温琴佐·维维亚尼Vincenzo Viviani

温琴佐·维维亚尼的父亲是Jacopo di Michelangelo Viviani，母亲是Maria Alamanno del Nente；两人都是贵族家庭的成员。维维亚尼在一所耶稣会学校学习，在那里他学习了人文学科。他的数学由Clemente Settimi教授，他是一位Scolopian修士，也是伽利略的朋友。Settimi很快看到了维维亚尼的非凡智慧，并于1638年在宫廷将他引见给托斯卡纳大公Ferdinando II de' Medici。宫廷在里窝那，所以维维亚尼不得不从佛罗伦萨长途跋涉。在路上，他没有浪费时间，而是花了许多小时研究欧几里得的Elements的前三卷。他向托斯卡纳宫廷做了演示，解释了Elements第一卷中的前十六个命题。Ferdinando的宫廷数学家Famiano Michelini随后给他出了一个难题，他自信地回应了。Ferdinando大为赞赏，并为维维亚尼提供了月薪，使他能够购买数学书籍。他还安排维维亚尼与伽利略会面，后者住在佛罗伦萨附近Arcetri的别墅中，他被天主教会软禁在那里。尽管被软禁，伽利略仍然是Ferdinando的托斯卡纳宫廷数学家。

伽利略此时已完全失明，对维维亚尼的知识和能力印象深刻。1639年，他将维维亚尼带回家中作为同伴、学生和合作者，维维亚尼一直担任这一角色，直到伽利略于1642年1月去世。维维亚尼在此期间从伽利略那里学到了很多，与他一起研究物理学和几何学。然而，正如维维亚尼自己所述，他们的关系远远超出了科学家和助手的关系，变得更像父子关系。他写道（例如见[4]）：-

在《关于两门新科学的对话与数学论证》意外出版后不久，伽利略先生允许我进入他在阿切特里的别墅，他当时住在那里。我得以从我们智慧的对话和他珍贵的教导中受益，他也很乐意让我在数学研究中——我最近才开始这项研究——求助于他自己的声音，以解决我因智力天生薄弱而经常遇到的疑惑和困难。

1641年10月，埃万杰利斯塔·托里拆利从罗马搬来后，在阿切特里的别墅与伽利略和维维亚尼会合。伽利略去世后，埃万杰利斯塔·托里拆利被任命接替伽利略的职位，担任费迪南多二世·德·美第奇的托斯卡纳宫廷数学家，维维亚尼继续与他合作。埃万杰利斯塔·托里拆利最为著名的是作为第一个制造持续真空并发现气压计原理的人。1643年，埃万杰利斯塔·托里拆利提出了一个实验，该实验将证明大气压决定液体在充满液体然后倒置在该液体上的管子中上升的高度。1644年，维维亚尼作为埃万杰利斯塔·托里拆利的合作者，进行了这项实验，该实验证明了一项重大科学进展，并导致了气压计的发展。当埃万杰利斯塔·托里拆利于1647年去世时，维维亚尼被任命填补佛罗伦萨设计学院的讲师职位，担任该职位两年。大公还任命维维亚尼为宫廷美第奇家族的数学教师，并担任河流管理局的工程师，他终身担任这一职位[4]：-

虽然作为宫廷数学家的职责常常因大公的意愿、托斯卡纳宫廷的旅行以及其他思想家的出现而变化，但他一生都因对美第奇宫廷的职责而忙碌。对他而言特别疲惫的是他作为工程师的角色，这需要大量骑马旅行，并不止一次导致生病。

然而，维维亚尼一生的主要目标是让伽利略的记忆永存，他想通过出版他的文集来实现这一目标。Michael Segre写道[15]：-

维维亚尼 心中有一项宏伟的计划，要出版 伽利略 著作的版本，其中拉丁文著作将被译成意大利文，反之亦然，并且他一生中收集了大量与 伽利略 相关的材料……但他从未完成这一雄心勃勃的计划，主要是因为他过于追求完美，对自己收集的材料从未完全满意，也不愿停止收集而开始出版。出于大致相同的原因，维维亚尼 自己的大部分科学工作也未能出版，而 维维亚尼 所希望看到的 伽利略 著作版本，直到他去世两个世纪后才在 Favaro 的主持下问世。然而，如果没有 维维亚尼 收集的材料，Favaro 几乎不可能出版他的国家版。

在 [1] 中，Natucci 提出是教会阻止了 伽利略 著作的出版，并对意大利的科学造成了严重打击。然而，Segre 在 [15] 中反驳了这一观点，他写道：-

……教会在这一时期意大利科学衰落中的作用，尽管无疑很大，但有时被夸大了。在 维维亚尼 的情况下，我几乎没有发现干涉的证据。早在 1656 年，他就拥有足够的材料来出版一部杰出的 伽利略 著作版本，即使在宗教裁判所设定的限制范围内；他的文件证明，主要是他自己的犹豫阻止了他完成这项工作。

维维亚尼 一生中的主要兴趣之一是古希腊数学。早在 1646 年，在与 埃万杰利斯塔·托里拆利 合作的同时，他还在进行一项恢复 大阿里斯泰俄斯 工作的计划。帕普斯 将一部已失传的题为 Five Books concerning Solid Loci 的著作归功于 大阿里斯泰俄斯。（Solid Loci 是圆锥曲线的希腊术语。）然而，帕普斯 指出了该著作中的命题，维维亚尼 根据 帕普斯 的这些引用重建了原著。这是 维维亚尼 一生中大部分时间都在从事的项目。1673 年，他出版了他的重建第一版，但他继续研究，他的最终成果 De locis solidis secunda divinatio geometrica in quinque libros iniuria temporum amissos tristaei senioris geometrae Ⓣ（一位资深数学家恢复的《Solid loci》五卷）直到 1701 年才出版，即他去世前两年。

维维亚尼对一部希腊文著作的另一次复原，因多种原因而引人注目。这就是他对阿波罗尼奥斯的Conics第五卷的复原。当他开始复原时，这部八卷本著作只发现了前四卷，于是维维亚尼着手重构第五卷。到1656年，维维亚尼的工作已接近完成，此时吉奥万尼·阿方索·博雷利（一位托斯卡纳宫廷数学家同行）在佛罗伦萨的劳伦琴图书馆发现了阿波罗尼奥斯的Conics前七卷的阿拉伯文版本。博雷利将手稿带到罗马，由Abrahamus Ecchellensis译成拉丁文。1659年，阿拉伯文译本和维维亚尼的复原同时出版。维维亚尼的著作题为De maximis et minimis geometrica Divinatio Ⓣ（几何极大与极小的占卜），无疑是在他完全不知道阿波罗尼奥斯著作译本的情况下写成的。当然，有趣的是看看维维亚尼能多么忠实地重构阿波罗尼奥斯的书，因为现在复原和原著都已可得。维维亚尼做得非常出色，他最大的“错误”是他能够比阿波罗尼奥斯本人更深入地洞察。意识到维维亚尼在某种意义上比受人尊敬的阿波罗尼奥斯是更好的几何学家，使他立刻在欧洲各学术中心声名鹊起。他作为数学家的声誉在整个欧洲都很高。1666年，法国路易十四邀请他担任Académie Royale的职位，波兰的弗瑞兹·约翰二世卡齐米日也于1666年邀请维维亚尼担任他的天文学家。大公不愿失去维维亚尼，任命他为自己的数学家。维维亚尼接受了这一职位，拒绝了路易十四和约翰二世卡齐米日的邀请。

1657年，维维亚尼成为大公新成立的科学院——实验科学院的首批成员之一，该科学院于当年6月正式成立。然而，在此之前，该科学院已非正式存在了一段时间，维维亚尼组织了一群自然哲学家进行各种科学实验。其中一项在实验科学院正式成立之前进行的实验与声音的速度和传播有关。1656年10月10日，维维亚尼与博雷利一起使用摆作为计时装置，证明声音传播两倍固定距离所需时间是传播原距离所需时间的两倍。换句话说，其速度是均匀的。两天后，在佛罗伦萨的宫殿里，他再次使用摆，通过测量在Petraia别墅发射的大炮的闪光与声音之间的时间差来测量声速。他们得到了350米/秒的值，这比皮埃尔·伽桑狄之前获得的478米/秒的值要好得多（目前公认的值是0°C时331.29米/秒）。科学院正式成立后，维维亚尼参与了他们进行的大部分实验工作，例如用埃万杰利斯塔·托里拆利的气压计进行的实验以及与水的冻结特性有关的实验。

维维亚尼 的另一部重要著作是 Quinto libro degli Elementi d'Euclide: ovvero Scienza universale delle proporzioni spiegata colla dottrina del Galileo, con nuov'ordine distesa Ⓣ（欧几里得《几何原本》第五卷：用伽利略理论解释的普遍比例科学……）。这是另一次重建，这次是 欧几里得 的第五卷，建立在 1641 年与 埃万杰利斯塔·托里拆利 和 伽利略 已经完成的工作基础上。在序言中，他承认自己深受 伽利略 的恩惠，自称“伽利略 的最后弟子”（见 [4]）：-

事实是，由于我的好运，我是他最后的弟子，因为在他生命的最后三年里，他一直是我的老师，而在所有在他临终时在场的人中（除了两位神父，还包括埃万杰利斯塔·托里拆利、他的儿子Vincenzio 伽利略以及他家里的其他人），只有我活到了最后。

第一版于1674年以简本形式出版，随后于1676年4月15日印行了增订第二版。Natucci写道[1]：-

以古人的严谨和冗长，维维亚尼将附录用于几何问题，其中一个是关于trisection of an angle的，通过使用圆柱螺线或摆线来解决；另一个是倍立方问题，借助圆锥曲线或三次$xy^{2} = k$来解决。

他还出版了Diporto Geometrico Ⓣ（几何的乐趣）（1676年）和Enodatio problematum universis geometris propositorum Ⓣ（所提议事物的几何学的所有问题，看哪，odatio）（1677年）。其中第一本包含了十二个几何问题的解答，这些问题是由Cristoforo Sadler作为挑战发表的。看来维维亚尼觉得这些问题很容易，并不认为值得发表他的解答，但大公的兄弟Leopoldo亲王强烈鼓励他这样做。由于他一生都是工程师，维维亚尼也发表了工程学方面的著作。特别是他出版了Discorso intorno al difendersi da' riempimenti e dalle corrosione de' fiumi Ⓣ（关于防御河流淤积和侵蚀的讲话）（1687年）。

我们尚未评论维维亚尼最为人所知的事情，即他的Racconto istorico della vita di Galileo GalileiⓉ（伽利略的历史）。1650年后不久，卡洛·马诺莱西开始准备伽利略著作的一个版本。由于Dialogue on the Great World Systems的出版被天主教会禁止，它永远不可能包含所有著作，但出版其余著作的计划得到了广泛支持。莱奥波尔多王子要求维维亚尼为马诺莱西版本的伽利略作品写一篇生平概述。然而，事情并未按计划进行，尽管维维亚尼写了他的Life of Galileo，但它并未出现在马诺莱西于1655-56年在博洛尼亚出版的伽利略的Collected Works版本中。看来维维亚尼和美第奇家族觉得马诺莱西的版本不够宏大，不足以纪念伽利略。维维亚尼想出版他自己的伽利略的Collected Works版本，因此他保留了他所写的Life of Galileo，打算将其纳入他希望编辑的伽利略的Collected Works出版物中。1659年，维维亚尼写了第二篇关于伽利略作品Lettera di Vincenzio Viviani al Principe Leopoldo de' Medici intorno all'applicazione del pendolo all'orologio的文章。这两篇文章在维维亚尼生前均未出版，但他的Racconto istorico于1717年出版。它成为伽利略后来的传记作者的主要资料来源。然而，在20世纪，历史学家开始争论文章中的某些细节是维维亚尼编造的，特别是那些与伽利略进行的实验有关的细节。维维亚尼无疑非常仔细，并投入了大量精力收集关于伽利略生平的资料。如果他确实编造了某些事件，例如伽利略从比萨斜塔扔下重物，那么这些是为了说明伽利略的科学方法。参见[15]进行有趣的详细讨论。

尽管他在几何学方面有很高的造诣，维维亚尼在这一领域似乎很少做出原创性贡献。他确实确定了摆线的切线，但他并不是第一个成功做到这一点的人。然而，他热忱地相信几何学的价值，写道：-

思辨几何学是诚实获取有用、令人愉悦、美好和善良之物的唯一导师。几何学是唯一真正的科学，因为它从自身产生知识，无需原因的媒介。唯有几何学教导如何获得知识，甚至提醒人类理智——它是神圣理智的火花——作为一个通过最熟知的原则凭借自然之光而认识的人，如果它愿意，它能够在不欺骗自己或他人的情况下，认识关于受造宇宙的一切事物的存在和属性，以及上帝所安排的秩序，即数量、重量和度量。

然而，他生活的时代，微积分正开始被用来证明几何结果。哥特弗里德·威廉·莱布尼茨对伽利略留下的论文感兴趣，并于1689年11月底在佛罗伦萨会见了维维亚尼。在这次会面中，两人相处融洽，但在哥特弗里德·威廉·莱布尼茨于1692年发表Solutio problematis a Galilaeo propositi de linea catenariaⓉ（对伽利略提出的悬链线问题的解答）之后，维维亚尼对微分和积分的使用感到不满，他认为这不过是一种只能解决自身问题的游戏。因此，在1692年，他向欧洲数学家提出挑战，要求解决球面一部分的求积问题。在Aenigma geometricum de miro opificio testudinis quadrabilis hemisphaericaeⓉ（求半球与圆柱相交面积的几何问题）中，他询问如何从一个半球上切出四个相等的窗户，使得剩余表面能够精确求积。他的解答涉及使用四个直圆柱的交集，这些圆柱的底面与半球的底面相切。关于这个问题的讨论，见[12]。

我们还注意到，维维亚尼于1690年出版了欧几里得的Elements的意大利语版本。这由恩里科·贝蒂和弗朗切斯科·布廖斯基于1867年重印，当时他们正试图改进意大利的几何教学。维维亚尼去世时留下了一部几乎完成的关于固体阻力的著作，由格兰迪完成并出版。

贝尔纳·勒布耶·德·丰特奈尔[7]说，维维亚尼具有天真和朴素的举止，这是那些与书籍打交道多于与人打交道的人通常保持的。他和蔼可亲，谦虚，极其感恩，是一个迅速而忠实的朋友。

维维亚尼在佛罗伦萨建造了一座漂亮的房子，并将伽利略的半身像放在门上，两侧刻有铭文，以纪念他的名字。这座房子通常被称为“伽利略的房子”，尽管这是不正确的。铭文在文章[16]中给出。

Vincenzo Viviani's father was Jacopo di Michelangelo Viviani and his mother was Maria Alamanno del Nente; both were members of a noble families. Vincenzo studied at a Jesuit school where he learnt the humanities. He was taught mathematics by Clemente Settimi, a Scolopian friar and a friend of Galileo. Settimi quickly saw Viviani's exceptional intelligence and, in 1638, presented him at court to Ferdinando II de' Medici, Grand Duke of Tuscany. The court was in Livorno, so Viviani had to make the long journey from Florence. While on the road he did not waste his time but spent the many hours studying the first three books of Euclid's Elements. He made a presentation to the Tuscan Court explaining the first sixteen propositions in the first book of the Elements. Famiano Michelini, Ferdinando's Court mathematician, then set him a problem to which he responded confidently. Ferdinando was greatly impressed and provided a monthly salary for Viviani that enabled him to purchase mathematical books. He also arranged for Viviani to meet Galileo, who was living in his villa in Arcetri, near Florence, where he had been put under house arrest by the Catholic Church. Despite the house arrest, Galileo was still Ferdinando's Tuscan Court Mathematician.

Galileo, who by this time was totally blind, was very impressed by Viviani's knowledge and abilities. In 1639, he took Viviani into his home as a companion, student and collaborator, and Viviani continued in this role until Galileo died in January 1642. Viviani learnt much from Galileo over this period, working with him on physics and geometry. However, as Viviani relates himself, their relationship went well beyond that of scientist and assistant, becoming much more like that of a father and son. He wrote (see for example [4]):-

Soon after the unexpected publication of 'Discourses and mathematical demonstrations concerning the two new sciences', Signor Galileo allowed me into his villa in Arcetri where he was staying. I was able to benefit from our intelligent conversations and his precious teachings and he was content that in the study of mathematics, which I had only recently begun, I could turn to his own voice for the solution of those doubts and difficulties that I often found through the natural weakness of my intellect.

In October 1641, Galileo and Viviani were joined in the villa at Arcetri by Evangelista Torricelli when he moved from Rome. After Galileo's death, Torricelli was appointed to fill Galileo's post as Ferdinando II de' Medici's Tuscan Court Mathematician and Viviani continued his collaboration with him. Torricelli is most famed as the first person to create a sustained vacuum and to discover the principle of a barometer. In 1643 Torricelli proposed an experiment which would demonstrate that atmospheric pressure determines the height to which a fluid will rise in a tube filled with a liquid, then inverted over the same liquid. In 1644 Viviani, as Torricelli's collaborator, carried out the experiment which proved a major scientific advance and led to the development of the barometer. When Torricelli died in 1647, Viviani was appointed to fill the lectureship at the Accademia del Disegno in Florence, holding this post for two years. The Grand Duke also appointed Viviani as mathematics teacher to the Medici family at Court, and as engineer with the Uffiziali dei Fiumi, a position he held for the rest of his life [4]:-

While his responsibilities as a Court Mathematician often varied according to the desires of the Grand Duke, the travels of the Tuscan Court, and the arrival on the scene of other thinkers, all his life he was kept busy by his duties to the Medici Court. Particularly exhausting for him was his role of engineer, which required substantial travel on horseback and led to illness on more than one occasion.

The main thrust of Viviani's life, however, was to keep Galileo's memory alive and he wanted to do so by publishing his collected works. Michael Segre writes [15]:-

Viviani had in mind a grand edition of Galileo's works, in which the Latin works would be translated into Italian and vice versa, and throughout his life he collected an enormous quantity of material related to Galileo .... But he never brought this ambitious project to completion, mainly because he was too much of a perfectionist, never entirely satisfied with the material he had amassed and reluctant to stop collecting and begin publishing. For much the same reason, most of Viviani's own scientific work remained unpublished, and an edition of Galileo's works, as Viviani would have liked to see it, only appeared two centuries after his death, under Favaro's supervision. Favaro, however, could hardly have published his National Edition without the materials collected by Viviani.

In [1] Natucci suggests that it was the Church that prevented the publication of Galileo's works and caused a serious blow to science in Italy. Segre, however, counters that view when he writes [15]:-

... the role of the church in Italy's scientific decline during this period, though no doubt major, has sometimes been exaggerated. In Viviani's case I found little evidence of interference. As early as 1656 he had at his disposal enough material to publish an outstanding edition of Galileo's works, even within the limitations set by the Inquisition; his papers testify that primarily his own hesitations prevented him from completing the work.

Throughout his life, one of Viviani's main interests was in ancient Greek mathematics. As early as 1646, while collaborating with Torricelli, he was also working on a project to restore the work of Aristaeus the Elder. Pappus gave Aristaeus great credit for a work entitled Five Books concerning Solid Loci which had been lost. (Solid Loci is the Greek term for conic sections.) Pappus, however, indicated propositions from the work and Viviani reconstructed the original from these references by Pappus. It was a project that Viviani worked on for most of his life. In 1673 he published a first edition of his restoration but he continued to work on it and his final effort De locis solidis secunda divinatio geometrica in quinque libros iniuria temporum amissos tristaei senioris geometrae Ⓣ was only published in 1701, two years before his death.

Another restoration of a Greek text by Viviani is interesting for a number of reasons. This was his restoration of the fifth book of Apollonius's Conics. At the time he began the restoration only the first four books of this eight-book work had been found and Viviani set about reconstructing the fifth. By 1656 Viviani's work was quite close to completion when Giovanni Alfonso Borelli (a fellow Tuscan Court mathematician) discovered an Arabic version of the first seven books of Apollonius's Conics in the Laurentian Library in Florence. Borelli took the manuscript to Rome where it was translated into Latin by Abrahamus Ecchellensis. In 1659 both the translation from the Arabic and Viviani's restoration were published. Viviani's work was entitled De maximis et minimis geometrica Divinatio Ⓣ and was certainly written by him without any knowledge of the translation of Apollonius's work. It is interesting, of course, to see how faithfully Viviani was able to reconstruct Apollonius's book since now both the reconstruction and the original had become available. Viviani had done an excellent job, his biggest 'error' being that he had been able to penetrate deeper than Apollonius himself. The realisation that Viviani was, in some sense, a better geometer than the revered Apollonius, gave him instant fame throughout the centres of learning in Europe. His reputation as a mathematician was high throughout Europe. Louis XIV of France offered him a position at the Académie Royale in 1666, John II Casimir of Poland offered Viviani a post as his astronomer, also in 1666. The Grand Duke, not wishing to lose Viviani, appointed him as his mathematician. Viviani accepted this post and turned down the offers from Louis XIV and John II Casimir.

In 1657 Viviani became one of the first members of the Grand Duke's new academy, the Accademia del Cimento, which was formally founded in June of that year. However, the Academy had informally existed for some time before that with Viviani organising a group of natural philosophers to carry out a variety of scientific experiments. One such experiment, carried out before the formal founding of the Accademia del Cimento, related to the speed and propagation of sound. On 10 October 1656, together with Giovanni Borelli, Viviani used a pendulum as a timing device to show that sound travels twice a fixed distance in twice the time taken to travel the original distance. In other words, its velocity is uniform. Two days later, at the palace in Florence, he measured the velocity of sound by timing, again using a pendulum, the difference between the flash and the sound of a cannon fired at the villa of Petraia. They obtained the value of 350 metres per second, which is considerably better than the previous value of 478 metres per second obtained by Gassendi (the currently accepted value is 331.29 metres per second at 0°C. After the Academy was formally founded, Viviani was involved in most of the experimental work they carried out, such as experiments with Torricelli's barometer and experiments relating to properties of freezing water.

Another important work by Viviani is Quinto libro degli Elementi d'Euclide: ovvero Scienza universale delle proporzioni spiegata colla dottrina del Galileo, con nuov'ordine distesa Ⓣ. This was another restoration, this time of Euclid's fifth book, building on work already carried out with Torricelli and Galileo in 1641. In the Preface he acknowledges his great debt to Galileo, calling himself 'Galileo's last disciple' (see [4]):-

The fact is that through my good fortune, I am his last disciple, because he was my teacher continually during the last three years of his life, and from all of us who were present while he took his last breath (who apart from two priests, included Torricelli, his son Vincenzio Galilei, and others from his home), I alone have survived them all.

The first edition was published in a short form in 1674, then in an enlarged second edition was printed in 15 April 1676. Natucci writes [1]:-

With the rigour and prolixity of the ancients, Viviani devoted an appendix to geometric problems, among which was one on the trisection of an angle, solved by the use of the cylindrical spiral or of a cycloid; another was the problem of duplicating the cube, solved by means of conics or of the cubic $xy^{2} = k$.

He also published Diporto Geometrico Ⓣ (1676) and Enodatio problematum universis geometris propositorum Ⓣ (1677). The first of these contained solutions to twelve geometrical problems which had been published as challenges by Cristoforo Sadler. It appears that Viviani found these easy and did not consider it worthwhile to publish his solutions but Prince Leopoldo, brother of the Grand Duke, had strongly encouraged him to do so. As he was an engineer all his life Viviani also published on engineering. In particular he published Discorso intorno al difendersi da' riempimenti e dalle corrosione de' fiumi Ⓣ (1687).

We have not yet commented on the thing for which Viviani is best known, namely his Racconto istorico della vita di Galileo Galilei Ⓣ. Soon after 1650, Carlo Manolessi began preparing an edition of Galileo's writings. It could never have contained all the writings since publication of Dialogue on the Great World Systems was banned by the Catholic Church, but the project tp publish the rest had widespread support. Prince Leopoldo asked Viviani to write a sketch of Galileo's life for inclusion in Manolessi's edition of his works. However, things did not go as planned for although Viviani wrote his Life of Galileo it did not appear in the edition of Galileo's Collected Works published by Manolessi in Bologna 1655-56. It would appear that Viviani, and the Medici family, felt that Manolessi's edition was not grand enough to honour Galileo. Viviani wanted to publish his own edition of Galileo's Collected Works so he kept the Life of Galileo which he had written intending to include it in a publication of Galileo's Collected Works which he hoped to edit. In 1659 Viviani wrote a second essay on Galileo's work Lettera di Vincenzio Viviani al Principe Leopoldo de' Medici intorno all'applicazione del pendolo all'orologio. Neither essay was published in Viviani's lifetime, but his Racconto istorico was published in 1717. It became the main source of material for Galileo's later biographers. However, in the 20th century historians began to argue that certain details in the essay had been invented by Viviani, particularly those relating to the experiments carried out by Galileo. Viviani was certainly very careful and put in a great amount of effort collecting information about Galileo's life. If he did invent certain incidents, such as Galileo dropping weights from the leaning tower of Pisa, then they were done to illustrate Galileo's approach to science. See [15] for an interesting detailed discussion.

Despite his great expertise in geometry, Viviani seems to have done little original in this area, He did determine the tangent to the cycloid but he was not the first to succeed in this. He believed passionately in the value of geometry, however, writing:-

Speculative geometry is the unique teacher of the honest acquisition of what is useful, delightful, beautiful and good. Geometry is the only true science because it produces knowledge from itself without mediation of causes. Geometry alone teaches how to achieve knowledge and even reminds the human intellect - which is a spark of the divine one - that as a knower through the principles most known with the light of nature it can, if it so wishes, without deceiving itself or others, know the existence and properties of all things concerning the created universe and the order disposed by God, in number, weight, and measure.

However, he lived at a time when the calculus was beginning to be used to prove geometric results. Leibniz was interested in the papers left by Galileo and met Viviani in Florence in late November 1689. The two got on well during this meeting but after Leibniz published Solutio problematis a Galilaeo propositi de linea catenaria Ⓣ in 1692, Viviani became unhappy about the use of the differential and integral calculus which he believed to be nothing but a kind of game that could only solve its own problems. For this reason, in 1692, he challenged European mathematicians to solve a problem of the quadrature of a part of the surface of a sphere. In Aenigma geometricum de miro opificio testudinis quadrabilis hemisphaericae Ⓣ he asked how four equal windows could be cut from a hemisphere so that the remaining surface can be exactly squared. His solution involved using the intersection of four right cylinders, the bases of which are tangent to the base of the hemisphere. See [12] for a discussion of this problem.

We note also that Viviani published an Italian version of Euclid's Elements in 1690. This was reprinted in 1867 by Enrico Betti and Francesco Brioschi who, at that time, were trying to improve the teaching of geometry in Italy. On his death Viviani left an almost completed work on the resistance of solids which was completed and published by Guido Grandi.

Bernard de Fontenelle [7] says that Viviani had an innocence and simplicity of manners which persons commonly preserve who have less commerce with men than with books. He was affable, modest, grateful in the highest degree, and a fast and faithful friend.

Viviani built a handsome house in Florence, and placed the bust of Galileo over the door, with inscriptions on each side of it, to honour his name. This house was commonly called 'Galileo's house' although this is incorrect. The inscriptions are given in the article [16].