皮耶罗·德拉·弗朗西斯卡Piero della Francesca

皮耶罗·德拉·弗朗西斯卡出身于一个相当富裕的商人家庭，被公认为文艺复兴时期最重要的画家之一。尽管今天人们总是称他为弗朗西斯卡，但在他生前，他应该被称为弗朗西斯卡·迪·贝内代托·德·弗朗切斯基。他的父亲贝内代托·德·弗朗切斯基是一位皮革和羊毛商人，而他的母亲罗曼娜是羊毛商人伦佐·迪·卡洛·达·蒙特基的女儿，来自附近的蒙特基村。据认为，贝内代托和罗曼娜大约在1416年结婚，因为尽管没有找到结婚证书，但他们在当年交换了嫁妆。然而，贝内代托在1410年继承了一些钱和一所房子的份额，有一些证据表明他在这之后不久就结婚了，但如果这是真的，很难相信嫁妆会在六年后才交换。正如这些争论所表明的，没有确切的证据表明弗朗西斯卡的出生时间，来源给出的日期在1406年至1422年之间。然而，詹姆斯·贝克在[20]中为1420年6月提出了有力的论据，部分基于弗朗西斯卡是他父母的第三个儿子。我们对弗朗西斯卡的早年知之甚少，但鉴于他父亲的财富和地位（他拥有农田、葡萄园、城市房产和一座精美的家庭住宅），很明显他会在圣塞波尔克罗上学。

关于弗朗西斯卡受教育情况的细节仍然只是推测，但我们可以从他成年后所展现的技能中推断出一些性质。他用意大利语而非拉丁语撰写数学著作，这表明他可能没有在学校学过拉丁语。此外，很明显弗朗西斯卡的意大利语写作缺乏风格，相当简单和初级。这表明弗朗西斯卡在学校待的时间并不长，也证实了他缺乏拉丁语训练，因为那些能写流利拉丁语的人在写意大利语时会形成一种反映拉丁语结构的风格。然而，他成年后读过欧几里得的Elements，并对阿基米德的著作有所了解，而在弗朗西斯卡的时代没有意大利语版本，所以他后来一定学了足够的拉丁语来阅读这些著作。Paul Grendler写道[32]：-

他父亲的职业和社会地位可能决定了弗朗西斯卡就读于一所算盘学校和方言学校，也许只读了几年。这个男孩弗朗西斯卡可能接受了与成千上万其他具有相同背景的意大利人一样的教育，但这是一种令人惊讶地丰富而多样的教育。成年后的弗朗西斯卡可能学了足够的拉丁语，使他能够阅读带有专门词汇的拉丁语数学文本。关键似乎在于弗朗西斯卡终生都在学习。而他的学习对他很有用。人们有一种印象，弗朗西斯卡记住了一切——在学校读到的圣徒传说、算盘学校的数学、他对古代几何学的创造性探索——他所有的学习都丰富了他的绘画。

James Banker的研究大大增加了我们对弗朗西斯卡在圣塞波尔克罗生活的了解（见[3]和[19]）。Banker发现的文件表明，弗朗西斯卡在1430年代担任Antonio d'Anghiari的助手。Antonio来自距博尔戈圣塞波尔克罗约7公里的安吉亚里村，但搬到了博尔戈圣塞波尔克罗，在那里完成了他的第一项委托，并于1430年5月27日获得报酬。弗朗西斯卡的父亲Benedetto是这次付款的见证人。次年六月，Benedetto收到了弗朗西斯卡“画蜡烛杆”工作的报酬。在接下来的几年里，Benedetto继续收到弗朗西斯卡所做工作的报酬，在此期间他显然在担任Antonio的助手。例如，一份日期为1432年12月29日的文件写道（见[19]）：-

来自安吉亚里、居住在博尔戈土地的画家Antonio di Giovanni师傅，承认并满意地成为上述博尔戈的Benedetto di 弗朗西斯卡 di Benedetto的真正债务人，金额为五十六弗罗林……作为上述Benedetto之子、画家Pietro的薪水和债务，以及上述Pietro自最近六月一日起为上述Antonio师傅提供劳动的工资，还有上述Benedetto为上述Antonio师傅的需要，为圣弗朗西斯科教堂主祭坛祭坛画的画作装饰而随时自由提供的借款，直至今日。

弗朗西斯卡在这份文件中被描述为“画家”，这一事实表明弗朗西斯卡此时已不是学徒，而是助手。根据我们上面给出的年代，他当时只有十二岁，当然正是当学徒的年龄，但在这个年龄就成为助手令人惊讶。真正的问题在于，尽管关于弗朗西斯卡青年时代的所有证据都引人入胜，但无论提出什么年代，似乎总是存在一些困难。文件显示，弗朗西斯卡的父亲继续收到弗朗西斯卡与Antonio d'Anghiari合作所做工作的报酬，时间是1435年、1436年和1437年，特别是为教堂绘制横幅。一份日期为1438年1月8日、同样由Benedetto签署的文件，结束了弗朗西斯卡与Antonio的合作关系，承认已收到弗朗西斯卡在博尔戈圣塞波尔克罗及其周边三座独立教堂和礼拜堂工作的报酬。Banker在[3]中概述了这些岁月对年轻的弗朗西斯卡的影响：-

1430年代圣塞波尔克罗的手工艺文化深刻地塑造了弗朗西斯卡的感受力。他源于这种文化的出身赋予了他几种力量和思维习惯，这些支撑了他一生，克服了他家庭在社会上介于工匠和商人地位之间的模糊性质。他在手工艺文化中的劳动对弗朗西斯卡的生活和艺术始终具有形成性影响，尽管他后来在绘画、数学和几何学方面取得了巨大成就，这些成就需要接触和参与精英文化。他作为工匠的形成岁月塑造了他心智的基础结构：他的语言实践、工匠方法以及对工匠成就的表现。

弗朗西斯卡 离开博尔戈圣塞波尔克罗，前往佛罗伦萨，1439年，他在那里协助多梅尼科·韦内齐亚诺绘制圣玛丽亚诺韦拉教堂中圣埃吉迪奥小教堂的壁画。保罗·沃森写道[2]：-

在佛罗伦萨，他可能研究了多纳泰罗和卢卡·德拉·罗比亚的雕塑、菲利波·布鲁内莱斯基的建筑以及马萨乔和弗拉·安杰利科的绘画，并且他可能阅读了人文主义者和建筑师莱昂·巴蒂斯塔·阿尔伯蒂关于绘画的理论著作。毫无疑问，多梅尼科·韦内齐亚诺引导他关注这些杰出人物，其作品体现了文艺复兴对色彩和光线作为绘画构图元素的强调。正是与佛罗伦萨早期文艺复兴艺术的这种接触，为弗朗西斯卡自己的风格奠定了基础。

尚不清楚弗朗西斯卡是否协助多梅尼科完成了整个工作，但我们知道他于1439年9月12日因参与绘制第一幅壁画而获得报酬。1441年6月，多梅尼科开始在小教堂绘制第二幅壁画，此时弗朗西斯卡被列为博尔戈圣塞波尔克罗镇议会的成员。有人指出，他当时可能实际上并不住在博尔戈，很可能继续协助多梅尼科。然而，他似乎于1441年左右在博尔戈圣塞波尔克罗的圣弗瑞兹·约翰福音传道者大教堂绘制了Baptism（尽管一些艺术史学家认为它是在20年后绘制的）。金兹堡在[9]中深入讨论了用于确定弗朗西斯卡绘画年代的各种论据。1445年，慈悲兄弟会，一个旨在帮助穷人和不幸者的行会，委托他在博尔戈圣塞波尔克罗的教堂绘制一幅祭坛画。委托规定该作品需在三年内完成，但直到1462年，即弗朗西斯卡开始十七年后才完成。弗朗西斯卡绘制的图像描绘了圣母在她的斗篷下保护行会成员。弗朗西斯卡只是断断续续地在这幅祭坛画上工作，在这些年里承担了许多其他委托。

你可以在THIS LINK看到San Sepolcro alterpiece。

同样在博尔戈圣塞波尔克罗，大约在这段时间，他画了Baptism of Christ，这幅画出色地运用了几何学，例如将鸽子置于构成画作顶部的圆的中心。博尔戈圣塞波尔克罗周围的山丘和城镇本身出现在场景的背景中。

你可以在THIS LINK看到Baptism of Christ。

在佩萨罗和安科纳工作后，大约1450年，他被博尔索·德·埃斯特邀请到费拉拉，在那里他[13]：-

……装饰了许多宫殿房间。然而，这些后来被老公爵埃尔科莱在翻新宫殿时摧毁。

同样在费拉拉期间，他在圣阿戈斯蒂诺的奥古斯丁会圣安德烈亚教堂绘制了壁画。1451年，他在里米尼，受西吉斯蒙多·潘多尔福·马拉泰斯塔委托。他绘制了一幅著名的壁画，描绘西吉斯蒙多在圣西吉斯蒙德前祈祷，由于这幅画有签名和日期，这是本传记中少数几个确定的日期之一。在里米尼期间，他会见了正在完成其著作De re aedificatoria的数学家和建筑师莱昂·巴蒂斯塔·阿尔伯蒂。莱昂·巴蒂斯塔·阿尔伯蒂，像弗朗西斯卡一样，对透视法深感兴趣，两人在里米尼期间必定就此主题交换了想法。弗朗西斯卡最著名的作品之一（许多人会说是他最著名的）是大约1457年开始的Story of the True Cross组画。玛丽莲·拉文[33]写道，这些壁画由阿雷佐最富有的家族（巴奇家族）委托，绘制在圣弗朗西斯科教堂：-

在托斯卡纳的阿雷佐镇，弗朗西斯卡绘制了二十世纪观众最为赞赏的文艺复兴壁画组。这些壁画描绘了真十字架的故事，是为圣方济各修道院教堂的圣坛区域委托创作的。装饰工作由一位受欢迎但相当工匠气的画家比奇·迪·洛伦佐于14世纪40年代末开始；很可能是在比奇于1452年去世后，弗朗西斯卡被召来。他在1466年12月20日之前的某个时候完成了这个项目……

这些壁画不仅显示弗朗西斯卡完全精通透视法，还显示出对光线的非凡处理。弗朗西斯卡利用光线，结合阴影和暗部，使人物具有三维感。光线还与透视法结合使用，以在所描绘的场景中创造深度的错觉。

你可以在THIS LINK看到True Cross frescos。

1458年8月教皇庇护二世成为教皇后不久，弗朗西斯卡被召到罗马。弗朗西斯卡在梵蒂冈绘制了许多壁画，后来被拉斐尔的作品取代时被毁，还在圣母大殿绘制了壁画。离开罗马后，弗朗西斯卡回到家乡，在那里承接了许多委托，然后前往洛雷托，与多梅尼科·韦内齐亚诺一起开始装饰圣器室的拱顶。瘟疫的爆发使他们在开始工作后不久就离开了。弗朗西斯卡的The Flagellation of Christ，约1460年在乌尔比诺绘制，现在被认为是杰作。玛丽莲·拉文写道[10]：-

尽管弗朗西斯卡的《鞭笞》今天是意大利十五世纪最著名和最受推崇的画作之一，但它在艺术史上的突出地位相对较新。它尺寸小，藏于偏僻的乌尔比诺镇，其存在在十八世纪末之前没有记录。当它在此后某个时候首次出现在艺术史文献中时，令人惊讶的是，弗朗西斯卡被认为更像技术员而非艺术家。……直到[20]世纪之交左右，[《鞭笞》]才开始被欣赏为伟大的艺术作品。随着浪漫主义态度的消退以及对后印象派和立体主义的批评性接受，它很快因早先被指责的那些品质而受到赞扬；其克制的表达、色彩的纯净，尤其是其构图的完美平衡。

场景以建筑为主，令人惊叹地运用透视法，既增加了现实主义，又将视线引向基督的形象，尽管他几乎被描绘在背景中。

你可以在THIS LINK看到Flagellation of Christ。

弗朗西斯卡还创作了其他一些引人注目的画作，包括约1563年在博尔戈圣塞波尔克罗绘制的Resurrection。在准备离开坟墓的基督形象背后，博尔戈圣塞波尔克罗周边的风景和城镇本身被描绘得惊人地准确。

你可以在THIS LINK看到Resurrection of Christ。

进一步的作品还有Madonna del Parto，这是一幅大约同一时期绘制的壁画，原位于蒙特基的圣玛丽亚迪莫门塔纳，约1470-75年绘制的Nativity，约1474年在乌尔比诺绘制的Madonna di Senigallia，以及Madonna and Child with Saints，这是一幅在《塞尼加利亚的圣母》之后不久绘制的蒙特费尔特罗祭坛画。

你可以在THIS LINK看到Madonna del Parto，在THIS LINK看到Madonna di Senigallia，在THIS LINK看到Montefeltro Altarpiece。

他还绘制了乌尔比诺公爵Federico da Montefeltro和乌尔比诺公爵夫人Battista Sforza的肖像。Federico右眼失明，因此弗朗西斯卡以侧面像呈现他，以免冒犯。尽管如此，公爵的画像仍以写实手法绘制，例如他的鹰钩鼻和疣都被描绘出来，但肖像仍显得极为优雅。背景风景极为细致却又微妙。

你可以在THIS LINK看到Urbino portraits。

弗朗西斯卡在家乡度过了生命的最后几年，1480年成为圣巴托洛梅奥兄弟会的会长。尽管一些历史学家对此有争议，但许多人接受瓦萨里在[13]中的说法，即弗朗西斯卡失明了。例如贝克写道[20]：-

有确凿证据表明他晚年失明，尽管几乎可以肯定这发生在1487年之后，当时他手写了一些笔记，勾勒出他最后遗嘱和遗言的细节。显然，失明对他的生活和艺术产生了巨大影响，使得任何约1488年之后的作品都无法确定年代。对批评者来说更具挑战性的是，他的视力可能缓慢恶化。如果这能被记录下来，我们或许能理解他后期作品中可能反映视力下降的特征……目前，没有证据能精确确定失明发生的时间，也没有人努力去识别他艺术中的视觉危机。

最后我们注意到，没有证据表明弗朗西斯卡曾经结过婚。当然，在他留存下来的遗嘱中，没有提到妻子或孩子。

我们现在给出由J V Field提供的关于弗朗西斯卡数学著作的描述。

在他自己的时代，他也以一位极其称职的数学家而闻名。在他的Lives of the most famous painters……[13]中，Giorgio Vasari（1511-1572）说弗朗西斯卡在早年就展现出数学才能，并继续写了“许多”数学论著。其中，已知有三部留存至今。它们所知的标题是：Abacus treatise（Trattato d'abaco）、Short book on the five regular solids（Libellus de quinque corporibus regularibus）和On perspective for painting（De prospectiva pingendi）。几乎可以肯定弗朗西斯卡用白话（他的母语方言是托斯卡纳语）写了这三部作品，而且这三部作品都采用与“实用数学”传统相关的风格，也就是说，它们主要由一系列有解答的例题组成，论述性文字相当少。

算盘论著类似于“算盘学校”中用于教学的作品。它涉及算术，从分数的使用开始，通过一系列标准问题，然后转向代数，并通过类似的标准问题，然后转向几何，并通过比标准更多的问题，然后（毫无预警地）提出一些完全原创的三维问题，涉及两个“阿基米德多面体”（现在称为截角四面体和截角八面体）。

在Short book on the five regular solids中又出现了四种阿基米德立体：截角立方体、截角八面体、截角二十面体和截角十二面体。（所有这些现代名称都归功于约翰内斯·开普勒（1619）。）弗朗西斯卡似乎是这六种立体的独立重新发现者。此外，他描述它们性质的方式清楚表明，他实际上发明了现代数学意义上的截角概念。

On perspective for painting是第一部处理透视数学的论著，透视是一种在绘画或雕塑浮雕等二维作品中产生第三维外观的技术。弗朗西斯卡决心表明，这种技术牢固地建立在他那个时代所理解的视觉科学之上。因此，他从一系列数学定理开始，其中一些取自欧几里得的光学著作（可能是通过中世纪资料），但也有一些是弗朗西斯卡自己的原创。其中一些定理具有独立的数学意义，但总体而言，这部著作被构想为一本教画家用透视法绘画的手册，而详细的绘画说明因其重复性而令人麻木。书中有许多图表和插图，但不幸的是，已知手稿中没有一部有弗朗西斯卡本人实际绘制的插图。

在文艺复兴时期，弗朗西斯卡的数学工作没有一部以他自己的名义出版，但它似乎以手稿形式相当广泛地流传，并通过被纳入他人的著作而产生影响。弗朗西斯卡的代数大部分出现在卢卡·帕西奥利的Summa（1494）中，他关于阿基米德立体的工作大部分出现在卢卡·帕西奥利的De divina proportione （1509）中，而弗朗西斯卡透视论著中较简单的部分则被纳入几乎所有后来面向画家的透视论著中。

Piero della Francesca, who came from a family of fairly prosperous merchants, is recognised as one of the most important painters of the Renaissance. Although today he is always known as Piero della Francesca, he would have been known in his lifetime as Piero di Benedetto de' Franceschi. His father, Benedetto de' Franceschi, was a leather and wool merchant, while his mother Romana was the daughter of the wool merchant, Renzo di Carlo da Monterchi, from the nearby village of Monterchi. It is thought that Benedetto and Romana married around 1416 for, although no marriage certificate has been found, they exchanged the dowry in that year. However, Benedetto inherited some money and a share of a house in 1410 and there is some evidence to suggest that he married shortly after this, but if this is true it is hard to believe that the dowry would only be exchanged six years later. As these arguments illustrate, there is no definite evidence when Piero was born, sources giving dates between 1406 and 1422. However, James Beck makes a good case in [20] for June 1420 based in part on Piero being the third of his parents' sons. We know little of Piero's early years but, given the wealth and status of his father (who owned agricultural land, vineyards, urban properties and a fine family home), it is clear that he would have attended school in San Sepolcro.

Details of Piero's schooling remain conjecture, but we can deduce something of its nature from the skills he displayed as an adult. He wrote his mathematics books in Italian rather than Latin, suggesting that he probably did not learn Latin at school. Also it is clear that Piero's Italian writing lacked style, and was rather simple and elementary. This suggests that Piero did not spend that long at school and it confirms his lack of Latin since those who could write fluent Latin developed a style when writing Italian that reflected Latin structure. However, as an adult he had read Euclid's Elements and had some knowledge of the works of Archimedes and, since no Italian versions were available in Piero's day, he must have learnt enough Latin in later life to read these works. Paul Grendler writes [32]:-

His father's occupation and social position probably determined that Piero della Francesca attended an abbaco and vernacular school, perhaps only for a few years. The boy Piero probably received the same schooling as did thousands of other Italians of his background, but it was a surprisingly rich and diverse education. The adult Piero may have learned enough Latin to enable him to read Latin mathematical texts with their specialised vocabulary. The key seems to be that Piero continued to learn throughout his life. And his learning worked for him. One has the impression that Piero retained everything - the saints' legends read in school, the mathematics of abbaco school, his creative explorations into ancient geometry - and that all his learning enriched his painting.

Our knowledge of Piero's life in San Sepolcro has been greatly increased by the research of James Banker (see [3] and [19]). Documents Banker has found show that Piero worked as the assistant of Antonio d'Anghiari in the 1430s. Antonio came from the village of Anghiari, about 7 km from Borgo San Sepolcro, but moved to Borgo San Sepolcro where he completed his first commission and was paid on 27 May 1430. Piero's father Benedetto was witness to the payment. In June of the following year Benedetto received payment for Piero's work "painting the poles of the candles". Benedetto received further payments for work carried out by Piero over the following years during which he was clearly working as an assistant to Antonio. For example one document dated 29 December 1432 states (see [19]):-

Master Antonio di Giovanni painter from Anghiari, inhabitant of the land of Borgo, was confessed and content to be the true debtor of Benedetto di Piero di Benedetto from aforesaid Borgo in the amount of fifty-six florins ... for the salary and debt of Pietro, painter, son of said Benedetto and the wages of the said Pietro for the labour lent to the said Master Antonio from the first of the month of the most recent June and also from the loan to him by the said Benedetto made freely on successive occasions and down to today for the exigencies of said master Antonio for the furnishings of the pictures of the altar-piece of the major altar of the church of S Francesco.

The fact that Piero is described as a "painter" in this document indicates that Piero was not an apprentice by this time but rather an assistant. By the dating we have given above, he was only twelve years old at the time, certainly the right age to be an apprentice but that he would be an assistant by this age is surprising. The real problem is that, fascinating as all the evidence about Piero's youth is, whatever dates are proposed, there always seems to remain some difficulties. The documents show that Piero's father continued to receive payments for work undertaken by Piero, working with Antonio d'Anghiari, in 1435, 1436, and 1437, in particular for painting banners for the Church. A document dated 8 January 1438, again signed by Benedetto, ends Piero's association with Antonio acknowledging that payment has been received for Piero's work on three separate churches and chapels in and around Borgo San Sepolcro. Banker, in [3], gives an overview of the influence of these years on the young Piero:-

The artisan culture of San Sepolcro in the 1430s profoundly informed Piero's sensibility. His origins in this culture endowed him with several strengths and habits of mind that sustained him throughout his life, overcoming the socially ambiguous nature of his family as it was perched between artisan and merchant status. His labour in the artisan culture remained formative in Piero's life and art, not withstanding his great later achievements in painting, mathematics, and geometry that required access to and participation in an elite culture. His formative years as an artisan moulded the understructure of his mind: his linguistic practices, artisan methods, and representation of artisan achievements.

Piero left Borgo San Sepolcro and went to Florence where, in 1439, he was assisting Domenico Veneziano in painting the chapel of Santa Egidio, in Santa Maria Novella. Paul Watson writes [2]:-

In Florence he probably studied the statuary of Donatello and Luca Della Robbia, the buildings of Filippo Brunelleschi, and the paintings of Masaccio and Fra Angelico, and he might have read a theoretical treatise on painting by the humanist and architect Leone Battista Alberti. Undoubtedly, he would have been directed to these luminaries by Domenico Veneziano, whose own works demonstrate a Renaissance emphasis on colour and light as elements of pictorial construction. It was this contact with the early Renaissance art of Florence that provided the foundation of Piero's own style.

It is not known whether Piero assisted Domenico for the whole work but we know he received payment for his contribution to painting the first mural on 12 September 1439. In June 1441 Domenico began work on a second fresco in the chapel and by this time Piero is listed as a member of the town council in Borgo San Sepolcro. It has been pointed out that he may not actually have been living in Borgo at this time and might well have continued to assist Domenico. However, it appears that he painted the Baptism in Borgo San Sepolcro around 1441 in the cathedral church of St John the Evangelist (although some art historians suggest it was painted 20 years later). Ginzburg in [9] discusses in depth the various arguments used in dating Piero's paintings. In 1445 the Compagnia della Misericordia, a guild whose aim was to help the poor and unfortunate, commissioned him to paint an altarpiece in their church at Borgo San Sepolcro. The commission stated that the work was to be completed within three years, but it was not finished until 1462, seventeen years after Piero began. The image painted by Piero depicts the Virgin protecting members of the guild under her cloak. Piero only worked intermittently on the altarpiece, undertaking many other commissions during these years.

You can see the San Sepolcro alterpiece at THIS LINK.

Also in Borgo San Sepolcro around this time he painted the Baptism of Christ, which uses geometry to great effect, for example in placing the dove at the centre of the circle forming the top of the painting. The hills around Borgo San Sepolcro and the town itself appear in the background of the scene.

You can see the Baptism of Christ at THIS LINK.

After undertaking work in Pesaro and Ancona, around 1450 he was invited to Ferrara by Borso d'Este where he [13]:-

... decorated many of the palace rooms. These, however, were afterwards destroyed by Duke Ercole the elder when he renovated the palace.

Also while in Ferrara he painted frescos in the Sant'Andrea Church of the Augustinians in Sant'Agostino. He was in Rimini in 1451 commissioned by Sigismondo Pandolfo Malatesta. He painted a famous fresco of Sigismondo praying before St Sigismund and, as this painting is signed and dated, it is one of the few definite dates in this biography. While in Rimini he met with the mathematician and architect Leone Battista Alberti who was completing his book De re aedificatoria. Alberti, like Piero, was deeply interested in perspective and the two must have exchanged ideas on this topic while in Rimini. One of Piero's most famous works (many would say his most famous) is the Story of the True Cross cycle, begun around 1457. Marilyn Lavin [33] writes about the frescos, commissioned by the richest family in Arezzo (the Bacci family), and painted in the church of San Francesco:-

In the Tuscan town of Arezzo, Piero della Francesca painted the Renaissance fresco cycle most admired by twentieth-century viewers. The frescoes represent the story of the True Cross, and were commissioned for the chancel area of the friary church of San Francesco. The decorations had been started by a popular, but rather workmanlike, painter Bicci di Lorenzo, in the late 1440s; it was probably after Bicci died in 1452 that Piero was called in. He completed the project sometime before December 20, 1466 ...

Not only do the frescos show that Piero is complete master of perspective but they also show a remarkable treatment of light. Piero uses light, combining shadow and shade, to make the figures 3-dimensional. Light is also used in conjunction with perspective to create the illusion of depth in the scenes depicted.

You can see the True Cross frescos at THIS LINK.

Piero was the called to Rome by Pope Pius II soon after he became pope in August 1458. Piero painted many frescos in the Vatican, later destroyed when replaced by works by Raphael, and in the Basilica di Santa Maria Maggiore. After leaving Rome, Piero returned to his home town where he undertook a number of commissions before he went to Loreto to begin decorating the vault of the sacristy with Domenico Veneziano. An outbreak of the plague caused them to leave soon after starting the work. Piero's The Flagellation of Christ, painted in Urbino around 1460, is now considered a masterpiece. Marilyn Lavin writes [10]:-

Although Piero della Francesca's Flagellation is today one of the most famous and highly regarded paintings of the Italian fifteenth century, its prominence in the history of art is relatively new. Small in scale and housed in the out-of-the-way town of Urbino, its very existence is unrecorded before the late eighteenth century. When it made its first appearance in art historical literature some time after, Piero was, surprisingly enough, considered more a technician than an artist. ... Only around the turn of [the 20th ] century did [the Flagellation] begin to be appreciated as a great work of art. With the waning of romantic attitudes and the critical acceptance of Post-Impressionism and Cubism, it was soon drawing praise for the very qualities that had earlier been censured; its restrained expression, purity of colour and, above all, the perfect equilibrium of its organisation.

The scene is dominated by architecture with stunning use of perspective to both add realism and also to draw the eye towards the figure of Christ although he is depicted almost in the background.

You can see the Flagellation of Christ at THIS LINK.

Piero produced other remarkable paintings, including the Resurrection painted in Borgo San Sepolcro around 1563. The landscape around Borgo San Sepolcro and the town itself is represented amazingly accurately behind the figure of Christ preparing to leave the tomb.

You can see the Resurrection of Christ at THIS LINK.

Further works are the Madonna del Parto, a fresco originally in Santa Maria di Momentana in Monterchi painted around the same time, the Nativity painted around 1470-75, the Madonna di Senigallia painted in Urbino around 1474, the Madonna and Child with Saints, a Montefeltro Altarpiece painted just after the Madonna of Senigalli.

You can see the Madonna del Parto at THIS LINK, the Madonna di Senigallia at THIS LINK and the Montefeltro Altarpiece at THIS LINK.

He also painted portraits of Federico da Montefeltro, Duke of Urbino, and Battista Sforza, Duchess of Urbino. Federico was blind in his right eye so Piero shows him in profile so as not to give offence. Despite this the Duke is painted with realism, for example his bent nose and wart are depicted, yet the portrait shows great elegance. The background landscape is remarkably detailed yet subtle.

You can see the Urbino portraits at THIS LINK.

Piero spent the last years of his life in his home town, becoming prior of the Confraternita di San Bartolomeo in 1480. Although some historians dispute this, many accept Vasari's claim in [13] that Piero went blind. For example Beck writes [20]:-

There is solid evidence that he became blind in later life, although almost certainly this occurred after 1487 when he made some hand-written notes blocking out the details of his last will and testament. Obviously, blindness had an enormous impact upon his life and his art, making it impossible to date any work whatsoever after ca. 1488. More challenging for the critic is the possibility that his eyesight deteriorated slowly. Were this documentable, we might understand features of his later work that could reflect a declining visual condition ... For the moment there is no evidence for fixing precisely when the blindness occurred nor have efforts been made to identify a visual crisis in his art.

Finally we note that no evidence exists that Piero ever married. Certainly no wife or children are mentioned in his will which has survived.

We now give a description, contributed by J V Field, of Piero's mathematical works.

In his own time he was also known as a highly competent mathematician. In his Lives of the most famous painters ... [13], Giorgio Vasari (1511-1572) says that Piero showed mathematical ability in his earliest youth and went on to write 'many' mathematical treatises. Of these, three are now known to survive. The titles by which they are known are: Abacus treatise (Trattato d'abaco), Short book on the five regular solids (Libellus de quinque corporibus regularibus) and On perspective for painting (De prospectiva pingendi). Piero almost certainly wrote all three works in the vernacular (his native dialect was Tuscan), and all three are in the style associated with the tradition of 'practical mathematics', that is, they consist largely of series of worked examples, with rather little discursive text.

The Abacus treatise is similar to works used for instructional purposes in 'Abacus schools'. It deals with arithmetic, starting with the use of fractions, and works through series of standard problems, then it turns to algebra, and works through similarly standard problems, then it turns to geometry and works through rather more problems than is standard before (without warning) coming up with some entirely original three-dimensional problems involving two of the 'Archimedean polyhedra' (those now known as the truncated tetrahedron and the cuboctahedron).

Four more Archimedeans appear in the Short book on the five regular solids: the truncated cube, the truncated octahedron, the truncated icosahedron and the truncated dodecahedron. (All these modern names are due to Johannes Kepler (1619).) Piero appears to have been the independent re-discoverer of these six solids. Moreover, the way he describes their properties makes it clear that he has in fact invented the notion of truncation in its modern mathematical sense.

On perspective for painting is the first treatise to deal with the mathematics of perspective, a technique for giving an appearance of the third dimension in two-dimensional works such as paintings or sculptured reliefs. Piero is determined to show that this technique is firmly based on the science of vision (as it was understood in his time). He accordingly starts with a series of mathematical theorems, some taken from the optical work of Euclid (possibly through medieval sources) but some original to Piero himself. Some of these theorems are of independent mathematical interest, but on the whole the work is conceived as a manual for teaching painters to draw in perspective, and the detailed drawing instructions are mind-numbing in their repetitiousness. There are many diagrams and illustrations, but unfortunately none of the known manuscripts has illustrations actually drawn by Piero himself.

None of Piero's mathematical work was published under his own name in the Renaissance, but it seems to have circulated quite widely in manuscript and became influential through its incorporation into the works of others. Much of Piero's algebra appears in Pacioli's Summa (1494), much of his work on the Archimedeans appears in Pacioli's De divina proportione (1509), and the simpler parts of Piero's perspective treatise were incorporated into almost all subsequent treatises on perspective addressed to painters.