Luca, Leonardo, Albrecht and the search for the third dimension.

Many of my more recent readers will not be aware that I lost a good Internet friend last year with the unexpected demise of the history of art blogger, Hasan Niyazi. If you want to know more about my relationship with Hasan then read the elegy I wrote for him when I first heard the news. Hasan was passionate about Renaissance art and his true love was reserved for the painter Raffaello Sanzio da Urbino, better known as Raphael. Today, 6th April is Raphael’s birthday and Hasan’s partner Shazza (Sharon) Bishop has asked Hasan’s friends in the Internet blogging community to write and post something today to celebrate his life, this is my post for Hasan.


I’m not an art historian but there were a couple of themes that Hasan and I had in common, one of these was, for example, the problem of historical dating given differing calendars. Another shared interest was the history of linear perspective, which is of course absolutely central to the history of Renaissance art but was also at the same time an important theme in Renaissance mathematics and optics. I have decided therefore to write a post for Hasan about the Renaissance mathematicus Luca Pacioli who played an important role in the history of linear perspective.


Luca Pacioli artist unknown

Luca Pacioli
artist unknown

Luca Pacioli was born in Sansepolcro in the Duchy of Urbino in 1445.

Duchy of Urbino  Henricus Hondius 1635

Duchy of Urbino
Henricus Hondius 1635

Almost nothing is known of his background or upbringing but it can be assumed that he received at least part of his education in the studio of painter and mathematician Piero della Francesca (1415 – 1492), who like Pacioli was born in Sansepolcro.

Piero della Francesca Self Portrait

Piero della Francesca
Self Portrait

Pacioli and della Francesca were members of what is now known as the Urbino school of mathematics, as was Galileo’s patron Guidobaldo del Monte (1545 – 1607). These three Urbino mathematicians together with, Renaissance polymath, Leone Battista Alberti (1404 – 1472) all played an important role in the history of linear perspective.


Leon Battista Alberti  Artist unknown

Leon Battista Alberti
Artist unknown

Whilst still young Pacioli left Sansepolcro for Venice where he work as a mathematics tutor. Here he wrote his first book, an arithmetic textbook, around 1470. Around this time he left Venice for Rome where he lived for several months in the house of Alberti, from whom he not only learnt mathematics but also gained good connections within the Catholic hierarchy. Alberti was a Papal secretary.

In Rome Pacioli studied theology and became a Franciscan friar. From 1477 Pacioli became a peripatetic mathematics teacher moving around the courts and universities of Northern Italy, writing two more arithmetic textbooks, which like his first one were never published.

Ludovico Sforza became the most powerful man in Milan in 1476, at first as regent for his nephew Gian Galeazzo, and then, after his death in 1494, Duke of Milan.

Ludovico Sforza Zanetto Bugatto

Ludovico Sforza
Zanetto Bugatto

Ludovico was a great patron of the arts and he enticed Leonardo to come and serve him in Milan in 1482. In 1496 Pacioli became Ludivico’s court mathematicus. Leonardo and Pacioli became colleges and close friends stimulating each other over a wide range of topics.


Leonardo Francesco Melzi

Francesco Melzi

Before he went to Milan Pacioli wrote his most famous and influential book his Summa de arithmetica, geometria, proportioni et proportionalità, which he published in Venice in 1494. The Summa, as it is generally known, is a six hundred-page textbook that covers the whole range of practical mathematics, as it was known in the fifteenth-century. Pacioli was not an original mathematician and the Summa is a collection of other peoples work, however it became the most influential mathematics textbook in Europe and remained so for almost the whole of the sixteenth-century. As well as the basics of arithmetic and geometry the Summa contains the first printed accounts of double entry bookkeeping and probability, although Pacioli’s account of determining odds is wrong. From our point of view the most important aspect of the Summa is that it also contains the first extensive printed account of the mathematics of linear perspective.


Pacioli Summa Title Page

Pacioli Summa
Title Page

According to legend linear perspective in painting was first demonstrated by Fillipo Brunelleschi (1377 – 1446) in Florence early in the fifteenth-century. Brunelleschi never published an account of his discovery and this task was taken up by Alberti, who first described the construction of linear perspective in his book De pictura in 1435. Piero della Francesca wrote three mathematical treatises one on arithmetic, one on linear perspective and one on the five regular Euclidian solids. However della Francesca never published his books, which seem to have been written as textbooks for the Court of Urbino where they existed in the court library only in manuscript. Della Francesca treatment of perspective was much more comprehensive than Alberti’s.

During his time in Milan, Pacioli wrote his second major work his Divina proportione, which contains an extensive study of the regular geometrical solids with the illustrations famously drawn by his friend Leonardo.


Leonardo Polyhedra


These two books earned Pacioli a certain amount of notoriety as the Summa contains della Francesca’s book on linear perspective and the Divina proportione his book on the five regular solids both without proper attribution. In his Lives of the Most Excellent Italian Painters, Sculptors, and Architects, from Cimabue to Our Timesthe Italianartist and art historian, Giorgio Vasari (1511 – 1574)


Giorgio Vasari Self Portrait

Giorgio Vasari
Self Portrait

accused Pacioli of having plagiarised della Francesca, a not entirely fair accusation, as Pacioli does acknowledge that the entire contents of his works are taken from other authors. However whether he should have given della Francesca more credit or not Pacioli’s two works laid the foundations for all future mathematical works on linear perspective, which remained an important topic in practical mathematics throughout the sixteenth and seventeenth centuries and even into the eighteenth with many of the leading European mathematicians contributing to the genre.

With the fall of Ludovico in 1499 Pacioli fled Milan together with Leonardo travelling to Florence, by way of Mantua and Venice, where they shared a house. Although both undertook journeys to work in other cities they remained together in Florence until 1506. From 1506 until his death in his hometown in 1517 Pacioli went back to his peripatetic life as a teacher of mathematics. At his death he left behind the unfinished manuscript of a book on recreational mathematics, De viribus quantitatis, which he had compiled together with Leonardo.

Before his death Pacioli possibly played a last bit part in the history of linear perspective. This mathematical technique for providing a third dimensional to two dimensional paintings was discovered and developed by the Renaissance painters of Northern Italy in the fifteenth century, one of the artists who played a very central role in bringing this revolution in fine art to Northern art was Albrecht Dürer, who coincidentally died 6 April 1528, and who undertook two journeys to Northern Italy explicitly to learn the new methods of his Italian colleagues.

Albrecht Dürer Self Portrait

Albrecht Dürer
Self Portrait

On the second of these journey’s in 1506-7, legend has it, that Dürer met a man in Bologna who taught him the secrets of linear perspective.  It has been much speculated as to who this mysterious teacher might have been and one of the favoured candidates is Luca Pacioli but this is highly unlikely. Dürer was however well acquainted with the work of his Italian colleagues including Leonardo and he became friends with and exchanged gifts with Hasan’s favourite painter Raphael.

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Filed under History of Mathematics, History of Optics, Renaissance Science, Uncategorized

5 responses to “Luca, Leonardo, Albrecht and the search for the third dimension.

  1. Pingback: Luca, Leonardo, Albrecht and the search for the...

  2. Thank you for providing us with some irrefutable links between mathematics and art. I will make sure to share this post with my skeptical son who is studying to be an engineer, with great delight. You have provided yet another unique link to Hasan’s many interests. Thanks.

  3. Great post! As an art historian, I am embarrassed to say that I don’t know much about Pacioli or the basis of what he wrote. (Perhaps that’s because I’m not a mathematician!) When the history of linear perspective is discussed in general terms, I think art historians tend to gravitate toward discussing names that are more familiar within our discipline (e.g. Brunelleschi, Alberti, della Francesca). I’ll pay more attention to Luca Pacioli’s name from now on!

  4. Hasan often mentioned you in his e-mails and I know that your relationship was very important to him. I was moved by your eulogy right after his death and I enjoyed your latest post on Luca Pacioli. I am sure that Hasan would have been pleased, since you and he shared so many interests–especially regarding the unity of art and science.

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