托马斯·哈里奥特Thomas Harriot

托马斯·哈里奥特是一位数学家和天文学家，他创立了英国代数学派。Fauvel和Goulding在[10]中这样描述他：-

……牛津培养出的最伟大的数学家……

然而他的名字直到最近才广为人知，甚至现在他的成就也未被大多数数学家充分认识。

我们对哈里奥特的青年时代知之甚少。事实上，已知的全部情况是，1577年12月20日星期五，他在牛津大学注册入学，官方记录中记载他时年十七岁，父亲为平民，出生地为牛津郡。正是从这一记录推断出他的出生日期为1560年，我们知道他的父亲是一位“平民”，但哈里奥特能进入牛津这一事实本身就意味着他不太可能来自最贫困的阶层。尽管对牛津郡的记录进行了广泛搜寻，仍未发现关于他出生或父母的进一步信息（尽管已确认了一些可能的亲属）。

在牛津大学读本科时，哈里奥特是圣玛丽学堂的学生。他与理查德·哈克卢特和Thomas Allen成为朋友，两人都是该大学的讲师，但不在圣玛丽学堂。哈里奥特于1580年毕业，前往伦敦。目前不清楚他最初几年在伦敦具体做了什么，但大概从1583年末起，他开始为沃尔特·雷利爵士效力。哈克卢特在1587年2月献给雷利的序言中写道（例如见[4]）：-

自从您认识到，航海技艺的才能——一个岛国的主要荣耀——如果得到数学科学的帮助，就能在我们中间达到辉煌，您就一直以极为慷慨的薪水，在家中供养哈里奥特，一个在这些研究中出类拔萃的人，以便通过他的帮助，您能在闲暇时获得那些高贵的科学……

哈里奥特写了一部名为Arcticon的著作，但从未出版，不幸的是，从未找到任何副本。这部作品本质上是他在这勒姆府讲授的课程，那是雷利在伦敦河岸街的住所，哈里奥特当时就住在那里。这些讲座是给雷利为参加新世界探险而聚集的水手们讲的。佩珀描述了哈里奥特在写Arcticon时在航海技术方面取得的进步[18]：-

……他解决了协调太阳和北极星观测以确定纬度的问题，引入了利用太阳振幅确定磁偏角的思想，并且在改进观测太阳或恒星高度的方法和仪器之外，还根据他自己的天文观测重新计算了太阳的赤纬表。……他给出了墨卡托问题的实用数值解法，很可能是通过添加正割……

正如罗奇在[24]中指出的：-

……当与新的仪器和观测实践相结合时，很明显雷利拥有欧洲最好的航海专业知识。

哈里奥特不仅作为航海教员受雇于罗利。他还参与了罗利探险队船只的设计，以及船只的建造和船员的挑选。他是罗利的会计，负责为探险队获取资金并保管所有账目。

雷利让船长菲利普·阿马达斯和亚瑟·巴洛于1584年对北卡罗来纳海岸外的罗阿诺克岛进行了一次探险。虽然没有直接证据表明哈里奥特进行了这次航行，但奎因在[23]中有说服力地论证他是进行这次初步勘测的人之一。哈里奥特确实参加了雷利在1585-86年组织的前往弗吉尼亚的航行。他于1585年4月9日乘坐“老虎”号从普利茅斯启航，他对4月19日日食的观测使现代科学家能够计算出当天船只的确切位置。哈里奥特在新世界期间做了许多笔记，尤其对当地居民的语言和习俗（特别是饮食习惯）感兴趣。这次航行的目的是殖民新世界，但未能实现这一目标。

德雷克正在与西班牙人进行海战，这时他得知他们打算阻止英国殖民者站稳脚跟。尽管德雷克与殖民者会合，但在1586年6月出现了猛烈的风暴，哈里奥特和大部分人匆忙返回了英格兰。哈里奥特连同德雷克的船只于1586年7月在朴茨茅斯登陆，他立即前往雷利那里报告探险情况。他于1588年出版了A Briefe and True Report of the New Found Land of Virginia，在书中他推荐吸烟，而他自己在弗吉尼亚学会了吸烟。然而，他也写了一篇关于这次航行的完整记述，但由于某种原因，他从未出版，尽管进行了艰苦的寻找，似乎已经丢失。

到哈里奥特返回时，雷利已将注意力转向爱尔兰。哈里奥特从1589年开始对雷利拥有的利斯莫尔庄园进行了测量。九年后，他仍在计算该庄园出租地块的英亩数。然而，政治局势即将发生变化，这将对哈里奥特产生戏剧性的影响。

早在1590年代，就已经有指控雷利无神论。这些指控针对的是雷利的学校以及“其校长的巫师”。哈里奥特觉得这是指他，于是他与约翰·迪伊讨论了这些指控（后者也觉得这些指控可能与他有关）。没有理由相信哈里奥特（或雷利）是无神论者，但可以肯定他们是自由思想家，而哈里奥特对世界的科学态度，至少可以说，被教会极为怀疑。除了指控引起的问题外，约翰·迪伊和哈里奥特在1590年代还讨论了科学和数学问题。

哈里奥特此时已从为罗利工作转为为诺森伯兰公爵亨利·珀西工作。公爵身边有一群学者朋友，其中许多人持有原子论观点。罗利的生活变得如此混乱，以至于哈里奥特寻求一位能为其科学追求提供更多稳定性的赞助人的支持。1595年，公爵将达勒姆的财产转让给哈里奥特，他因此社会地位上升，成为“乡绅”阶层的一员。哈里奥特后来还在康沃尔和诺福克拥有地产。达勒姆交易后不久，公爵让哈里奥特使用赛昂庄园（位于伦敦郊外基尤附近）的一所房屋，哈里奥特将其既用作住所又用作科学实验室。

我们确实从留存的手稿中得知，哈里奥特到1597年时已在赛昂从事光学的深入研究。尽管[4]中称他在1597年之前就发现了光的折射正弦定律，但实际上我们现在知道哈里奥特这一重要发现的精确日期是1601年7月。然而，与他所有其他数学发现一样，哈里奥特没有发表他的发现。然而有些讽刺的是，威理博·斯涅尔（该定律的发现现在被归功于他）并不是第一个发表这一结果的人。威理博·斯涅尔的发现是在1621年，大约在哈里奥特发现之后20年，但这一结果直到1637年勒内·笛卡儿将其付印才得以发表。

哈里奥特在1590年代确实研究过的光学问题之一是海什木问题。他给出了海什木问题的一个解答，其中涉及考虑一个等价问题，即由圆和绕圆上一点旋转的弦的直径之间形成的最大截距问题。[8]的作者推测哈里奥特可能使用了无穷小的技巧来证明这些问题的等价性，而且我们确实知道哈里奥特引入了后来被伊萨克·巴罗重新发现的思想。

光学并不是这一时期哈里奥特唯一关注的课题。16世纪90年代初，罗利曾要求他将数学技能应用于枪炮学。当时，关于抛体轨迹的观念仍然由亚里士多德的思想主导。哈里奥特将作用在抛体上的力分解为水平和垂直分量。他认识到空气阻力作用于整个飞行过程，而重力作用于垂直分量。他非常接近用向量分析解决求抛体速度的问题，并且肯定在1607年之前，他得出结论：抛体的路径是一条倾斜的抛物线。然而，他犯了一个错误，[4]：-

不知何故，他无法强迫自己放弃亚里士多德的观点，即较重的物体比轻的物体下落得更快。

哈里奥特在1600年之前开始研究的其他课题是化学问题。从1599年5月到1600年5月，他几乎整整一年都在集中研究化学，尽管他的实验是以一种新的科学精确度进行的，但他没有做出特别值得注意的发现。

罗利曾是伊丽莎白一世的特别宠臣，当她于1603年3月24日去世时，显然罗利的命运将会改变。也许不太明显的是，此时与罗利关系已不那么密切的哈里奥特也会遇到麻烦。詹姆斯一世成为国王，他很快将罗利视为反对其王位主张的人。诺森伯兰公爵亨利·珀西在伊丽莎白去世前几天就小心地通过一封支持信与詹姆斯建立了良好关系。7月，针对詹姆斯的阴谋被揭露，罗利被逮捕并被控叛国罪。

罗利试图自杀但失败了。随后他寻求哈里奥特的帮助以获取有利于他的证据。罗利被定罪并被判处绞刑。可怜的哈里奥特在判决中被单独指出为无神论者和邪恶影响。他帮助罗利的尝试基于基督教原则（他无疑信奉这些原则），但这反而损害了罗利，因为哈里奥特被视为一个为方便而利用基督教原则的无神论者。哈里奥特深受打击，在大约一年时间里没有进行新的科学工作，因为他试图接受正在发生的事情。罗利在最后一刻获得了死刑的缓刑，但被囚禁在伦敦塔。

另一个阴谋将导致更多的麻烦。1605年11月4日，Guy Fawkes和其他人因企图炸毁议会大厦而被捕。包括哈里奥特 Percy在内的另外四人，即Henry Percy的孙子，也作为主要阴谋者被捕。哈里奥特因涉嫌参与而被关押在Gatehouse监狱。他因被指控绘制了King James的星象图，企图利用魔法力量影响国王的未来而受到审讯。11月27日，哈里奥特的赞助人Henry Percy也被关进伦敦塔，直到1621年获释。似乎没有找到对哈里奥特不利的证据，尽管他在Gatehouse监狱待了一段时间，写了几封信请求释放，但他很可能在1605年底前就获得了自由。

一经释放，哈里奥特就回到了光学研究上。他现在考虑更复杂的系统，并从1605年初起聘请Christopher Tooke作为透镜研磨师。他在光方面的工作现在转向了光的色散。他开始发展彩虹理论，到1606年，约翰内斯·开普勒已经听说了哈里奥特在光学方面取得的显著成果。约翰内斯·开普勒写信给哈里奥特，但通信从未真正实现任何重要的思想交流。也许哈里奥特对他工作几乎给他带来的困难过于谨慎，或者也许他（如他向约翰内斯·开普勒声称的那样）如果健康允许，仍然打算发表他的成果。

一颗彗星的出现吸引了哈里奥特的注意，将他的科学思维转向了天文学。他于1607年9月17日从Ilfracombe观测到一颗彗星，后来被确认为爱德蒙·哈雷的彗星。约翰内斯·开普勒在六天前发现了这颗彗星，但最终是哈里奥特和他的朋友（及学生）William Lower的观测被弗里德里希·威廉·贝塞尔用来计算其轨道。

他的天文学重新回到前沿，哈里奥特继续进行英格兰最早的望远镜观测。1609年7月26日晚上9点，他绘制了当时月龄5天的月球，通过放大倍数为6的望远镜观察。一年后的1610年7月17日，他再次绘制了月球，此时他有一台放大倍数为10的望远镜。不久，他建造了一台放大倍数为20的望远镜，然后到1611年4月，他有了放大倍数为32的望远镜。

哈里奥特观测到了木星的卫星，尽管由于他的首次观测陈述道：-

我对新行星的首次观测。我只看到了一个，而且只有它一个

他一定已经知道伽利略的发现。正如他所有的科学发现一样，哈里奥特没有发表他的结果。这些对木星卫星的观测是在1610年10月17日至1612年2月26日之间进行的。

他是第一个发现太阳黑子的人，在1610年12月8日至1613年1月18日之间进行了199次观测。首次观测太阳黑子是在他观测木星卫星时进行的。从收集的数据中，他能够推断出太阳自转的周期。然而，大约在这个时候，他的科学工作基本结束了。他似乎因朋友的去世而精神低落，失去了继续研究的动力（尽管他没有发表成果，但这给他带来了很多麻烦）。

在1614年之后哈里奥特所做的少数几项工作中，有一项是他在1618年从锡永宫对另一颗彗星的观测（那年有三颗可见彗星，哈里奥特观测了第三颗）。1618年，1603年获得宽大处理由死刑改为监禁的罗利被处死。罗利在1618年10月29日被公开处决，哈里奥特在场目睹了这一事件。然而，此时哈里奥特已经患有鼻癌，这最终导致了他的死亡。

癌症似乎始于1613年左右，大约就是哈里奥特失去推进其数学与科学研究兴趣的时候。他在1615年咨询了顶尖专家，该专家写了一份关于这次咨询的报告。他将哈里奥特描述为（例如见[4]）：-

……一个有些忧郁的人。……左鼻孔的癌性溃疡侵蚀了他的鼻中隔，并且随着其大小而将嘴唇紧紧拉住并向上翻。它已逐渐深入鼻腔。病人遭受这一恶疾已有两年。

哈里奥特又遭受这一“恶疾”三年，之后癌症夺去了他的生命。

还有几项其他重大数学成就应归功于哈里奥特，我们应当提及。他展示了对数螺线作为球面上球极投影的斜驶线，他证明这种投影是conformal。等角航线是墨卡托地图上的直线，哈里奥特以极高的精度计算了它们。事实上，为了达到这种精度，哈里奥特引入了有限差分插值。

有一个有趣的历史问题，它直到最近才被解决，却起源于哈里奥特。罗利请哈里奥特解决关于堆叠炮弹的某些问题。在一份日期为1591年12月12日（星期日）的手稿上，哈里奥特列出了一个表格来回答罗利的问题。他展示了如果给定炮弹的数量，如何计算应放置在三角形、正方形或长方形底金字塔底部的炮弹数量。罗利提出了第二个问题，哈里奥特也回答了，即给定炮弹金字塔，计算堆中的数量。

然而，哈里奥特作为数学家不会止步于此。通过研究炮弹如何填充空间，他考虑了他所相信的物质原子论的含义。后来，在他与约翰内斯·开普勒关于原子理论的通信中，哈里奥特提到了堆积问题。约翰内斯·开普勒无法解决这个问题，但他相信如果每一层球体的中心位于下一层空隙的中心上方，就能达到最密堆积。这在直觉上似乎显而易见，但直到1998年密歇根大学的哈里奥特 Hales（借助数小时计算机生成的数据）才最终证明了这一猜想。

哈里奥特的工作中我们尚未描述的一部分是数学工作，在某些方面，他最为人所知的正是这部分工作，即他在代数方面的工作。他引入了简化的代数符号，他在方程理论方面的基础研究远远超前于时代。作为他解方程能力的例子，即使在根为负数或虚数时也是如此，我们重现他解一个四次方程的过程。所讨论的例子出自他的手迹，并重印于[3]中。

aaaa - 6aa + 136a = 1155
------------------------------------
    aaaa -2aa + 1 = 4aa -136a + 1156
           aa - 1 = 2a - 34

               33 = 2a - aa
          aa - 2a = -33
      aa - 2a + 1 = +1 - 33
            a - 1 = √-32
            1 - a = √-32
                a = 1 + √-32
                a = 1 - √-32
------------------------------------
           aa - 1 = 34 - 2a
          aa + 2a = 35
      aa + 2a + 1 = 1 + 35
            a + 1 = √36

      a = √36 - 1 = 5
------------------------------------
           -a - 1 = √36
     a = -√36 - 1 = -7

注意哈里奥特在配方法方面多么娴熟，他知道每当他取平方根时都有两个答案需要考虑，并且平等对待所有答案，无论正负、实虚。我们只对哈里奥特的符号做了一处轻微改动。在我们使用如今标准的符号=的地方，哈里奥特使用了插图：Harriotequal.gif ↗。

哈里奥特发明了一些今天仍在使用的符号。然而，表示“小于”的符号<和表示“大于”的符号>并非源自哈里奥特（尽管人们常常这样声称），而是由Artis Analyticae Praxis ad Aequationes Algebraicas ResolvendasⓉ（《分析术引论》，即通过它代数方程可以被求解的技艺）的编者引入的——哈里奥特本人使用了不同的符号。关于哈里奥特在多大程度上受到弗朗索瓦·韦达的影响，或者弗朗索瓦·韦达引入的记号与思想是否由他从哈里奥特那里学到，学界至今仍有争论。

正如从上面的例子所见，哈里奥特在方程求解方面做出了杰出的工作，他以一种使他的解法看起来像现代解法的方式识别出负根和复根。他观察到，如果$a, b, c$是一个cubic的根，那么该三次方程为$(x - a)(x - b)(x - c) = 0$。这是理解上的一个重大进步，哈里奥特随后将其推进到了更高次的方程。

尽管他远远超越了他的时代，但他的工作产生的影响远不及它本应产生的影响，因为正如我们上面反复指出的，他一生中没有发表任何数学著作。甚至他关于代数的著作Artis Analyticae Praxis ad Aequationes Algebraicas ResolvendasⓉ（《分析术引论》，即通过它代数方程可以被求解的技艺）（1631年）也是在他去世10年后才出版的，并且由那些没有充分领会其工作深度的人编辑。例如，它没有讨论负解。

Thomas Harriot was a mathematician and astronomer who founded the English school of algebra. He is described in [10] by Fauvel and Goulding as:-

... the greatest mathematician that Oxford has produced ...

yet his name has only recently become widely known, and even now his achievements are not fully appreciated by most mathematicians.

We know very little of Harriot's youth. In fact all that is known is that on Friday 20 December 1577 he matriculated at the University of Oxford with an entry in the official records giving his age as seventeen, his father as a plebeian, and his birthplace Oxfordshire. It is from this record that his date of birth is deduced to be 1560 and we know that his father was a "commoner" but the very fact that Harriot was entering Oxford means that it is unlikely that he came from the poorest classes. Despite extensive searches of the Oxfordshire records, no further information concerning his birth or parentage has been found (although a number of possible relatives have been identified).

As an undergraduate at Oxford, Harriot was a student at St Mary's Hall. He became friends with Richard Hakluyt and Thomas Allen, both lecturers at the university, but not at St Mary's Hall. Harriot graduated in 1580 and went to London. It is not clear exactly what he did in his first few years there but, probably from late 1583, he entered Sir Walter Raleigh's service. Hakluyt, dedicating a preface to Raleigh in February 1587, wrote (see for example [4]):-

Ever since you perceived that skill in the navigator's art, the chief ornament of an island kingdom, might attain its splendour amongst us if the aid of the mathematical sciences were enlisted, you have maintained in your household Thomas Harriot, a man pre-eminent in those studies, at a most liberal salary, in order that by his aid you might acquire those noble sciences in your leisure hours ...

Harriot wrote a text called Arcticon which was never published and unfortunately no copies have ever been found. This work was essentially his lecture course given at Durham House, Raleigh's lodgings in The Strand in London, where Harriot lived at this time. The lectures were given to the seamen who were being gathered by Raleigh to participate in his expeditions to the New World. Pepper describes the advances in navigational techniques made by Harriot by the time he wrote Arcticon [18]:-

... he solved the problem of reconciling the sun and pole star observations for determining latitude, introduced the idea of using solar amplitude to determine magnetic variation and, as well as improving methods and devices for observation of solar or stellar altitudes, he recalculated tables for the sun's declination on the basis of his own astronomical observations. ... he produced a practical numerical solution of the Mercator problem, most probably by the addition of secants ...

As Roche notes in [24]:-

... when combined with new instruments and observational practices it is clear that Raleigh had the best navigational expertise in Europe.

It was not only as a navigational instructor that Raleigh employed Harriot. He was involved with the design of the ships for Raleigh's expeditions as well as being involved in the construction of the vessels and selecting the seamen. He was Raleigh's accountant, being responsible for obtaining funding for the expeditions and keeping all the accounts.

Raleigh had the captains Philip Amadas and Arthur Barlowe make an expedition to Roanoke Island off the coast of North Carolina in 1584. Although there is no direct evidence that Harriot made this voyage, Quinn in [23] argues convincingly that he was one of those making this preliminary survey. Harriot was certainly on a voyage to Virginia organised by Raleigh in 1585-86. He sailed from Plymouth on 9 April 1585 on board the Tiger and his observations of a solar eclipse on 19 April have allowed modern scientists to compute the exact position of the ship on that day. Harriot made many notes during his time in the New World, being particularly interested in the language and customs (particularly the eating habits) of the inhabitants. The object of the voyage was to colonise the New World but it was not successful in this aim.

Drake was engaged in sea battles with the Spanish when he learnt that they intended to prevent the British colonists becoming established. Although Drake met up with the colonists, in June 1586 there were severe storms and there was a hurried return to England by Harriot and most of the party. Harriot, together with Drake's ships, landed at Portsmouth in July 1586 and he went immediately to Raleigh to report on the expedition. He published A Briefe and True Report of the New Found Land of Virginia in 1588, a book in which he recommends the smoking of tobacco which he himself had learnt to do in Virginia. However, he also wrote a full account of the voyage which, for some reason, he never published and, despite strenuous attempts to find a copy, seems lost.

By the time Harriot had returned, Raleigh had turned his attention to Ireland. Harriot carried out surveys of the Lismore estate, which was owned by Raleigh, beginning in 1589. Nine years later he was still involved in working out the acreage of plots being leased on the estate. However the political situation was about to change and this would have dramatic implications for Harriot.

Already in the 1590s there were allegations against Raleigh of atheism. The charges were against Raleigh's school and "the conjurer that is master thereof". Harriot felt that this was a reference to him and he discussed the allegations with John Dee (who also felt that the charges might relate to him). There is no reason to believe that Harriot (or Raleigh) were atheists but certainly they were free thinkers and Harriot's scientific approach to the world was, to say the least, viewed with great suspicion by the church. As well as problems caused by allegations, Dee and Harriot discussed scientific and mathematical matters in the 1590s.

Harriot had now moved from working for Raleigh to working for Henry Percy, Duke of Northumberland. The Duke had around him a circle of friends who were scholars, many of whom held a atomistic views. Raleigh's life became so chaotic that Harriot had sought the support of a patron who could provide more stability for his scientific pursuits. In 1595 the Duke made property in Durham over to Harriot and he moved up the social ladder becoming a member of the "landed gentry". Harriot also later held estates in Cornwall and Norfolk. Not long after the Durham transaction, the Duke gave Harriot the use of one of the houses on the estate at Syon (near Kew outside London) which Harriot used both as a residence and as a scientific laboratory.

We certainly know from manuscripts which survive that Harriot was engaged in deep studies of optics at Syon by 1597. Although in [4] it states that he had discovered the sine law of refraction of light before 1597, in fact we now know that the precise date of Harriot's important discovery was July 1601. As with all his other mathematical discoveries, however, Harriot did not publish his findings. It is somewhat ironical, however, that Snell (to whom the discovery of this law is now attributed) was not the first to publish the result. Snell's discovery was in 1621, about 20 years after Harriot's discovery, but the result was not published until Descartes put it in print in 1637.

One of the optical problems which Harriot did study in the 1590s was Alhazen's problem. He gave a solution to Alhazen's problem which involved considering an equivalent problem, namely the problem of the maximum intercept formed between a circle and a diameter of a chord rotating about a point on a circle. The author of [8] conjectures that Harriot may have used infinitesimal techniques in demonstrating the equivalence of these problems, and certainly we know that Harriot introduced ideas later rediscovered by Barrow.

Optics was not the only topic to occupy Harriot during this period. He had been asked by Raleigh in the early 1590s to apply his mathematical skills to the science of gunnery. At this time ideas of the trajectory taken by a projectile were still dominated by Aristotle's thinking. Harriot resolved the forces acting on the projectile into horizontal and vertical components. He understood that air resistance acted throughout the whole flight, and that gravity acted on the vertical component. He came very close to a vector analysis solution of the problem of finding the velocity of the projectile and, certainly by 1607, he came to the conclusion that the path of the projectile was a tilted parabola. He made one error, however, [4]:-

Somehow, he could not force himself to abandon the Aristotelian idea that heavier bodies fell at a faster rate than lighter ones.

Other topics which Harriot began to work on before 1600 were problems of chemistry. He worked intensively on chemistry for almost exactly a year from May 1599 to May 1600 and, although his experiments were conducted with a new scientific precision, he made no discoveries of particular note.

Raleigh had been a particular favourite of Elizabeth I and, when she died on 24 March 1603, it was clear that Raleigh's fortunes would change. Perhaps it is less clear that Harriot, by this time not so closely associated with Raleigh, would find problems too. James I became king and he quickly saw Raleigh as someone opposed to his claims to the throne. Henry Percy, the Duke of Northumberland, had taken care to put himself on a good footing with James with a letter of support for him only days before Elizabeth died. In July plots were discovered against James and Raleigh was arrested and charged with high treason.

Raleigh attempted suicide but failed. He then sought Harriot's help in obtaining evidence on his behalf. Raleigh was convicted and sentenced to death by hanging. Poor Harriot was singled out in the judgement as being an atheist and an evil influence. His attempts to help Raleigh had been based on Christian principles (to which undoubtedly he adhered) but this had rather damaged Raleigh as Harriot was seen an atheist using Christian principles for convenience. Harriot was devastated and for about a year undertook no new scientific work as he tried to come to terms with what was happening. Raleigh received a last-minute reprieve from the death sentence but was imprisoned in the Tower of London.

Another plot was to lead to further trouble. On 4 November 1605 Guy Fawkes and others were arrested for attempting to blow up the Houses of Parliament. Four others, including Thomas Percy, the grandson of Henry Percy, were also arrested as the main conspirators. Harriot was held on suspicion of being involved and imprisoned in the Gatehouse. He was interrogated on the charge that he had cast a horoscope of King James in an attempt to use magical powers to influence the King's future. On 27 November Henry Percy, Harriot's patron, was also put in the Tower where he remained until 1621 when he was released. No evidence seems to have been found against Harriot and, although he remained in the Gatehouse for some while writing several letters requesting his release, he was a free man probably by the end of 1605.

As soon as he was released, Harriot returned to his work on optics. He now considered more complex systems and employed Christopher Tooke as a lens grinder from early 1605. His work on light now moved to the dispersion of light into colours. He began to develop a theory for the rainbow and, by 1606, Kepler had heard of the remarkable results on optics achieved by Harriot. Kepler wrote to Harriot, but the correspondence never really achieved any significant exchange of ideas. Perhaps Harriot was too wary of the difficulties that his work had nearly brought on him, or perhaps he did (as he claimed to Kepler) still intend to publish his results if his health permitted.

The appearance of a comet attracted Harriot's attention and turned his scientific mind towards astronomy. He observed a comet on 17 September 1607 from Ilfracombe which would later be identified as Halley's Comet. Kepler had discovered the comet six days earlier but it would be the observations of Harriot and his friend (and student) William Lower which eventually were used by Bessel to compute its orbit.

His astronomy brought back to the fore, Harriot went on to make the earliest telescopic observations in England. On 26 July 1609 at 9 p.m. he sketched the Moon which was at that time 5 days old, viewing it through a telescope with a magnification of 6. He sketched the Moon again a year later on 17 July 1610, by this time he had a telescope giving him a magnification of 10. Soon he had constructed a telescope with a magnification of 20, then by April 1611 he had a 32 magnification telescope.

Harriot observed the moons of Jupiter, although since his first sighting states:-

My first observation of the new planets. I saw but one and that alone

he must have already been aware of Galileo's discovery. As with all his scientific discoveries, Harriot did not publish his results. These observations of Jupiter's moons were made between 17 October 1610 and 26 February 1612.

He was the first to discover sunspots, making 199 observations between 8 December 1610 and 18 January 1613. The first observation of sunspots was made while he was observing Jupiter's moons. From the data he collected he was able to deduce the period of the Sun's rotation. However, around this time his scientific work basically came to an end. He seems to have had his spirits brought low by the deaths of his friends and lost the spirit to continue research (which had brought him much trouble despite his lack of publications).

Of the few pieces of work done by Harriot after 1614, one was his observation of another comet in 1618 (there were three visible comets that year and Harriot observed the third) from Syon House. In 1618 Raleigh, who had been shown the clemency of imprisonment in 1603 rather than death, was put to death. Raleigh was executed on 29 October 1618 in a public execution, with Harriot present to witness the event. However, by this time Harriot was already suffering from the cancer of the nose which eventually led to his death.

The cancer seems to have started around 1613, about the time when Harriot lost interest in pushing forward his mathematical and scientific research. He consulted the top specialist in 1615 who wrote report on the consultation. He described Harriot as (see for example [4]):-

... a man somewhat melancholy. ... A cancerous ulcer in the left nostril eats up the septum of his nose and in proportion to its size holds the lips hard and turned upwards. It has gradually crept well into the nose. This evil the patient has suffered the last two years.

Harriot would suffer this "evil" for a further three years before the cancer took his life.

There are a few other major mathematical achievements due to Harriot which we should mention. He exhibited the logarithmic spiral as the stereographic projection of a loxodrome on a sphere, a projection he proved to be conformal. The loxodromes are the straight lines on the Mercator map, which Harriot computed with great precision. In fact in order to achieve this degree of precision, Harriot introduced finite-difference interpolation.

There is an interesting history to a problem which has only recently been solved, yet originated with Harriot. Raleigh asked Harriot to solve certain problems regarding the stacking of cannonballs. On a manuscript dated 12 December 1591 (Sunday), Harriot set out a table to answer Raleigh's questions. He shows how, if the number of cannonballs is given, one can compute the number of cannonballs to be placed in the base of a pyramid with a triangular, square or oblong base. Raleigh posed a second question, which Harriot also answered, namely given the pyramid of cannonballs, compute the number in the pile.

Harriot was too much the mathematician to stop there, however. From a study of how the cannonballs could fill space, he considered the implications for the atomic theory of matter which he believed in. Later, in his correspondence with Kepler about atomic theory, Harriot mentioned the packing problem. Kepler could not solve the problem but he believed that the densest packing of spheres would be attained if in each layer the centres of the spheres were above the centres of the holes in the layer below. This seems intuitively obvious, but resisted proof until 1998 when Thomas Hales of the University of Michigan (with the help of hours of computer generated data) finally proved the conjecture.

The one part of Harriot's work which we have not yet described is the mathematical work for which, in some ways, he is best known, namely his work on algebra. He introduced a simplified notation for algebra and his fundamental research on the theory of equations was far ahead of its time. As an example of his abilities to solve equations, even when the roots are negative or imaginary, we reproduce his solution of an equation of degree 4. The example in question is in his own handwriting and reproduced in [3].

aaaa - 6aa + 136a = 1155
------------------------------------
    aaaa -2aa + 1 = 4aa -136a + 1156
           aa - 1 = 2a - 34

               33 = 2a - aa
          aa - 2a = -33
      aa - 2a + 1 = +1 - 33
            a - 1 = √-32
            1 - a = √-32
                a = 1 + √-32
                a = 1 - √-32
------------------------------------
           aa - 1 = 34 - 2a
          aa + 2a = 35
      aa + 2a + 1 = 1 + 35
            a + 1 = √36

      a = √36 - 1 = 5
------------------------------------
           -a - 1 = √36
     a = -√36 - 1 = -7

Note how expert Harriot is in completing the square, knowing that whenever he takes a square root there are two answers to consider, and treating all answers equally whether positive or negative, real or imaginary. We have only made one slight change to Harriot's notation. Where we have used the now standard symbol  =  Harriot used插图：Harriotequal.gif ↗.

Harriot invented certain symbols which are used today. However, the symbols < for "less than" and > for "greater than" were not due to Harriot (as is often claimed), but were introduced by the editor of Artis Analyticae Praxis ad Aequationes Algebraicas Resolvendas Ⓣ - Harriot himself used different symbols. There is still scholarly debate on how much Harriot was influenced by Viète, or whether notation and ideas introduced by Viète were learnt by him from Harriot.

As we have seen from the example above, Harriot did outstanding work on the solution of equations, recognising negative roots and complex roots in a way that makes his solutions look like a present day solution. He made the observation that if $a, b, c$ are the roots of a cubic then the cubic is $(x - a)(x - b)(x - c) = 0$. This is a major step forward in understanding which Harriot then carried forward to equations of higher degree.

Although he was far ahead of his time, his work had far less influence than it should have done since, as we have remarked repeatedly above, he published no mathematical work in his lifetime. Even his work on algebra Artis Analyticae Praxis ad Aequationes Algebraicas Resolvendas Ⓣ (1631) was published 10 years after his death and was edited by people who did not fully appreciate the depth of his work. For example, it does not discuss negative solutions.