14

LEON BATTISTA ALBERTI AND LEONARDO DA VINCI

Leon Battista Alberti (February 14, 1404–April 25, 1472) has been heralded as the “universal man” of the early Renaissance and described as “the prophet of the new grand style in art” inaugurated by Leonardo da Vinci.1 His range of abilities was astounding.

Alberti was born in Genoa, the son of a wealthy Florentine banker, Lorenzo Alberti. His mother, Bianca Fieschl, was a widow from Bologna. When he was very young the family moved to Venice, where his father ran the family bank. A ban (a common political occurrence in those days) on the family was lifted in 1428, leaving the young Alberti free to return to Florence.

He benefited from the finest education available. From 1414 to 1418 he studied classics at the famous school of Gasparino Barzizza in Padua and later attained his master’s in law at the University of Bologna. In 1430 he moved to Rome, where he prepared legal briefs for Pope Eugenius IV and met Nicholas of Cusa who was prime minister. In June 1434, Eugenius IV was forced to leave Rome for Florence because of a disagreement with the Church Council. Alberti joined him and was appointed canon of Santa Maria del Fiore when the cathedral was near completion. In Florence, he was introduced to both Filippo Brunelleschi (1377–1446) and Paolo Toscanelli, who had assisted Brunelleschi with the mathematics for the cathedral dome. Alberti became lifelong friends with both and part of the group of friends and admirers surrounding Toscanelli.2

Before moving to Florence, Alberti had written treatises on the use and disadvantages of the study of letters; two dialogues, Deiphira and Ecatonfilea (love scenes); a thesis, Intercenale; a book about the family, Della famiglia; and a life of Saint Potitus, Vitas Potiti.

After 1434, however, he began producing a range of works in mathematics, astronomy, architecture, and cryptography.3 His biographer Joan Gadol describes Alberti’s influence:

[Most astronomers considered] “the maximum declination of the sun in our days is 24 degrees and 2 minutes, but my teachers [Peurbach] and I have ascertained with instruments that it is 23 degrees and 28 minutes and I have often heard Magister Paolo the Florentine and Battista Alberti say that by diligent observation they found that it did not exceed 23 degrees and 30 minutes, the figure I have decided to register in our table.” 4

This description is significant for several reasons. First, Regiomontanus, disciple of Toscanelli and a very accomplished astronomer, credits Alberti as one, as well. Second, he depicts the astronomers arguing about two minutes of declination, which means they must have had very accurate instruments to determine the altitude of the sun at its meridian passage at noon. Third, it suggests they had solved the declination problem with all that implies. Finally, and most important of all, it tells us that they are working on the obliquity of the ecliptic.

Gadol considered that Alberti’s entirely new knowledge of the universe, which he had gained from Toscanelli, enabled him to develop many of his ideas by using an astrolabe—in architecture, in perspective, even in cryptography.

At least a decade before Alberti’s great works on painting and sculpture, De pictura (1435), which he translated into an Italian version, Della pittura, the following year, and De statua (ca. 1446), Florentine artists had been experimenting with perspective. However, the current consensus seems to be that Brunelleschi, Masaccio, and Donatello were intuitive geniuses who developed the costruzione legittima, a method of determining perspective with the use of pinhole cameras and mirrors, but did not know the mathematics of the costruzione abbreviata developed later by Alberti.

Before considering Alberti’s great works, perhaps one should consider how so many brilliant people appeared on the European stage at the same time. Toscanelli, Regiomontanus, Alberti, Francesco di Giorgio, and Leonardo da Vinci revolutionized European thought—in knowledge of the universe and the solar system, in astronomy, mathematics, physics, architecture, cartography, surveying, town planning, sculpture, painting, even cryptography. How did they all appear in the same small area of northern Italy? Did God wave a magic wand over Tuscany?

Undoubtedly, one reason was money. In the 1430s Venice was the wealthiest city in Europe, followed by Paris and Nuremburg. Venetian wealth spilled into Florence. The Medicis were the richest family in Europe. They made their money from banking, a part of which involved lending out money and charging interest for doing so—usury in the eyes of the Church. To atone for their sins, the Medicis sponsored a whole range of religious works—building and embellishing first chapels and later hospitals and libraries. They engaged the best artists to paint frescoes of the stars and planets. They employed people to search out books and maps and scholars to translate books of the ancients.

There were many scholars to employ. Italy boasted some of the oldest European universities—Bologna was nearly as old as Paris—and there were many of them. Tuscany probably had a higher proportion of postgraduates (to use a modern term) than anywhere else on earth. To those who could not afford a university education, the Church offered a free alternative. The religious orders, first Benedictines, then Cistercians, Franciscans, Dominicans, and Jesuits, offered not only a first-class religious training but a practical one for daily life. Benedictines not only prayed but ran highly successful and profitable farms pioneering research into animal husbandry, crop improvements, honey production, fish and poultry breeding, even genetic engineering. Benedictines in time became bankers to small farms, so improving agriculture. As one religious order followed another, the quality of education continuously improved, culminating in the superb education that the Jesuits brought to peoples of the New World. Benedictines, Cistercians, Franciscans, and Dominicans all had their principal bases in Burgundy and northern Italy.

This was the loam in which the seeds of Chinese ideas and inventions were propagated. We should not underestimate the pollination of ideas that resulted from the continuous intellectual interchange among these geniuses. Toscanelli and Regiomontanus collaborated on world maps; determining the declination of the sun; changes in the obliquity of the ecliptic; comets; spherical trigonometry; torquetums; and astronomical instruments. Alberti exchanged ideas on astronomy, mathematics, and trigonometry with Regiomontanus and Toscanelli, on locks and canals with Francesco di Giorgio, and on raising sunken ships with Francesco and Taccola. Nicholas of Cusa discussed astronomy with Toscanelli, Alberti, and Regiomontanus. Members of the group dedicated their books to one another.

They prayed at the same cathedral, Santa Maria del Fiore, ate at the mensa in Florence’s Palazzo Vecchio, and dined with the Medicis. Nicholas of Cusa’s home in Rome was the gathering place for men of influence and science—including Bruni, Alberti, Regiomontanus, and Toscanelli. There were several occasions at which Alberti and Nicholas of Cusa met over the years; during the Council of Florence—Alberti was at Ferrara with Eugenius IV, as was Nicholas of Cusa. The historian Giovanni Santinello draws a number of parallels between Alberti’s writings on beauty, art, and perspective and Nicholas of Cusa’s.5

De Pictura

Alberti’s masterpiece, De pictura, is generally accepted by art historians of the Renaissance as the most important book on painting ever written. Leonardo da Vinci repeatedly refers to it, sometimes quoting it word for word. It seems appropriate to analyze how Alberti came to write the book, not least because of its impact on the development of Leonardo’s genius and the book’s influence on the future course of the Renaissance. In my opinion, Alberti would have realized from his and Toscanelli’s study of the Shoushiastronomical calendar that the earth traveled in an ellipse around the sun while rotating on its axis and that the planets also rotated round the sun in ellipses, and this would have been a seismic shock. That Alberti knew how the solar system worked is evidenced by his painting in the San Lorenzo Baptistry of the heavens of the sun, moon, and stars on July 6, 1439, at noon. Not only did this new knowledge overturn the authority of Ptolomy and Aristotle, but it knocked over the entire hierarchical order of the universe and replaced it with a conception of a harmonious and, above all, mathematical world order. Mathematics brought systematic order into the plan of the heavens and revealed a connection between astronomical data and physical research—quite literally a shattering revelation. If the workings and motions of the heavens could be explained in a mathematical rather than a religious context, then surely architecture, engineering, painting, even cryptography could also be explained by mathematics—hence De pictura,which gives the first rational and systematic exposition of the rules for perspective. To quote Joan Gadol again:

[Alberti’s] major accomplishment of this Florentine period (1434–1436) was theoretical. By bringing his humanistic and mathematical learning to bear upon the practice of painting and sculpture, Alberti fathered the new, mathematically inspired techniques of these arts and developed the aesthetic implications of this renascent artistic reliance upon geometry.

The sculptural counterpart to the theory of perspective appeared somewhat later in Della Statua. Treating the statue as another kind of geometric imitation of nature, he devised an equally ingenious method of mensuration for the sculptor and worked out the first Renaissance canon of proportions.6

Alberti, as Joan Gadol so succinctly writes, went beyond the bounds of astronomy to determine its relation with mathematics and then mathematics to develop painting and architecture, cartography and surveying—even engineering design.

Leonardo da Vinci made great use of Della pittura [the Italian translation of De pictura] in his own treatise on painting, using the same terms, and ideas, even some of Alberti’s phrases. For example, Leonardo says the perspective picture is to look as if it were drawn “on a glass through which the objects are seen” (Gadol), which was a term used by Alberti; and then again when defining painter’s perspective as “a sort of visual geometry.” Leonardo follows Alberti’s theory and principles in every detail: “The sciences have no certainty except when one applies one of the mathematical sciences”…and again, to quote Leonardo, “painting must be founded on sound theory and to this perspective is the guide and gateway.” Jakob Burckhardt portrayed Alberti in The Civilization of the Renaissance in Italy as a truly universal genius and considered Leonardo da Vinci was to Alberti as finisher to the beginner.

Leonardo’s use of perspective to create sublime paintings and architecture, and to illustrate his mechanical drawings, is his legacy to mankind.

Alberti’s intellectual achievements were truly awesome. As Grayson art historian of medieval Italy, so clearly explains, he introduced the concept of the picture plane as a window on which the observer can see the scene lying beyond it and thus laid the foundations of linear perspective. Alberti then codified the basic geometry so that linear perspective became mathematically coherent.

He wrote a ten-volume architectural treatise covering all aspects of Renaissance architecture—town planning, building designs, water and sewage treatment, public spaces, methods of construction. De re aedificatoria (On the art of building) became a standard reference book that spread Renaissance building techniques throughout Italy.

He drew the stars on the ceiling of the San Lorenzo Baptistry as they were seen on July 6, 1439, probably assisted by his friend Toscanelli. He collaborated with Toscanelli and Regiomontanus in helping determine Regiomontanus’s declination of the sun, the obliquity of the ecliptic, and the change in its obliquity. He composed the first European treatise on cryptography, “De componendis cifris.”

Could one man really cover such a vast array of subject matter ranging from the invention of polyalphabetic substitutes and the cryptic code to new mathematical models for treating perspective?

Alberti was, like Regiomontanus, Toscanelli, Di Giorgio, and Taccola, remarkably reticent in crediting others for the source of his inspiration. Of obvious interest to me was any possible link between Alberti and Zheng He’s delegation’s visit to Florence in 1434, not least because Alberti as notary to Pope Eugenius IV would have attended meetings between the pope and the Chinese. Moreover, Alberti’s writings before 1434 were on domestic themes—his explosion of astronomical, mathematical, and cartographic works all came after 1434.

I started my search by looking into Alberti’s work on cryptography, in particular Chinese cryptography of the early fifteenth century. Zheng He would have been likely to have used cryptography for transmitting intelligence reports to the emperor and to his admirals and captains. I could find no translated works.

Then, when researching Regiomontanus’s life and works, as recounted in the previous chapter, I came upon the curious fact that Regiomontanus had mastered the Chinese remainder theorem, unique to China at the time. His source for this (as far as I know, the unique source) was the Shu-shu Chiu-chang of Ch’in Chiu-shao, published in 1247, which contains a detailed explanation of the Ta-Yen rule.

The Shu-shu Chiu-chang is a massive book, the Chinese equivalent to Alberti’s De re aedificatoria, but published two centuries earlier. With feverish excitement I hurried off to the British Library and read Needham’s description of this work—a bombshell; as far as I could see, the genesis of Alberti’s work in relation to perspective contained in Ludi matematici is in the Chinese book. It is clear to me that both Alberti and his friend Regiomontanus may have had access to this book, which contained not only rules for perspective and the Chinese remainder theorem (for cryptographic analysis) but all aspects of town planning. On our 1434 website are pictures taken from Alberti’s Ludi matematici and Ch’in Chiu-shao’s book side by side, describing ways of measuring height, depth, distance, and weight by mathematical and geometric means.

Let us start with the basic stages of Alberti’s work on perspective, the building blocks for his works De statua and De pictura.

As a first stage: Alberti draws a large rectangle like a window frame, through which he can see the subject he wishes to paint or create. For the second stage, he selects the largest human he wishes to paint seen through the picture frame. The height of this person is divided into three equal parts, which form the basic unit of measurement, called a braccia.

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The Chinese, and later the Sienese engineers, used very similar methods for constructing towers and measuring their heights.

In the third stage, he makes the center point of the picture frame, which should be no higher than three braccia above the ground.

In the fourth stage, he divides the base line into braccia.

In the fifth stage: He draws straight lines from this center point to each of the braccia on the base line.

For illustrations of the above, please visit our website.

Now to compare where Alberti has gotten with the Chinese method illustrated in the Shu-shu Chiu-chang.

The first comparison is illustrated by the method for finding the height of a tower (as explained in Alberti’s Ludi matematici, ca. 1450):

Stick an arrow or a rod into the ground (c-d) so as to form a straight perpendicular line along which to take sightings to the tower (a-b). Mark the rod with wax where the line of sight to the top of the tower crosses it (f). The triangle formed by the arrow, ground and eye is the geometric counterpart of the triangle formed by the tower, ground and eye (abc) hence it can be used to find the height of the tower (ab). ab divided by bc equals fc divided by ce.

This is how Alberti “discovered” the rules of projection, which since then have formed the basis of perspective for sculptors and painters.

However, Alberti had not made an original discovery. The same explanation from Liu Hui in the third century is illustrated in the Shu-shu Chiu-chang. In this book the calculations are called “the method of double differences,” that is, the properties of right-angled triangles. There are illustrations depicting methods for calculating the heights of islands seen from the sea; the height of a tree on a hill; the size of a distant walled city; the depth of a ravine; the height of a tower; the breadth of a river mouth; the depth of a transparent pool. This trigonometry was invented by Euclid, and Alberti could have obtained his ideas from him as well as from the Chinese—he never acknowledged his sources.

However, the links between Chinese sources and Alberti go much further than trigonometry. Alberti used the same instruments as Toscanelli and adopted similar mathematics. Alberti’s method of perspective was brilliant. He realized that perspective was determined not only by the size of the object viewed and its distance from the beholder but also by the height of the observer relative to the viewed object and the angle from which the viewer was looking at the object. In short, each figure in a crowd when the crowd is viewed in depth would need a different rule of perspective.

By now I was beginning to feel uncomfortable about the amount of knowledge that it seemed Florentine mathematicians had copied from the Chinese—Taccola, Francesco di Giorgio, and Alberti from the Shu-shu Chiu-chang for mathematics, surveying, perspective cartography, and cryptography; Regiomontanus from Guo Shoujing’s work on spherical trigonometry, Toscanelli and Nicholas of Cusa for Guo Shoujing’s work on astronomy. I could explain one or two Chinese manuals coming into the hands of Venetians and Florentines—but this many, in so many different fields? It seemed too much of a coincidence—too good to be true! On the other hand, there was Toscanelli’s evidence about the transfer of knowledge that was unquestionably true—evidenced by maps, which do not lie.

It seemed sensible at this stage to see the original books in China, not only Needham’s accounts. Could these have been taken out of context in some way? Perhaps there were also many Chinese inventions that had never been copied by Europeans. Perhaps those that were was just a huge coincidence. Ian Hudson, who has been in charge of our research team and website for five years, volunteered to go to China to inspect the original books that I believed Europeans had copied—by visiting libraries in mainland China and Hong Kong.

He found there were, as far as we can see, no anomalies—first it seemed everything that Taccola, di Giorgio, Regiomontanus, Alberti, and Leonardo da Vinci had “invented” was already there in Chinese books, notably ephemeris tables, maps, mathematical treatises, and the production of civil and military machines. So how was the transfer effected? I had many sleepless nights of worry before the penny dropped—all of these books were reproduced in parts of the Yongle Dadian, which Zheng He would have carried. Zheng He’s representatives would have undoubtedly told the pope and Toscanelli about the Yongle Dadian—as evidenced by Toscanelli’s comment, China was indeed ruled by “astronomers and mathematicians of great learning.”

Alberti also applied his mathematical ability to surveying, and is cited by many as being the father of modern surveying. Here again, he makes a complete break with the past. His map of Rome bears almost no relation to Ptolemy’s system of mapping. He rejects Ptolemy’s rectangular coordinates and uses the astrolabe to find the relative positions of points on the ground, just as a navigator would—he takes sightings from more than one vantage point. As Joan Gadol says, “He first set forth these ideas in Descriptio urbis Romae, the brief Latin treatise written in the 1440s.” Gadol believes Alberti’s Descriptio urbis Romae and Ludi matematici were among the earliest works in surveying land areas by sightings and mapping by scale pictures. He believes Regiomontanus, Schöner, and Waldseemüller followed Alberti’s work.

Leonardo’s map of Pisa and the mouth of the Arno is thought to be the first modern map to show contours of land by using different shades of color. Leonardo followed Alberti in the principles used in surveying, as he did in rules of perspective.

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