together with his result on the ratio of the two, to be inscribed on his tomb. Cicero was in Sicily in 75 BC and he writes how he searched for Archimedes tomb (see for example [1]):-

... and found it enclosed all around and covered with brambles and thickets; for I remembered certain doggerel lines inscribed, as I had heard, upon his tomb, which stated that a sphere along with a cylinder had been put on top of his grave. Accordingly, after taking a good look all around ..., I noticed a small column arising a little above the bushes, on which there was a figure of a sphere and a cylinder... . Slaves were sent in with sickles ... and when a passage to the place was opened we approached the pedestal in front of us; the epigram was traceable with about half of the lines legible, as the latter portion was worn away.

It is perhaps surprising that the mathematical works of Archimedes were relatively little known immediately after his death. As Clagett writes in [1]:-

Unlike the Elements of Euclid, the works of Archimedes were not widely known in antiquity. ... It is true that ... individual works of Archimedes were obviously studied at Alexandria, since Archimedes was often quoted by three eminent mathematicians of Alexandria: Heron, Pappus and Theon.

Only after Eutocius brought out editions of some of Archimedes works, with commentaries, in the sixth century AD were the remarkable treatises to become more widely known. Finally, it is worth remarking that the test used today to determine how close to the original text the various versions of his treatises of Archimedes are, is to determine whether they have retained Archimedes' Dorian dialect.