Luca Pacioli divine proportion 1509. Luca Pacioli “On Divine Proportion”

The search for our origin is the juice of that sweet fruit that brings so much satisfaction to the mind of philosophers.

Luca Pacioli (1445–1517)
Only a few great painters in human history were also gifted mathematicians. However, the expression “Renaissance Man” means in our vocabulary a person who embodied the Renaissance ideal of the broadest outlook and education. So the three most famous artists of the Renaissance - the Italians Piero della Francesca (c. 1412-1492) and Leonardo da Vinci and the German Albrecht Dürer, also made very significant contributions to mathematics. Perhaps it is not surprising that the mathematical research of all three was related to the golden ratio. The most active mathematician of this brilliant trio of virtuosos was Piero della Francesca. The writings of Antonio Maria Graziani, who was related to Piero's great-grandsons and purchased the artist's house, indicate that Piero was born in 1412 at Borgo Sansepolcro in central Italy. His father Benedetto was a successful tanner and shoemaker. Almost nothing more is known about Piero's childhood, but documents have recently been discovered that make it clear that before 1431 he spent some time as an apprentice to the artist Antonio D'Angiari, whose works have not reached us. Towards the end of the 1430s, Piero moved to Florence, where he began to collaborate with the artist Domenico Veneziano. In Florence, the young artist became acquainted with the works of early Renaissance artists - including Fra Angelico and Masaccio - and with the sculptures of Donatello. He was particularly impressed by the majestic serenity of Fra Angelico's works. religious themes, and his own style reflects this influence in everything related to chiaroscuro and color. In subsequent years, Piero worked tirelessly in a variety of cities, including Rimini, Arezzo and Rome. The figures by Pierrot were either distinguished by architectural severity and monumentality, as in “The Flagellation of Christ” (now the painting is kept in the National Gallery of Marche in Urbino; ​​Fig. 45), or they seemed to be a natural continuation of the background, as in “The Baptism” (currently located in the National Gallery in London; Fig. 46). The first art historian Giorgio Vasari (1511–1574) in his “Lives of the most famous painters, sculptors and architects” writes that Piero with early youth showed remarkable mathematical abilities, and is credited with writing “numerous” mathematical treatises. Some of them were created in old age, when the artist, due to infirmity, could no longer paint. In a dedicatory letter to Duke Guidobaldo of Urbino, Piero mentions one of his books, written “so that his mind would not become rigid from disuse.” Three works of Pierrot on mathematics have reached us: “ De Prospectiva pingendi"("On perspective in painting"), " Libellus de Quinque Corporibus Regularibus"("Book about five regular polyhedra") and " Trattato d ’Abaco"("Treatise on Accounts").

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The treatise On Perspective (mid 1470 - 1480) contains many references to Euclid's Elements and Optics, as Piero della Francesca decided to prove that the technique of conveying perspective in painting was based entirely on the mathematical and physical properties of visual perspective. In the paintings of the artist himself, perspective is a spacious container that is in full accordance with the geometric properties of the figures contained in it. In fact, for Pierrot, painting itself primarily came down to “showing bodies of reduced or enlarged size on a plane.” This approach is clearly visible in the example of “The Flagellation” (Fig. 45 and 47): this is one of the few paintings of the Renaissance where the perspective is built and worked out very carefully. As he writes contemporary artist David Hockney in his book The Secret Knowledge ( David Hockney. Secret Knowledge, 2001), Pierrot paints figures “as he believes they should be, and not as he sees them.”

On the occasion of the 500th anniversary of Piero's death, scientists Laura Geatti from the University of Rome and Luciano Fortunati from the National Research Council in Pisa carried out a detailed analysis of the Flagellation using a computer. They digitized the entire picture, determined the coordinates of all points, measured all distances and compiled full analysis perspectives based on algebraic calculations. This allowed them to accurately determine the location of the “vanishing point” where all the lines leading to the horizon from the viewer intersect (Fig. 47), thanks to which Pierrot was able to achieve the “depth” that makes such a strong impression.

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Pierrot's book on perspective, distinguished by its clarity of presentation, became the standard guide for artists attempting to draw plane figures and geometric solids, and those sections of it that are not overloaded with mathematics (and are more understandable) were included in most subsequent works on perspective. Vasari claims that Piero received a solid mathematical education and therefore "better than any other geometer understood how best to draw circles in regular bodies, and it was he who shed light on these questions" ( hereinafter per. A. Gabrichevsky and A. Benediktov). An example of how carefully Pierrot developed the method of drawing a regular pentagon in perspective can be seen in Fig. 48.

In both his Treatise on Abacus and his Book of the Five Regular Polyhedra, Pierrot poses (and solves) many problems involving the pentagon and the five Platonic solids. It calculates side and diagonal lengths, areas and volumes. Many solutions are also based on the golden ratio, and some of Pierrot’s techniques testify to his ingenuity and originality of thinking.

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Piero, like his predecessor Fibonacci, wrote the Treatise on Abacus mainly to provide his businessman contemporaries with arithmetic “recipes” and geometric rules. In the world of commerce at that time there was no unified system of weights and measures, or even agreements on the sizes and shapes of containers, so it was impossible to do without the ability to calculate the volume of figures. However, Pierrot's mathematical curiosity took him far beyond topics that were reduced to everyday needs. Therefore, in his books we also find “useless” problems - for example, calculating the length of an edge of an octahedron inscribed in a cube, or the diameter of five small circles inscribed in a circle of a larger diameter (Fig. 49). To solve the last problem, a regular pentagon is used, and therefore the golden ratio.

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Pierrot's algebraic research was mainly included in a book published by Luca Pacioli (1445–1517) entitled “ Summa de arithmetica, geometria, proportioni et proportionalita"("Code of knowledge in arithmetic, geometry, proportions and proportionality"). Piero's works on polyhedra, written in Latin, were translated into Italian by the same Luca Pacioli - and again included (or, to put it less delicately, simply stolen) in his famous book on the golden ratio entitled “On Divine Proportion” (“On Divine Proportion” (“On Divine Proportion”) Divina Proportione »).

Who was he, this contradictory mathematician Luca Pacioli? The greatest plagiarist in the history of mathematics - or still the great popularizer of mathematical science?

The unsung hero of the Renaissance?

Luca Pacioli was born in 1445 in the same Tuscan town of Borgo Sansepolcro where Piero della Francesca was born and kept a workshop. Moreover, Luca received his primary education in Pierrot’s workshop. However, unlike other students who showed aptitude for painting - some of them, for example, Pietro Perugino, were destined to become great painters - Luca turned out to be more inclined towards mathematics. Piero and Pacioli maintained friendly relations in the future: evidence of this is that Piero depicted Pacioli in the form of St. Peter of Verona (Peter the Martyr) on the “Altar of Montefeltro”. While still a relatively young man, Pacioli moved to Venice and became a mentor there to the three sons of a wealthy merchant. In Venice, he continued his mathematical education under the guidance of mathematician Domenico Bragadino and wrote the first book on arithmetic.

In 1470, Pacioli studied theology and became a Franciscan monk. Since then, it has become customary to call him Fra Luca Pacioli. In subsequent years, he traveled widely, teaching mathematics at universities in Perugia, Zadar, Naples and Rome. At that time, Pacioli probably also taught for some time Guidobaldo Montefeltro, who in 1482 was to become Duke of Urbino. Perhaps the best portrait of a mathematician is a painting by Jacopo de Barbari (1440–1515), depicting Luca Pacioli giving a geometry lesson (Fig. 50, the painting is in the Capodimonte Museum in Naples). On the right on Pacioli's book " Summa» rests one of the Platonic solids - the dodecahedron. Pacioli himself, in a Franciscan cassock (also similar to a regular polyhedron, if you look closely), copies a drawing from Book XIII of Euclid’s Elements. A transparent polyhedron called a rhombicuboctahedron (one of the Archimedean solids, a polyhedron with 26 faces, 18 of which are squares and 8 of which are equilateral triangles), hanging in the air and half filled with water, symbolizes the purity and eternity of mathematics. The artist managed to convey with amazing skill the refraction and reflection of light in a glass polyhedron. The identity of Pacioli's student depicted in this painting has been the subject of controversy. In particular, it is assumed that this young man is Duke Guidobaldo himself. English mathematician Nick MacKinnon put forward an interesting hypothesis in 1993. In his article “Portrait of Fra Luca Pacioli”, published in “ Mathematical Gazette” and based on very solid research, MacKinnon concludes that this is a portrait of the great German painter Albrecht Dürer, who was very interested in both geometry and perspective (and we will return to his relationship with Pacioli a little later). Indeed, the student's face bears a striking resemblance to Dürer's self-portrait.

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In 1489, Pacioli returned to Borgo Sansepolcro, having received some privileges from the Pope himself, but was greeted with jealous hostility by the local religious establishment. For about two years he was even forbidden to teach. In 1494, Pacioli went to Venice to print his book " Summa", which was dedicated to Duke Guidobaldo. " Summa"by nature and scope (about 600 pages) is a truly encyclopedic work, where Pacioli brought together everything that was known at that time in the field of arithmetic, algebra, geometry and trigonometry. In his book, Pacioli does not hesitate to borrow problems about the icosahedron and dodecahedron from Piero della Francesca’s “Treatise” and other problems in geometry, as well as in algebra, from the works of Fibonacci and other scientists (however, he usually expresses gratitude to the author, as expected). Pacioli admits that his main source is Fibonacci, and says that where there are no references to someone else, the works belong to Leonardo of Pisa. Interesting section " Summa» – double entry accounting system, a method that allows you to track where money came from and where it went. This system was not invented by Pacioli himself, he only brought together the techniques of the Venetian merchants of the Renaissance, but it is believed that this is the first book on accounting in the history of mankind. And so it was that Pacioli’s desire to “enable the businessman to immediately receive information about his assets and financial obligations” earned him the nickname “Father of Accounting,” and in 1994, accountants around the world celebrated the quincentenary of “ Summa"in Sansepolcro, as this city is now called.

In 1480, the place of the Duke of Milan was actually taken by Ludovico Sforza. In reality he was only regent to the real duke, who was then only seven years old; this event brought an end to the period of political intrigue and murder. Ludovico decided to decorate his court with artists and scientists and in 1482 invited Leonardo da Vinci to the “college of ducal engineers.” Leonardo was very interested in geometry, in particular its practical application in mechanics. According to him, “Mechanics is a paradise among mathematical sciences, since it is it that gives birth to the fruits of mathematics.” And subsequently, in 1496, it was Leonardo who most likely ensured that the Duke invited Pacioli to the court as a mathematics teacher. Leonardo undoubtedly studied geometry from Pacioli, and instilled in him a love of painting.

While in Milan, Pacioli completed work on a three-volume treatise, On Divine Proportion, which was published in Venice in 1509. The first volume, " Compendio de Divina Proportione” (“Compendium on Divine Proportion”), contains a detailed summary of all the qualities of the golden ratio (which Pacioli calls “divine proportion”) and a study of the Platonic solids and other polyhedra. On the first page of “On the Divine Proportion,” Pacioli somewhat pompously declares that this is “a work necessary for all inquisitive, clear human minds, in which anyone who loves to study philosophy, perspective, painting, sculpture, architecture, music and other mathematical disciplines will find a very subtle, elegant and charming teaching and will receive pleasure from a variety of questions affecting all the secret sciences.”

Pacioli dedicated the first volume of his treatise “On the Divine Proportion” to Ludovico Sforza, and in the fifth chapter he lists five reasons why, in his opinion, the golden ratio should be called nothing other than the divine proportion.

2. Pacioli sees a similarity between the fact that the definition of the golden ratio includes exactly three lengths (AC, CB and AB in Fig. 24), and the existence of the Holy Trinity - Father, Son and Holy Spirit.

3. For Pacioli, the incomprehensibility of God and the fact that the golden ratio is an irrational number are equivalent. Here is how he writes: “Just as the Lord cannot be properly defined and cannot be comprehended through words, so our proportion cannot be conveyed in comprehensible numbers and expressed through any rational quantity, it will forever remain a mystery, hidden from everyone, and mathematicians call it irrational.”

4. Pacioli compares the omnipresence and immutability of God with self-similarity, which is associated with the golden ratio: its value is always unchanged and does not depend on the length of the segment, which is divided in the appropriate proportion, or with the size of a regular pentagon, in which the ratios of lengths are calculated.

5. The fifth reason shows that Pacioli held even more Platonic views on being than Plato himself. Pacioli argues that just as God gave life to the universe through the quintessence reflected in the dodecahedron, so the golden ratio gave life to the dodecahedron, since it is impossible to build a dodecahedron without the golden ratio. Pacioli adds that it is impossible to compare the other Platonic solids (symbols of water, earth, fire and air) with each other without relying on the golden ratio.
In the book itself, Pacioli constantly rants about the qualities of the golden ratio. He sequentially analyzes 13 so-called “effects” of “divine proportion” and attributes to each of these “effects” epithets such as “inherent”, “unique”, “wonderful”, “supreme”, etc. For example, that “effect” that golden rectangles can be inscribed in the icosahedron (Fig. 22), he calls “incomprehensible.” He stops at 13 “effects,” concluding that “this list must be completed for the salvation of the soul,” since it was 13 people who sat at the table during the Last Supper.

There is no doubt that Pacioli was very interested in painting, and the purpose of creating the treatise “On Divine Proportion” was partly to hone the mathematical basis of the fine arts. On the very first page of the book, Pacioli expresses his desire to reveal to artists the “secret” of harmonic forms through the golden ratio. To ensure the attractiveness of his work, Pacioli enlisted the services of the best illustrator any writer could dream of: Leonardo da Vinci himself provided the book with 60 drawings of polyhedra, both in the form of “skeletons” (Fig. 51) and in the form of solid bodies (Fig. 52). There was no question of gratitude - Pacioli wrote about Leonardo and his contribution to the book like this: “The best painter and master of perspective, the best architect, musician, a man endowed with all possible virtues - Leonardo da Vinci, who invented and executed a series of schematic images of regular geometric bodies " The text itself, admittedly, does not achieve its stated high goals. Although the book begins with sensational tirades, what follows is a fairly ordinary set of mathematical formulas, carelessly diluted with philosophical definitions.

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The second book of the treatise “On Divine Proportion” is devoted to the influence of the golden ratio on architecture and its manifestations in the structure of the human body. Pacioli's treatise is largely based on the work of the Roman architect Marcus Vitruvius Pollio (c. 70–25 BC). Vitruvius wrote:
Center point human body- this is, naturally, the navel. After all, if a person lies face down on his back and spreads out his arms and legs, and a compass is placed on his navel, then his fingers and toes will touch the circumscribed circle. And just as the human body fits into a circle, so you can get a square out of it. After all, if we measure the distance from the soles to the top of the head, and then apply this measure to the outstretched arms, it will turn out that the width of the figure is exactly equal to the height, as in the case of flat surfaces shaped like a perfect square.
Renaissance scholars considered this passage further proof of the connection between the natural and geometric basis of beauty, and this led to the creation of the concept of the Vitruvian Man, which Leonardo so beautifully depicted (Fig. 53, the drawing is currently kept in the Accademia Gallery in Venice). Similarly, Pacioli's book begins with a discussion of the proportions of the human body, "since in the human body one can find proportions of all kinds, revealed by the will of the Almighty through the hidden secrets of nature."

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In the literature you can often find statements that Pacioli supposedly believed that the golden ratio determines the proportions of all works of art, but in reality this is not at all the case. When talking about proportion and external structure, Pacioli mainly refers to the Vitruvian system based on simple (rational) fractions. Writer Roger Hertz-Fischler has traced the origins of the common misconception that the golden ratio was Pacioli's canon of proportions: it goes back to a false statement made in the 1799 edition of the History of Mathematics by French mathematicians Jean Etienne Montucle and Jerome de Lalande ( Jean Etienne Montucla, Jerome de Lalande. Histoire de Mathématiques).

The third volume of the treatise On Divine Proportion (a short book in three parts on the five regular geometric solids) is essentially a literal translation into Italian of Piero della Francesca's Five Regular Polyhedra, written in Latin. The fact that Pacioli never mentions that he is just a translator of the book evoked heated condemnation from the art historian Giorgio Vasari. Vasari writes about Piero della Francesca:

R. Emmett Taylor (1889–1956) published a book in 1942 entitled “There Is No King's Way. Luca Pacioli and his time" ( R. Emmett Taylor. No Royal Road: Luca Pacioli and His Times). In this book, Taylor treats Pacioli with great sympathy and defends the view that, based on style, Pacioli probably had nothing to do with the third volume of the treatise On Divine Proportion, and this work is only attributed to him.

Whether this is true or not is unknown, but it is certain that if it were not for printed works of Pacioli, ideas and mathematical constructions of Piero, which were not published in printed form, probably would not have gained the fame that they received as a result. Moreover, until the time of Pacioli, the golden ratio was known under terrifying names like “extreme and mean ratio” or “proportion having a mean and two extremes,” and this concept itself was known only to mathematicians.

The publication of "On the Divine Proportion" in 1509 caused a new outbreak of interest in the topic of the golden ratio. Now the concept was examined, as they say, with a fresh look: since a book was published about it, it means it is worthy of respect. The very name of the golden section turned out to be endowed with a theological and philosophical meaning ( divine proportion), and this also made the golden ratio not just a mathematical question, but a topic into which intellectuals of all sorts could delve, and this diversity only expanded over time. Finally, with the advent of Pacioli’s work, artists also began to study the golden ratio, since now it was talked about not only in openly mathematical treatises - Pacioli spoke about it in such a way that this concept could be used.

Leonardo's drawings for the treatise “On Divine Proportion,” drawn (as Pacioli put it) “by his indescribable left hand,” also had a certain impact on the readership. These were probably the first images of polyhedra in a schematic, skeletal form, which made it easy to imagine them from all sides. It is possible that Leonardo drew the polyhedrons from wooden models, since the documents of the Council of Florence record that the city purchased a set of Pacioli's wooden models in order to put them on public display. Leonardo not only drew diagrams for Pacioli's book, we see sketches of all kinds of polyhedrons everywhere in his notes. At one point Leonardo gives an approximate method for constructing a regular pentagon. The fusion of mathematics with fine art reaches its peak in " Trattato della pittura"("Treatise on Painting"), which was compiled by Francesco Melzi, who inherited Leonardo's manuscripts, according to his notes. The treatise begins with a warning: “No one who is not a mathematician should read my works!” – you can hardly find such a statement in modern textbooks on fine arts!

Drawings of geometric bodies from the treatise “On Divine Proportion” also inspired Fra Giovanni da Verona to create works in technology intarsia. Intarsia is a special type of wood inlay on wood, creating complex flat mosaics. Around 1520, Fra Giovanni created inlaid panels depicting the icosahedron, almost certainly using Leonardo's schematic drawings as a model.

The paths of Leonardo and Pacioli crossed several times even after the completion of the treatise “On Divine Proportion”. In October 1499, both fled Milan when it was captured by the French army of King Louis XII. Then they stopped briefly in Mantua and Venice and settled for a while in Florence. During the period when they were friends, Pacioli created two more works on mathematics that glorified his name - a translation into Latin of Euclid's Elements and a book on mathematical entertainment that remained unpublished. Pacioli's translation of the Elements was an annotated version based on an earlier translation by Giovanni Campano (1220–1296), which was printed in Venice in 1482 (this was the first printed edition). Achieve the publication of a collection of entertaining mathematics problems and sayings " De Viribus Quantitatis"("On the Abilities of Numbers") Pacioli was never able to do this during his lifetime - he died in 1517. This work was the fruit of collaboration between Pacioli and Leonardo, and Leonardo's own notes contain quite a few problems from the treatise " De Viribus Quantitatis ».

Of course, Fra Luca Pacioli was glorified not by the originality of his scientific thought, but by his influence on the development of mathematics in general and on the history of the golden section in particular, and these merits of his cannot be denied.

Melancholy

An interesting combination of artistic and mathematical interests was also characteristic of another great thinker of the Renaissance - the famous German painter Albrecht Durer.

Dürer is often considered the greatest German artist of the Renaissance. He was born on May 21, 1471 in the imperial city of Nuremberg in the family of a jeweler who worked tirelessly. Already at the age of 19, Albrecht showed remarkable talent as a painter and woodcarver and noticeably surpassed his teacher, the best Nuremberg painter and book illustrator Michael Wolgemut. Therefore, Dürer went traveling for four years and during this time came to the conviction that mathematics - “the most accurate, logical and graphically verified of all sciences” - should be an important component of the visual arts.

Upon returning, he stayed in Nuremberg for only a short time, but during this time he managed to marry Agnes Frey, the daughter of a successful artisan, and then went on a trip again - to Italy - in order to expand his horizons in both mathematics and the fine arts. Apparently, he fully achieved this goal during his visit to Venice in 1494–1495. The meeting with the founder of the Venetian school of painting Giovanni Bellini (c. 1426–1516) made an indelible impression on the young artist; he admired Bellini until the end of his days. At the same time, Dürer met Jacopo de Barbari, the same one who painted the portrait of Luca Pacioli (Fig. 50), and as a result studied Pacioli’s works on mathematics and its significance in the fine arts. In particular, de Barbari showed Dürer how to construct male and female figures using geometric methods, and this pushed Dürer to study the proportions and movement of the human body.

Perhaps Dürer met with Pacioli in person - this was in Bologna during his second visit to Italy (1501-1507). In a letter from that time, he mentions that the trip to Bologna was undertaken “for the sake of art, since there is a person there who will teach me secret art prospects." The mysterious “man from Bologna,” according to many interpreters, is Pacioli, although other names have been proposed, for example, the outstanding architect Donato di Angelo Bramante (1444–1514) and the architectural theorist Sebastiano Serlio (1475–1554). During the same trip to Italy, Dürer again met Jacopo di Barbari. However, Dürer's second visit was overshadowed by paranoid suspicions: he was afraid that other artists, envious of his fame, would harm him. In particular, he refused invitations to dinners for fear that someone would try to poison him.

From 1495 Dürer demonstrated a serious interest in mathematics. He studied the Elements for a long time (he acquired a Latin translation in Venice, although he did not know Latin very well), Pacioli’s works on mathematics and fine arts, and authoritative works on architecture, proportions and perspective by the Roman architect Vitruvius and the Italian architect and theorist Leon Baptista Alberti (1404 –1472).

Dürer's contribution to the history of the golden ratio consists of both written works and works of fine art. In 1525 his main treatise “ Unterweisung der Messung mit dem Zirkel und Richtscheit"("Treatise on Measurements with Compass and Ruler"), one of the first books on mathematics published in Germany. In this essay, Dürer complains that so many artists are ignorant of geometry, “without which no one can be or become a perfect artist.” The first of the four books that make up the Treatise gives detailed recommendations on how to construct various curves, including the logarithmic (equiangular) spiral, which, as we have already seen, is closely related to the golden ratio. The second book contains exact and approximate methods for constructing various polygons, including two methods for constructing a regular pentagon (one exact, the other approximate). The fourth book discusses the Platonic solids, as well as other polyhedra - some of which Dürer invented himself - and the theory of perspective and chiaroscuro. Dürer's book is not intended as a textbook on geometry; in particular, he gives only one example of a proof. On the contrary, Dürer always begins with practical application and then lists the most basic theoretical information. The book also contains the first examples of developments of polyhedra. A development is a drawing on a plane that depicts the surface of a polyhedron in such a form that it can be cut out and folded into a three-dimensional polyhedron from the resulting figure. A drawing of a dodecahedron (associated, as we know, with the golden ratio), made by Dürer, can be seen in Fig. 54.

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An interest in engraving and woodcarving, combined with an interest in mathematics, is reflected in Dürer's enigmatic allegorical work Melancholy I (Fig. 55). This is one of three exquisite engravings (the other two are entitled "Knight, Death and the Devil" and "St. Jerome in his cell"). It is assumed that Durer created this engraving during an attack of melancholy after the death of his mother. The central figure of “Melancholia” is a winged woman, sitting on a stone parapet in complete despair and apathy. In her right hand she holds a compass, the legs of which are open, as if for measurements. Almost everything depicted in this engraving is endowed with complex symbolic meaning, and entire articles are devoted to its interpretation. For example, it is believed that the pot on the hearth in the middle left and the scales at the top are symbols of alchemy. The “magic square” at the top right (that is, a square in which the sum of the numbers in each row, column, diagonally, and the sum of the numbers in the four corners and the sum of the four central numbers is equal to 34 - by the way, this is the Fibonacci number), apparently symbolizes mathematics (Fig. 56). The middle two numbers in the bottom row are 1514, the date the engraving was created. Probably the magic square is a consequence of Pacioli’s influence, since in Pacioli’s treatise “ De Viribus» a whole series of magic squares is given. Apparently, the main meaning of the engraving with all its geometric shapes, keys, bat, seascape and so on is the melancholy that gripped the artist or thinker, mired in doubts and thoughts about what he is doing, and meanwhile time - an hourglass at the top - does not stand still.

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The strange polyhedron on the left in the middle has been the subject of serious discussion and various attempts at reconstruction. At first glance, it appears to be a cube with two opposite corners cut off (which has provoked some Freudian interpretations), but in reality this is not the case. Most researchers agree that this is the so-called rhombohedron ( geometric body with six faces, each of which is a rhombus, see fig. 57), cut so that it can fit into a sphere. It rests on one of the triangular faces, with its front pointing directly at the magic square. The angles of the polyhedron's face have also been the subject of controversy. Many scientists assume that they were 72 degrees, which would connect the figure with the golden ratio (see Fig. 25), but the Dutch crystallographer K. G. Macgillavry concluded, based on an analysis of perspective, that the angles were 80 degrees. The mysterious properties of this geometric body are perfectly described in an article by T. Lynch, published in 1982 in “ Journal of the Warburg and Courtauld Institutes" This is the conclusion the author comes to: “Since the depiction of polyhedra was considered one of the main tasks of perspective geometry, Dürer, wanting to prove his knowledge in this area, could hardly find a better way for this than to place on his engraving a geometric body, so new and, perhaps even unique, and leave it to other geometers to decide what it is and where it came from.”

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With the exception of the authoritative work of Pacioli and the research of the artists Leonardo and Durer at the intersection of mathematics and fine arts, nothing particularly new happened in the history of the golden ratio in the 16th century. Although many mathematicians, including Raphael Bombelli (1526–1572) and François Foy (Flussates) (1502–1594), relied on the golden ratio to solve a wide variety of problems, including those involving the regular pentagon and the Platonic solids, more interesting applications our relationships appeared only at the very end of this century. However, the works of Pacioli, Dürer and other scientists revived interest in the teachings of Plato and Pythagoras. Renaissance thinkers suddenly saw a real opportunity to connect mathematics and rational logic with the structure of the universe - in the spirit of Plato's worldview. Concepts like “divine proportion,” on the one hand, built bridges between mathematics and the structure of the universe, and on the other, provided a connection between physics, theology and metaphysics. And this enchanting mixture of mathematics and mysticism was especially vividly embodied in his ideas and works by none other than Johannes Kepler.

Mysterium Cosmographicum

Johannes Kepler is remembered mainly as an outstanding astronomer, who left us, among other things, the three laws of planetary motion that bear his name. However, Kepler was also a talented mathematician, a subtle metaphysician and prolific writer. He was born at a time of great political upheaval and religious wars, which radically influenced his education, life, and thinking. Kepler was born on December 27, 1571 in Germany, in the imperial city of Weil der Stadt, in the house of his grandfather Sebald. Johann's father Heinrich, a hired soldier, spent almost all of his son's childhood on campaigns, and during his short stays, according to Kepler, he behaved “offensively, harshly and quarrelsome.” When Kepler was about sixteen, his father left home and was never seen again. Apparently, he took part in some kind of sea voyage as part of the fleet of the Kingdom of Naples and died on the way home. Consequently, Kepler was raised mainly by his mother Katharina, who worked in the hotel that her father kept. Katarina herself was an odd woman, rather unpleasant, who collected herbs and was convinced of their magical healing properties. A combination of circumstances - personal grievances, unfortunate gossip and greed - ultimately led to the fact that Katharina, already in old age, in 1620, was arrested on charges of witchcraft. Such accusations were not uncommon at the time; between 1615 and 1629, at least 38 women were executed for witchcraft in Weil der Stadt. Kepler was already a famous person at the time of his mother’s arrest, and the news of his mother’s trial caused him “indescribable grief.” In fact, he took over her defense in court and enlisted the help of the law faculty of the University of Tübingen. The trial was long, but in the end the charges against Katharina Kepler were dropped, mainly due to her own testimony, given under threat terrible torture: Katarina stubbornly denied her guilt. This story conveys the atmosphere in which Kepler's scientific work took place and the prevailing mentality of the time. Kepler was born into a society that, just half a century earlier, had experienced Martin Luther's departure from the Catholic Church and his declaration that the only thing God needs from a person is faith. This society had yet to plunge into the bloody madness of the Thirty Years' War. One can only be amazed how Kepler, a man from such an environment, who had such ups and downs, such a turbulent life, was able to make a discovery that many consider to be the true birth of modern science.

Kepler began his scientific research while still at school at the Maulbronn monastery, and then, in 1589, won a scholarship from the Duke of Württemberg and was given the opportunity to attend the Lutheran seminary at the University of Tübingen. He was most interested in two topics, theology and mathematics; in his mind they were closely connected. Astronomy was at that time considered a part of mathematics, and Kepler's mentor in astronomy was the eminent scientist Michael Maestlin (1550–1631); Kepler maintained contact with him even after leaving Tübingen. During formal teaching, Mestlin, of course, taught only the traditional Ptolemaic, geocentric system, according to which the Moon, Mercury, Venus, Sun, Mars, Jupiter and Saturn revolve around a stationary Earth. However, Mestlin was well aware of the heliocentric system of Nicolaus Copernicus, information about which was published in 1543, and privately discussed the merits of this system with his favorite student Kepler. According to the Copernican system, six planets (including the Earth, but excluding the Moon, which was no longer considered a planet, but a “satellite”) revolve around the Sun. In much the same way that from a moving car you can only observe the relative motion of other cars, in the Copernican system the motion of the planets in many ways simply reflects the motion of the Earth itself.

It seems that Kepler immediately liked the Copernican system. The fundamental idea of ​​this cosmology, according to which the central Sun is surrounded by a sphere of fixed stars, with some space remaining between the Sun and the sphere, exactly corresponded to Kepler's idea of ​​​​the universe. Kepler was a deeply religious man and believed that the Universe is a reflection of the Creator. The unity of the Sun, stars and the space between them was for him a symbolic likeness of the Holy Trinity - Father, Son and Holy Spirit.

When Kepler graduated with honors from the Faculty of Fine Arts and was ready to complete his theological education, an event occurred that changed his choice of profession: he became not a pastor, but a mathematics teacher. The Protestant Seminary in the Austrian city of Graz asked the University of Tübingen to recommend a replacement for one of its mathematics teachers who had died suddenly, and the university chose Kepler. In March 1594, Kepler went on an unwilling journey to Graz in the Austrian province of Styria; It took a whole month to travel.

Realizing that fate had imposed on him a career as a mathematician, Kepler became determined to fulfill his Christian duty as he imagined it: to comprehend the creation of the Lord, the structure of the Universe. Therefore, he studied the translations of the Elements and the works of the Alexandrian geometers Apollonius and Pappus. Based on the basic principle of the Copernican heliocentric system, Kepler decided to find answers to two main questions: why there are exactly six planets and what determines exactly such distances between planetary orbits. Questions of “why” and “what” were new to astronomy. Unlike his predecessors, who were content simply to note the observed positions of the planets, Kepler sought to derive a theory that would explain everything. Kepler explained his new approach, reaching a new level of curiosity very beautifully:

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The answer to the first two questions that occupied Kepler is given in his first treatise entitled “ Mysterium Cosmographicum"("Cosmographic Riddle"), which was published in 1597. The full title given on the title page of the book (Fig. 59; although the publication date is 1596, the book was not published until the following year) reads: “A Preliminary Introduction to Cosmographic Speculations, Containing the Universal Riddle of the Delightful Proportions of the Celestial Spheres, and the True and Genuine Causes their Size, Number and Periodic Movement of the Heavens, proved by the Five Regular Geometric Solids.”

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The answer to the question why there are exactly six planets was given to Kepler very simply: because there are exactly five regular Platonic solids. If we consider that they define the gaps between the planets, we get six gaps, counting the outer spherical boundary - the heavens with fixed stars. Moreover, Kepler's model is designed to answer the question of the size of orbits. Here is how the scientist himself writes:
The earthly sphere is the measure of all other orbits. Draw a dodecahedron around it. The sphere surrounding it will be the sphere of Mars. Describe a tetrahedron around Mars. The sphere surrounding it will be the sphere of Jupiter. Describe a cube around Jupiter. The sphere surrounding it will be the sphere of Saturn. Now fit the icosahedron into the Earth's orbit. The sphere inscribed in it will be the sphere of Venus. Inscribe the octahedron in the orbit of Venus. The sphere inscribed in it will be the sphere of Mercury. So much for the justification for the number of planets.
In Fig. 60 shows a diagram from “ Mysterium Cosmographicum", illustrating Kepler's cosmological model. Kepler explains at some length why he draws specific parallels between the Platonic solids and the planets based on their geometric, astrological and metaphysical properties. He arranged the geometric bodies based on their relationship to the sphere, suggesting that the difference between the sphere and the other geometric bodies reflected the difference between the creator and the creation. In a similar way, a cube is characterized one and only angle - right. For Kepler, this symbolized loneliness, which is associated with Saturn, etc. Generally speaking, astrology was so important for Kepler because “Man is the crown of the Universe and of all creation,” and the metaphysical approach was justified by the fact that “mathematical properties are causes physical, since God from the very beginning of time contained within himself mathematical objects as simple divine abstractions that served as prototypes for various quantities on the material level.” The position of the Earth was chosen to separate bodies that can be stood upright (cube, tetrahedron and dodecahedron) from bodies that “float” (octahedron and icosahedron).

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The distances between the planets obtained from this model in some cases completely coincided with reality, and in others they differed noticeably, although the difference was no more than 10%. Kepler was unshakably convinced of the correctness of his model and attributed the inconsistencies to errors in orbital measurements. He sent copies of his book to various astronomers for their comments and suggestions; among them was one of the most outstanding scientists of that time, the Dane Tycho Brahe (1546–1601). One copy even fell into the hands of the great Galileo Galilei (1564–1642), who told Kepler that he was also confident in the correctness of Copernicus’s model, but admitted with chagrin that “to a great many people, for such are the number of fools,” Copernicus “seems a worthy subject for ridicule and booing."

Needless to say, Kepler's cosmological model, based on the Platonic solids, was not only completely wrong, but also insane even by the standards of the scientist's contemporaries. The discovery of Uranus (the next planet after Saturn, counted from the Sun) in 1781 and Neptune (the next planet after Uranus) in 1846 put the final nail in the coffin of this stillborn idea. However, the importance of Kepler's model in the history of science cannot be underestimated. As the astronomer Owen Gingerich noted in an article devoted to the biography of Kepler: “Rarely in history has it happened that such an erroneous book has directed the further course of science in such a correct direction.” Kepler relied on the Pythagorean idea of ​​the universe, and mathematicians would call this a great step forward. He developed mathematical model Universe, which, on the one hand, was based on the observational data available at that time, and on the other hand, could be refuted subsequent observations. These are the necessary components of the “scientific method” - an organized approach to explaining observed facts based on a model of nature. Ideal scientific method begins with the collection of facts, then a model is proposed, and then what it predicts is tested through either artificial experiments or further observations. Sometimes this process is described in three words: induction, deduction, verification. In 1610, Galileo used his telescope to discover four more celestial bodies in the solar system. If it had been proven that these were planets, Kepler's theory would have been dealt a mortal blow during the scientist's lifetime. However, to Kepler's great delight, the new bodies turned out to be satellites of Jupiter, similar to our Moon, and not new planets orbiting the Sun.

Modern physical theories, aimed at explaining the existence of all elementary (subatomic) particles and the basic interactions between them, are also based on mathematical symmetry and in this sense are very similar to the theory of Kepler, who relied on the symmetrical qualities of the Platonic solids to explain the number and properties of planets. Kepler's model had one more thing in common with the modern fundamental theory of the Universe: both theories are inherently reductionist, that is, they strive to explain many phenomena with a small number of physical laws. For example, Kepler's model derives both the number of planets and the properties of their orbits from the Platonic solids. Likewise, modern theories - such as string theory - rely on fundamental entities (strings) that are very small (more than a billion billion times smaller than an atomic nucleus), from which all the properties of elementary particles are derived. Strings - like a violin string - vibrate and produce various "tones", and all known elementary particles merely embody these tones.

While in Graz, Kepler became interested in the golden ratio, which led to another interesting result. In October 1597, the scientist wrote to his former teacher Mestlin about the following theorem: “If on a segment divided in extreme and mean ratio, a right triangle is constructed so that the right angle lies on the perpendicular drawn at the point of division, then the smaller leg will be equal to the larger segment divided segment." The drawing for this theorem is presented in Fig. 61. Segment AB is divided by point C in the golden ratio. Kepler constructs a right triangle A.D.B. with the hypotenuse AB so that the right angle D lies on the perpendicular drawn from the golden section point C. He then proves that BD(short leg right triangle) is equal to AC (the longer segment of the line segment divided in the golden ratio). In addition to the use of the golden section, such a triangle is also notable for the fact that pyramid researcher Friedrich Reber cited it in 1855 to prove one of the false theories that suggested the use of the golden section in the construction of pyramids. Reber did not know about Kepler’s works, but he used a similar structure to confirm his opinion about the most important role of “divine proportion” in architecture.

Publication " Mysterium Cosmographicum“became the reason for Kepler’s acquaintance with Tycho Brahe; The meeting place, which took place on February 4, 1600, was Prague, at that time the residence of the Holy Roman Emperor. As a result of this meeting in October of the same 1600, Kepler moved to Prague and became an assistant to Tycho Brahe (because of his Lutheran faith, he was forced to leave Catholic Graz). After Brahe's death on October 24, 1601, Kepler became court mathematician.

Tycho left a lot of observations, especially related to the orbit of the planet Mars, and Kepler, relying on these data, discovered the first two laws of planetary motion, named after him. Kepler's first law states that the orbits of the known planets around the Sun are not circles, but ellipses with the Sun at one of the foci (Fig. 62; for clarity, the ellipse is elongated much more than it actually is). The ellipse has two points, the so-called foci, such that the sum of the distances of any point of the ellipse to both foci is always constant. Kepler's Second Law states that a planet moves fastest when it is closest to the Sun (this point is called perihelion), and slowest at its farthest point (aphelion), so that the line connecting the planet to the Sun traces (sweeps out) equal area for equal periods of time (Fig. 62). The question of what makes Kepler's laws valid has been a major unsolved mystery in science for nearly seventy years after Kepler published his laws. It took the genius of Isaac Newton (1642–1727) to conclude that the planets are held in orbit by gravity. Newton explained Kepler's laws using equations where the laws describing the motion of bodies were combined with the law of universal gravitation. He showed that elliptical orbits with variable speed (according to Kepler's laws) provide the only possible solution to these equations.

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Kepler's heroic efforts to calculate the orbit of Mars (many hundreds of sheets of arithmetic calculations and their interpretations, which he himself called “my military campaign against Mars”), according to many researchers, mark the birth of modern science. In particular, at one point Kepler discovered a circular orbit that matched almost all of Tycho Brahe's observations. However, in two cases this orbit predicted positions that differed from observations by about a quarter of the angular diameter of the full moon. Kepler wrote about this: “If only I had assumed that we could neglect these eight minutes [of arc], I would have included my hypothesis in the corresponding 16th chapter. But since it is impermissible to neglect them, it turns out that these eight minutes pointed the way to a complete reform of astronomy.”

Kepler's years in Prague bore rich fruit in both astronomy and mathematics. In 1604, he discovered a "new" star, now known as Kepler's Supernova. A supernova is a powerful explosion in which a star, whose end is near, throws off its outer shells, which move at speeds of tens of thousands of kilometers per second. In our home galaxy, the Milky Way, such an outbreak, according to scientists, should occur on average once every hundred years. Indeed, Tycho Brahe discovered a supernova in 1572 (Tycho Brahe's Supernova), and Kepler discovered his in 1604. However, since then, for unknown reasons, there have been no other supernovae in the Milky Way (except for another that apparently occurred in the 1660s, but went undetected). Astronomers joke that this lack of supernovae is most likely due to the fact that there have been no great astronomers since Tycho Brahe and Kepler.

In June 2001, I visited Prague, in the house where Kepler lived, at 4 Charles Street. Nowadays it is a busy shopping street, and there is a rusty plaque above number 4, which states that Kepler lived here from 1605 to 1612, easy to miss. The owner of the store located directly below Kepler's apartment did not even know that one of the greatest astronomers in history lived here. True, there is a small armillary sphere with Kepler's name carved on it in the dull courtyard, and another memorial plaque hangs near the mailboxes. However, Kepler's apartment is not marked at all and is not open to the public - now it is just a residential apartment, of which there are many on the upper floors above shops, and it is occupied by an ordinary family.

Kepler's mathematical works brought several bright touches to the history of the golden ratio. In the text of a letter that Kepler wrote in 1608 to a Leipzig teacher, we find that he discovered the relationship between the Fibonacci numbers and the golden ratio. He also reports this discovery in an essay where he studies why snowflakes have a six-pointed shape. Kepler writes:

Consider the square with side 8 (with area 82 = 64) in Fig. 63. Now let’s cut it into four parts along the marked lines. From these four pieces you can make a rectangle (Fig. 64) with sides 13 and 5 - that is, with an area of ​​65! Where did the extra square come from?! The answer to this paradox is that the puzzle pieces do not actually fit perfectly along the long diagonal of the rectangle, but rather a long, narrow parallelogram that is not visible because of the thick line marking the long diagonal in Fig. 64, and its area is just enough for the area of ​​one unit square. Of course, 8 is a Fibonacci number, and its square 82 = 64 differs by 1 from the product of two adjacent Fibonacci numbers (3 × 5 = 65): a property that Kepler discovered.

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You may have already noticed that Kepler refers to the golden ratio as “the divine proportion, as modern geometers call it.” All of Kepler's scientific research is colored by a combination of rational reasoning and Christian beliefs. Kepler was a Christian naturalist and considered it his duty to understand not only the structure of the Universe, but also the intentions of its Creator. He built his hypothesis about the solar system under the influence of a strong craving for the number 5, adopted from the Pythagoreans, and wrote about the golden ratio as follows:
The peculiarity of this ratio is that a similar proportion can be built from the whole and the larger part, and what used to be for the most part, now becomes smaller, and what was previously a whole now becomes a larger part, and their sum has the ratio of the whole. This happens ad infinitum, and the divine proportion is always preserved. I believe that this geometric proportion served as an idea for the Creator when He created like from like in His own image and likeness - and this also happens ad infinitum. I see the number five in almost all flowers that pave the way for fruit, that is, creation, and which exist not for their own sake, but for the sake of being followed by fruit. Almost all the flowers of fruit trees can be included here; Lemons and oranges should probably be excluded, although I have not seen their flowers and judge only by the fruits or berries, which are divided not into five, but into seven, eleven or nine segments. However, the embodiment of the number five in geometry, that is, the regular pentagon, is constructed through divine proportion, which I would like to [presumably consider] the prototype of Creation. Moreover, [it] is also observed between the movements of the Sun (or, as I suppose, the Earth) and Venus, which stands at the pinnacle of the generating power of the ratio 8 and 13, which, as we will hear, comes very close to the divine proportion. Finally, according to Copernicus, the sphere of the Earth is located midway between the spheres of Mars and Venus. The proportion between them can be obtained from the dodecahedron and icosahedron, both of which in geometry are derived from divine proportion - however, the act of creation takes place on our Earth.

Now let us consider how the images of man and woman arise from divine proportion. In my opinion, the reproduction of plants and the procreation of animals consist of the same relationship as a geometric proportion, a proportion expressed by parts of a segment, or an arithmetic or numerically expressed proportion.

The relatively calm and professionally fruitful period of life in Prague ended for Kepler in 1611, when a series of misfortunes befell him. First, his son Friedrich died of smallpox, then his wife Barbara died of a contagious fever brought by the Austrian occupiers. In the end, Emperor Rudolf abdicated the throne in favor of his brother Matthias, known for his intolerant attitude towards Protestants. Therefore, Kepler was forced to move to Linz, in the territory of modern Austria.

The crowning achievement of Kepler's works, created in Linz, was the publication in 1619 of his second main work in cosmology - " Harmony Mundi"("Harmony of the World").

Let us remember that for Pythagoras and the Pythagoreans, music and harmony were the first argument in favor of the fact that cosmic phenomena can be described mathematically. Consonant tones were generated only by those strings whose lengths corresponded to simple fractions. The ratio 2:3 sounded like a fifth, 3:4 like a fourth, etc. It was believed that a similar harmonic arrangement of planets also generates “music of the spheres.” Kepler was very familiar with this concept, having read almost the entire book of his father Galileo Galilei Vincenzo's Dialogues on Ancient and Modern Music, although he did not agree with some of Vincenzo's ideas. Since he was also convinced that he had created a comprehensive model of the solar system, he was even able to calculate small "motifs" for different planets (Fig. 65).

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Since Kepler believed that “before the beginning of things, geometry was as eternal as the Divine Mind,” The Harmony of the World was largely devoted to geometry. One aspect of this work was especially important for the history of the golden ratio - I mean Kepler's research in the field of geometric parquet.

Parquet in geometry is a pattern or structure consisting of “tiles” of one or more shapes that completely cover the plane without leaving gaps - like a mosaic of tiles on the floor. We will see in Chapter 8 that some of the mathematical concepts seen in such "parquets" are closely related to the golden ratio. Although Kepler was not aware of all the mathematical intricacies of parquet, his interest in the relationships between different geometric figures and his reverence for the regular pentagon, which embodies divine proportion most clearly, allowed him to create an interesting work on parquet. Kepler was especially interested in the congruence (“fitting” to each other) of geometric figures and bodies such as polyhedra and polygons. In Fig. 66 shows an example from “Harmony of the World”. This parquet pattern is made up of four figures - and all of them are related to the golden ratio: these are regular pentagons, pentagrams, decagons and double decagons. For Kepler, this is the epitome of “harmony,” since in Greek the word means “conformity with one another.”

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It is interesting that two more people showed interest in parquet before Kepler, who also played an important role in the history of the golden ratio (and were already mentioned on the pages of our book): Abu-l-Wafa and the artist Albrecht Durer. Both of them considered patterns of figures with five-ray symmetry (an example from Dürer’s sketches is shown in Fig. 67).

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The fifth book of “Harmonies of the World” contains the most significant result of Kepler’s astronomical research - the Third Law of Planetary Motion. Here all his painful thoughts about the size of the orbits of different planets and the periods of their revolution around the Sun were fully expressed. Twenty-five years of work have been concentrated in an amazingly simple law: the squares of the periods of revolution of the planets around the Sun are related like the cubes of the semi-major axes of the planets’ orbits, and this ratio is the same for all planets (the semi-major axis is half the long axis of the ellipse, see Fig. 62). Kepler discovered this fundamental law, which served as Newton's starting point for formulating the law of universal gravitation when The Harmony of the World was already in print. Unable to contain his jubilation, the scientist announced: “I stole the golden vessels of the Egyptians in order to build an altar to my Lord far from Egypt.” The essence of the law naturally follows from the law of universal gravitation: the closer the planet is to the Sun, the greater the force of gravity, which is why planets that are closer to it are forced to rotate faster, otherwise they will fall into the Sun.

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In 1626, Kepler moved to Ulm and completed work on the Rudolf Tables - at that time they were the most detailed and accurate astronomical tables in history. When I was at the University of Vienna in June 2001, I was shown the first edition of the tables, stored in the observatory library (147 copies have survived to this day). The frontispiece of the book (Fig. 68) symbolically depicts the history of astronomy, and in the lower left corner there is perhaps Kepler's only self-portrait (Fig. 69). It shows Kepler working by candlelight under a vignette listing his main publications.

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Kepler died at noon on November 15, 1630 and was buried in Regensburg. Even after death, fate did not leave him alone, as if his stormy life was not enough: wars wiped his grave off the face of the earth. Fortunately, a sketch of the tombstone, which was made by a friend of Kepler, has been preserved, and it also contains an epitaph for the scientist:
I measured the heavens, now I measure the shadows of the Earth.

My spirit lived in heaven, but here the shadow of my body lies.
Nowadays, it is perhaps impossible to imagine a scientist as original and prolific as Kepler. It must be understood that this man suffered unimaginable suffering: in particular, in 1617–1618, he lost three children in less than six months. Perhaps the English poet John Donne (1572–1631) said it best about him in his pamphlet “Ignatius and His Conclave”: Kepler “made it his duty to see that nothing new happened in the heavens without his knowledge.”

Luca Pacioli and his treatise

"On Divine Proportion"

A. I. SHCHETNIKOV

Biographical sketch

LUCA PACIOLI (LUCA PACIOLI or PACIOLLO) was born in 1445 into the poor family of BARTOLOMEO PACIOLI in the small town of Borgo San Sepolcro, located on the coast

Tiber, on the border of Tuscany and Umbria, and which at that time belonged to the Florentine Republic. As a teenager, he was sent to study in the studio of the famous artist

PIERO DELLA FRANCESCA (c. 1415–1492), who lived in the same town. Studying in the workshop did not make him an artist, but it did develop excellent taste, and most importantly, here he first became familiar with mathematics, which deeply interested his teacher. Together with his teacher, LUCA often visited the court of FEDERICO DE MONTEFELTRO, Duke of Urbino.

Here he was noticed by the great Italian architect LEON BATISTA ALBERTI (1404–1472), who in 1464 recommended a young man to the wealthy Venetian merchant ANTONIO DE ROMPIANZI as a home teacher.

In Venice, LUCA taught the sons of his patron and studied himself, attending lectures by the famous mathematician DOMENICO BRAGADINO at the Rialto School. In 1470, he compiled his first book, a textbook of commercial arithmetic. In the same year he left Venice and moved to Rome, where he was received by ALBERTI and settled in his house. However, two years later PACIOLI left Rome and took monastic vows, becoming a Franciscan.

After being tonsured, brother LUKA lives for some time in his homeland in San Sepolcro. From 1477 to 1480 he taught mathematics at the university in Perugia. Then, for eight years, he lived in Zara (now Zadar in Croatia), where he studied theology and mathematics, sometimes traveling to other cities in Italy on order business. During these years, PACIOLI began to write the main work of his life - the encyclopedic Sum of Arithmetic, Geometry, Relations and Proportions. In 1487 he was again invited to take the chair in Perugia. In subsequent years he lives in Rome, Naples, Padua.

On October 12, 1492, PIERO DELLA FRANCESCA dies. The following year, PACIOLI's work on the Summa was finally completed. With this manuscript he comes to Venice, where in November 1494 this book, dedicated to the young GUIDO UBALDO DE MONTEFELTRO (1472–1508), who became the Duke of Urbino in 1482 after the death of his father, is published.

It is noteworthy that the book was written not in the usual Latin for scientific works, but in Italian. Some authors can read that LUCA wrote his treatises in Italian because he did not receive the appropriate education and did not speak Latin perfectly. However, he was a Master of Theology, and Latin was the only language of theological treatises; he taught mathematics at various universities, and there all subjects were taught in Latin; and he also translated the entire EUCLID from Latin into Italian (although this translation was never published). Therefore, although he did not speak humanistic Latin, school Latin was his everyday language.

Therefore, the reason why he preferred the Italian language to Latin was due to Dr. LUCA PACIOLI AND HIS TREATISE “ON THE DIVINE PROPORTION” 2nd.

Here is what LUKE himself says about this in his dedication to the Summa (written in both Italian and Latin):

The correct understanding of difficult terms among Latinists ceased due to the fact that good teachers became rare. And although the style of Cicero or even higher would be better suited for Your Ducal Highness, I believe that not everyone will be able to take advantage of this source of eloquence. So, taking into account the interests of the general benefit of your respectful subjects, I have decided to write my essay in the native local language, so that both educated and uneducated alike can enjoy these activities.

In the preface to the Summa, PACIOLI talks about those people, thanks to communication with whom he developed the conviction that mathematics considers “a universal law applicable to all things.” He talks about astronomy, oh scientific approach to architecture, embodied in the works of VITROVIUS and ALBERTIE, about numerous painters who developed the art of perspective, “which, if examined carefully, would be an empty place without the use of mathematical calculations,” among which stands out “the king of our time in painting” PIERO DELLA FRANCESCA, o wonderful sculptors. These are those masters “who, using calculations in their works with the help of a level and compass, brought them to extraordinary perfection.” PACIOLI also speaks of the importance of mathematics for music, for cosmography, for trade, for the mechanical arts, for military affairs.

The sum of arithmetic, geometry, relations and proportions is a vast encyclopedic work, printed on 300 folio pages. The first part, with 224 sheets, is devoted to arithmetic and algebra, the second, with 76 sheets, is devoted to geometry. The numbering of sheets in both parts begins again. Each part is divided into sections, sections into treatises, treatises into chapters.

The arithmetic part of the Summa sets out methods for performing arithmetic operations; this part is based on numerous Books of the Abacus that belonged to different authors. Algebraic problems solved in the Summa do not go beyond the range of problems on linear and quadratic equations considered in Arabic treatises on “algebra and almukabala”; in Europe, these tasks were known from the Book of the Abacus of LEONARDO OF PISA (1180–1240). Among the problems that attracted the attention of mathematicians of subsequent generations, noteworthy is the problem of dividing the bet in an unfinished game, which LUKA himself solved incorrectly. Perhaps PACIOLI's most significant innovation was the systematic use of syncopated algebraic notation - a kind of precursor to the subsequent symbolic calculus. The book contains a table of coins, weights and measures adopted in different parts of Italy, as well as a guide to Venetian double-entry bookkeeping. As for the geometric part of the Summa, it follows the Practical Geometry of LEONARDO OF PISA.

In the first half of the 90s, PACIOLI lived in Urbino. It is from this era that the painting by JACOPO DE BARBARI dates back, in which PACIOLI is depicted accompanied by an unknown young man. Various hypotheses have been put forward regarding the identity of this young man. The most plausible assumption seems to be that this is Duke GUIDO UBALDO, patron of PACIOLI.

LUCA PACIOLI AND HIS TREATISE “ON THE DIVINE PROPORTION” 3

Portrait of LUCA PACIOLI and an unknown young man.

Painting by JACOPO DE BARBARI (Naples, National Museum) In 1496, the Department of Mathematics was established in Milan, and PACIOLI offered to occupy it. Here he gives educational lectures to students and public lectures to everyone. Here, at the court of Duke LODOVICO MORO SFORZA (1452–1508), he became close to LEONARDO DA VINCI. In LEONARDO's notebooks there are notes: “Learn to multiply roots from maestro LUCA,” “ask your brother from Borgo to show you a book about scales.” PACIOLI carried out calculations for LEONARDO on the weight of the gigantic equestrian monument of FRANCESCO SFORZA. In Milan, PACIOLI wrote the message On Divine Proportion, addressed to Duke LODOVICO SFORZA, and LEONARDO illustrated it. The treatise was completed on December 14, 1498. Several handwritten copies of the treatise, handed to the rulers, were accompanied by a set of regular polyhedra and other geometric bodies, which Brother LUK says that he made them with his own hand. (He wrote about models of regular polyhedra back in the Summa.) Two manuscripts of this treatise have been preserved - one in the Public Library in Geneva, the second in the Ambrosian Library in Milan.

In 1499 the French army occupied Milan and the Duke of SFORZA fled; LEONARDO and LUCA soon left the city. In subsequent years, LUCA PACIOLI lectured in Pisa (1500), Perugia (1500), Bologna (1501–1502) and Florence (1502–1505). In Florence, he is patronized by PIETRO SODERINI, Gonfaloniere for life of the Republic.

However, not all of PACIOLI's works have been published, and so he goes to Venice again. Here in 1508 he published the Latin translation of EUCLID, owned by GIOVANNI CAMPANO from Novara. This translation, made back in 1259 Arabic, was already published in 1482 and then reprinted several times, but the edition was replete with typos and errors. PACIOLI edited the translation; According to this edition, supplied with numerous comments, he gave his university lectures. However, the publication turned out to be unclaimed, since in 1505 BARTOLOMEO ZAMBERTI published a new translation of the Elements, made directly from the Greek original.

In 1509, another book by PACIOLI was published in Venice: Divina proportione. Opera a tutti glingegni perspicaci e curiosi necessaria. Ove ciascun studioso di Philosophia, Prospectiva,

LUCA PACIOLI AND HIS TREATISE “ON THE DIVINE PROPORTION” 4

This printed edition includes a number of texts. The publication is prefaced by an appeal to the Florentine gonfaloniere PIETRO SODERINI. The first part (33 sheets) contains a message on divine proportion, as well as a treatise on architecture, on the proportions of the human body and on the principle of constructing the letters of the Latin alphabet. It is followed by the Book in three separate treatises on regular bodies (27 sheets), of which the first treatise considers flat figures, the second - regular bodies inscribed in a sphere, the third - regular bodies inscribed in each other. Next are graphic tables printed on one side of the sheet: the proportions of the human face (1 sheet), the principle of constructing the letters of the Latin alphabet (23 sheets), images of architectural elements (3 sheets), images of regular and other bodies made on the basis of LEONARDO’s drawings (58 sheets ), and, finally, the “tree of proportions and proportionality” - a drawing that PACIOLI already provided in the Summa (1 sheet).

In his message On Divine Proportion, LUCA PACIOLI says that, as an old man, it is time for him to retire to “count the years in a sunny place.” This request was heard, and in 1508 he became the locum tenens of the monastery in his native San Sepolcro. However, in December 1509, two monks of his monastery handed over a letter to the general of the order, in which they pointed out that “Maestro LUCA is an unfit man to govern others” and asked to be relieved of his administrative duties.

But they did not find support from their superiors, and in February 1510 LUCA PACIOLI became the full prior of his native monastery. However, strife within the monastery continued further.

IN last years throughout his life, brother LUKA continued to lecture occasionally; he was invited to Perugia in 1510 and to Rome in 1514, the last invitation coming from the new Pope LEO X. LUCA PACIOLI died at the age of 72, on June 19, 1517 in Florence.

Review of the message “On Divine Proportion”

In LUCA PACIOLI's message On Divine Proportion the following contents are highlighted:

Introduction (Chapters 1–4). Divine qualities, definition and mathematical properties of the proportion that arises when dividing a value in an average and extreme ratio (chap. 5–23). About regular bodies, why there cannot be more than five of them and how each of them fits into a sphere (chapters 24–33). About how regular bodies fit into each other (chap.

34–46). About how a sphere fits into each of these bodies (chapter 47). About how truncated and superstructured bodies are obtained from regular bodies (chapters 48–52). About other bodies inscribed in a sphere (chap. 53–55). Sphere (chap. 56–57). On columns and pyramids (chap. 58–69). On the material forms of represented bodies and their perspective images (chapter 70). Glossary (chapter 71).

LUCA PACIOLI AND HIS TREATISE “ON THE DIVINE PROPORTION” 5

a square is on the larger part equal to a rectangle, the sides of which are the whole and the smaller part.

Brother LUKE justifies the special value and distinction of the relation of “divine proportion” among other relations with arguments of a metaphysical and theological nature. The uniqueness and immutability of this proportion is compared with the uniqueness and immutability of God, its three members - with the three hypostases of the Holy Trinity, the irrationality of the relationship - with the incomprehensibility and inexpressibility of God. But in addition to these arguments, there is one more: this proportion is associated with the procedures for constructing a regular flat pentagon, and solid dodecahedron and icosahedron. But PLATO in the Timaeus regarded the five regular bodies as the five elements of which the Universe is composed. Thus, PACIOLI’s metaphysical constructions combine the motives of Christian theology and Platonic cosmology.

Books XIII and XIV of Euclid's Elements. In total, he considers thirteen such properties, connecting this number with the number of participants in the Last Supper. Here is an example of one of these properties:

“Let a straight line be divided in proportion, having a middle and two edges, then if to the larger part one adds half of the entire proportionally divided line, then it will necessarily turn out that the square of the sum will always be five times, that is, 5 times greater than the square of the indicated half.” He accompanies all these properties with the same numerical example, when the length of the whole segment is 10, and its parts are: the smaller 15 is 125, and the larger 125 is 5. The example with algebraic division 10 in the average and extreme ratio was borrowed by LUCA PACIOLI from LEONARDO OF PISA (1180–1240), and the last - from ABU KAMIL (850–930) and AL-KHWAREZMI (787–850). The calculation of the roots of the corresponding quadratic equation itself is not performed in the treatise: here LUKA refers to his own Summa, where this result was obtained “according to the rules of algebra and almukabala.” And in general, the genre of message he chose is predetermined by the fact that PACIOLI gives all the results without proof, although this evidence is undoubtedly known to him.

Following this, PACIOLI considers the five Platonic solids. First, he proves the theorem that there are exactly five of these bodies, and no more. Then he gives the construction of all five bodies inscribed in a given sphere in the following order: tetrahedron, cube, octahedron, icosahedron, dodecahedron. Next, the proportion between the sides of these bodies inscribed in the same sphere is considered, and a number of theorems are given about the relationships between their surfaces.

It then considers some of the ways in which one regular body can be inscribed in another. Finally, the theorem is discussed that a sphere can also be inscribed in every regular body.

Now PACIOLI leaves EUCLID for a while and moves on to new material. Namely, he considers bodies that can be obtained from regular bodies by “truncation” or “superstructure”. Bodies that are obtained from regular bodies by truncation are

LUCA PACIOLI AND HIS TREATISE “ON THE DIVINE PROPORTION” 6

PACIOLI's superstructured regular and superstructured truncated bodies are not the same as the KEPPLER stellate polyhedra studied in subsequent mathematics. KEPLER solids are obtained by extending the planes of the original polyhedra; PACIOLI bodies - by constructing on each face of the original polyhedron a pyramid, the sides of which are equilateral triangles. PACIOLI gives an interesting theorem that in a superstructured icosidodecahedron the five vertices of the triangular pyramids and the vertex of the pentagonal pyramid lie in the same plane; the omitted proof is “raised by the subtlest practice of algebra and almukabala to a rare point.”

Next, we consider the “body with 72 bases,” which EUCLID used as an auxiliary in the last two sentences of the XII book of the Elements; this body is sometimes called the “CAMPANO sphere” in the literature (Fig. 2). PACIOLI claims that the shape of this body served as the geometric basis for the dome of the Pantheon in Rome and for the vaults of a number of other buildings.

–  –  –

Following this, PACIOLI says that by truncation and superstructure an infinite number of polyhedral forms can be obtained, and proceeds to consider the sphere, once again touching on the inscription of regular bodies into it.

LUCA PACIOLI AND HIS TREATISE “ON THE DIVINE PROPORTION” 7

PACIOLI further writes that the handwritten copies of the treatise given to the Duke and his relatives are accompanied by tables with perspective drawings made by LEONARDO DA VINCI, as well as “material forms” of all the bodies mentioned in it. The designs and shapes of the polyhedrons were made in two versions - solid, with solid flat edges, and hollow, with only edges. Whether LEONARDO made his drawings purely by calculation or from life, we do not know. Some of the drawings were made with an error noticeable to the eye, but this can be explained both by the inaccuracy of the calculations and by a change in the point from which the depicted body was viewed. The message ends with a glossary, which once again explains the special terms used in the text.

The golden ratio in “ancient” and “new” aesthetics Numerous popular and specialized books and articles devoted to the problem of proportions in art consider the golden ratio as the “most perfect”

proportions, and this perfection is interpreted in these books mainly psychologically: a rectangle with a “golden” aspect ratio is considered the most pleasant for visual perception, etc. In these publications it is customary to consider various works of fine art and architectural monuments created by masters of antiquity and the Renaissance as examples confirming this thesis.

It should be noted that not a single text has reached us from antiquity in which the division of magnitude in the average and extreme ratio would be discussed as a formative principle in the fine arts and architecture. It seems that such texts did not exist at all. For comparison, we can consider the so-called musical proportion 12: 9 = 8: 6, which sets the structure musical harmony. This proportion, discovered by the Pythagoreans, is mentioned in dozens of ancient texts devoted to the theory of music, both special and general philosophical. It would be strange if the golden ratio played a similar role in architecture, sculpture and painting, but ancient authors did not have any evidence of this.

All ancient texts that discuss the division of magnitude in average and extreme ratios are purely mathematical treatises in which this construction is considered solely in connection with the construction of a regular pentagon, as well as two regular Platonic solids - the icosahedron and the dodecahedron (for a review of these texts, see

HERZ-FISHLER 1998). It is true that the interest in regular bodies, and thereby in the golden ratio, was not purely mathematical: after all, PLATO, following the Pythagoreans, began to consider the five regular bodies as the elementary foundations of the universe, putting the tetrahedron in correspondence with fire, the cube with earth, the octahedron with air, the icosahedron - water, and he associated the shape of the dodecahedron with the Universe as a whole. In this regard, of course, we can talk about the aesthetic significance of the golden ratio, as A.F. LOSEV did in his writings;

but this “aesthetics” itself is not at all psychological, but cosmological in nature.

LUCA PACIOLI AND HIS TREATISE “ON THE DIVINE PROPORTION” 8

Exactly the same views are characteristic of JOHANN KEPLER and other Renaissance authors who were interested in the golden ratio and the role of regular polyhedra in the “harmony of the world.” So, looking in their writings for some concept of the golden ratio associated with the aesthetics of works of art is a completely futile exercise, since it simply wasn’t there.

The fate of Pacioli's works. Question about plagiarism After PACIOLI's death, his writings were not remembered for too long. An era of grandiose scientific achievements was approaching, when new results began to be valued primarily in science, and PACIOLI’s books were reviews of what had been done in earlier times. GIROLAMO CARDANO (1501–1576) called PACIOLI a compiler, in which, from his point of view, he was quite right. However, another outstanding mathematician of this era, RAFAEL BOMBELLI (1526–1573), said that PACIOLI was the first after LEONARDO OF PISA “who shed light on the science of algebra.”

The revival of interest in the personality and writings of PACIOLI dates back to 1869, when the Summa fell into the hands of the Milanese professor of mathematics LUCINI, and he discovered in it a Treatise on Accounts and Records. After this discovery, PACIOLI began to be looked at as the founder of the science of accounting, and it was this treatise that turned out to be the most sought-after part of his legacy, translated many times into other languages, including Russian.

However, soon after the first publications of the Treatise on Accounts and Records, heated debates broke out among researchers about whether LUCA PACIOLI was its actual author. It was doubted whether a person far from commercial affairs could have compiled such a treatise. And if he couldn’t, then shouldn’t we assume that plagiarism was committed here?

It still seems that the accusation of plagiarism in this case is unjustified.

PACIOLI never says that he invented double-entry bookkeeping; he only describes its norms “according to Venetian custom.” But if we open any modern accounting manual, it will be exactly the same normative description, without references to its predecessors. And if PACIOLI describes the accounting system based on some manuscript he read, then he didn’t come up with the rules for multiplication by column either, but in this case no one can accuse him of plagiarism

LUCA PACIOLI AND HIS TREATISE “ON THE DIVINE PROPORTION” 9

Another serious accusation of plagiarism was brought against PACIOLI back in 1550, when GIORGE VASARI (1511–1572) in his book Lives of Famous Painters, Sculptors and Architects, in the chapter dedicated to PIERO DELLA FRANCESCA, wrote the following:

And although the one who should have tried with all his might to increase his glory and fame, for he had learned everything he knew from him, tried, like a villain and a wicked man, to destroy the name of PIERO, his mentor, and to seize for himself the honors that should have belonged to PIERO alone, releasing under his own name, namely brother LUCA of Borgo, all the works of this venerable old man.

The mathematical works of PIERO DELLA FRANCESCA were considered lost for a long time. However, in 1903 J. PITTARELLI discovered in the Vatican Library the manuscript of Petri Pictoris Burgensis de quinque corporibus regularibus (“PETER, artist from Borgo, on the five regular bodies”). Somewhat later, two more manuscripts by PIERO were discovered: Perspective in painting (De perspectiva pingendi) and On the abacus (De abaco). At the same time, it was established that the found Latin manuscript On the Five Regular Bodies and three Italian treatises on regular bodies in the printed edition of De Divina Proportione were two close versions of the same text.

The surviving handwritten book by PIERO On the Five Regular Solids is dedicated to GUIDO UBALDO DE MONTEFELTRO, Duke of Urbino. He received the ducal title in 1482 after the death of his father. PIERO died in 1492. Therefore, the copy of the book that has reached us was completely rewritten in the interval between 1482–1492. However, the book itself could have been created earlier. LUCA PACIOLI in the Summa (VI, I, II) says that PIERO wrote the book on perspective in Italian, and the Latin translation was done by his friend MATTEO DAL BORGO. In the same way, the Latin text of the book On the Five Regular Bodies could have been born. In any case, it is natural to regard the Italian text subsequently published by PACIOLI as the original one.

As for this publication in the appendix to the edition of the Divine Proportion, its full title is as follows: Libellus in tres partialis tractatus divisus quinque corpore regularium e dependentium active per scrutationis. D. Petro Soderino principi perpetuo populi florentinia. M. Luca Paciolo, Burgense Minoritano particulariter dicatus, feliciter incipit (“A book, divided into three separate treatises, on the five regular and dependent [on] bodies, successively considered. To Mr. PETER SODERINI, permanent leader of the Florentine people. M[ aestro] LUCA PACIOLI, minor from Borgo, dictated in parts, begins happily”).

This title really says nothing about any relation of PIERO DELLA FRANCESCA to the treatise. But PACIOLI also designates his own “authorship” in a very strange way. Namely, he says that this book was dictated to him particularlyiter dicatus, “in parts (or in part?)” - and nothing more.

It makes you think. After all, LUCA PACIOLI in his writings does not at all look like a person who sought to shamelessly appropriate other people's results.

So in Section I of Chapter I of the Summa he writes:

LUCA PACIOLI AND HIS TREATISE “ON THE DIVINE PROPORTION” 10

A similar notice is found in Chapter IV of the Divine Proportion:

First of all, I will notice that whenever I write “the first in the first,” “the fourth in the second,” “the tenth in the fifth,” “20 in 6,” and so on until the fifteenth, the first digit should always be understood as the number of the sentence, and under the second - the number of the book of our philosopher EUCLID, who is recognized by everyone as the head of this faculty. Thus, speaking about the fifth in the first, I am talking about the fifth sentence of his first book, and also about the other separate books that make up a whole book about the elements and principles of Arithmetic and Geometry. But when another work of his or a book by another author is mentioned, this work or this author is called by name.

We should not forget that during those periods when LUKA lived in his hometown, he had the opportunity to communicate with PIERO directly. It is natural to think that the meetings of the two mathematicians were quite frequent, and their communication was meaningful. The themes of the book About the Five Regular Bodies were almost certainly discussed in these conversations, and therefore both of them could to some extent look at it as their own, regardless of who gave it its final form.

We also know nothing about the influence of the work of the German astronomer and mathematician JOHANN MÜLLER (1436–1476), better known by his Latin name REGIOMONTANUS, on PIERO DELLA FRANCESCA and LUCA PACIOLI. But he lived a lot in Italy and died in Rome, so Italian mathematicians could be familiar with him and his manuscripts. Among his works was the treatise De quinque corporibus aequilateris, quae vulgo regularia nuncupantur, quae videlicet eorum locum impleant naturalem et quae non contra commentatorem Aristotelis Averroem (“On the five equilateral bodies, usually called regular, namely, which of them fill the natural place, and which are not, against AVERROES, commentator on ARISTOTLE").

It has not survived to this day, but REGIOMONTANUS gives an overview of it in another of his works. This treatise examined the construction of regular bodies, their transformations into each other, and their volumes were calculated. It also contained the idea found in PACIOLI that by successively changing regular bodies one can obtain an unlimited number of semi-regular ones.

Further, the first printed book on mathematics was published in 1475. PIERO DELLA FRANCESCA still lived in the world of manuscripts, and the younger LUCA PACIOLI spent his mature years in the world of printed books. The manuscript could be rewritten for someone else's own use, but each time in one copy. Its copyist performs a godly deed simply because he prolongs the life of the manuscript and prevents it from perishing. The same is true when a surviving manuscript turns into a printed book.

We can now return to the question of plagiarism with an assessment more in keeping with the frame of reference of the time. It seems that in the era when PIERO DELLA FRANCESCA and LUCA PACIOLI lived, the question of authorship simply did not arise. (The Middle Ages, by the way, did not know authorship at all: can we say who was the “author” of the beautiful Gothic cathedrals? This very formulation of the question is obviously meaningless. - So in the Elements of EUCLID, most of the results were copied from other mathematical books, but For some reason we are not indignant at this and do not accuse EUCLID of plagiarism.) PIERO himself was interested in mathematics, and not in fame in the coming centuries. In the prelude of LUCA PACIOLI AND HIS TREATISE “ON THE DIVINE PROPORTION” 11 words to his Latin book, he writes that it will be his “pledge and monument,” but not among his descendants in general, but among his Ducal Highness.

As for authorship as an indication of who was the first to make such and such a discovery, the ontological moment is important here. A mathematician discovers some hitherto unknown bodies, and COLUMBUS at the same time discovers new countries. But COLUMBUS is not the “author” of these countries, and in the same way the mathematician is not the “author” of the bodies he discovered.

And after all, when COLUMBUS organized his expedition, his goal was the new countries themselves, and not the memory of descendants that he discovered them.

Luca Pacioli and the formation of the Institute of Expertise Addressing the message On Divine Proportion to the Duke of Milan LODOVICO SFORZA, LUCA PACIOLI nowhere recommends himself like this: “I am a mathematician because I can get new mathematical results" No, he talks about himself completely differently: “I am a mathematician because I know mathematics and can teach it to others.”

Here comes DANTE in Divine Comedy called ARISTOTLE “the teacher of those who know,” and LUKE quotes this quote for a reason. To clarify this argument, let us make the following comparison. The doctor knows medicine and therefore can treat. A lawyer knows the law and can therefore be a lawyer. But a mathematician knows mathematics - so what next? Can he teach her? But a doctor and a lawyer can also teach their sciences - which is why there are medical and law faculties at the university. But who can a mathematician be outside the field of study? What skill sets him apart from other people and makes him needed by someone? An astronomer can calculate the movements of celestial bodies and draw up horoscopes. An architect can build a beautiful villa, a military builder can build an impregnable fortress.

Artists create beautiful works that delight the eye. And a mathematician - what good can he be?

Let's see how LUKE himself answers this question. First of all, he insists that mathematics, as the most exact science, is the foundation and touchstone for all other sciences.

“In [our treatise] we speak of lofty and refined things, which truly serve as a test and test crucible for all refined sciences and disciplines: after all, from them flow all other speculative actions, scientific, practical and mechanical; and without prior acquaintance with them, it is impossible for a person to either know or act, as will be shown... As ARISTOTLE and AVERROES confirm, our mathematical sciences are the truest and stand on the first level of rigor, followed by the natural ones.”

From praising mathematics as such, he moves on to praising mathematicians:

“The prudent know the proverb: Aurum probatur igni et ingenium mathematicis. That is, gold is tested by fire, and the insight of the mind by mathematical disciplines. This statement tells you that the good mind of mathematicians is the most open to every science, because they are accustomed to the greatest abstraction and subtlety, since they have always considered what is beyond sensible matter. As the Tuscan proverb says, these are the ones who split the hair in flight” (Chapter II).

But in itself, “consideration of what is outside of sensible matter” is unlikely to interest the rulers to whom LUKE addresses.

Therefore, he moves from ideal things to real things, and argues that mathematics is a necessary foundation of military art and architecture:

LUCA PACIOLI AND HIS TREATISE “ON THE DIVINE PROPORTION” 12

“They call themselves architects, but I have never seen in their hands the outstanding book of our most worthy architect and great mathematician VITRIVUS, who compiled a treatise on architecture with best descriptions any building. And those whom I marvel at write on water and build on sand, hastily wasting their art: after all, they are architects only in name, for they do not know the difference between a point and a line and do not know the difference between angles, without which it is impossible to build well... However There are also those who admire our mathematical disciplines, introducing true guidance for all buildings in accordance with the writings of the above-mentioned VITROVIUS. The deviation from it is noticeable if you look at what our buildings are like, both church and secular: which are curved and which are skewed” (chap. XLIV).

In modern language, LUKA recommends himself to the Duke as an expert, and in matters not strictly mathematical (the Duke does not need such an expert at all), but purely applied ones, which are directly related to the preservation of power (military affairs) and prosperity (architecture). As for the ability to obtain new mathematical results, in this era it was not yet considered as a necessary distinctive quality of a high-class mathematician, remaining accidental and not an essential feature of the latter.

Literature GLUSHKOVA F. R., GLUSHKOV S. S. Geometric part of Pacioli’s “Summa”. History and methodology of natural sciences, 29, 1982, p. 57–63.

COLLINS R., RESTIVO S. Pirates and politicians in mathematics. Domestic notes, 2001, № 7.

OLSHKI L. History of scientific literature in new languages. In 3 volumes. M.–L.: GTTI, 1933–34. (Reprint: M.: MCIFI, 2000.) SOKOLOV Y. Luca Pacioli is a man and a thinker. In the book: PACCIOLI ONION. Treatise on accounts and records. M.: Statistics, 1974.

YUSHKEVICH A. P. History of mathematics in the Middle Ages. M.: Fizmatgiz, 1961.

ARRIGHI G. Piero della Francesca e Luca Pacioli. Rassegna della questione del plagio e nuove valutazioni. Atti della Fondazione Giorgio Ronchi, 23, 1968, p. 613–625.

BIAGIOLI M. The social status of Italian mathematicians, 1450–1600. History of Science, 27, 1989, p. 41–95.

BERTATO F. M. A obra “De Divina Proportione” (1509) de Fr Luca Pacioli. Anais do V Seminrio Nacional de Histria da Matemtica, Rio Claro, 2003.

BIGGIOGERO G. M. Luca Pacioli e la sua “Divina proportione”. Rendiconti dell"Istituto lombardo di scienze e lettere, 94, 1960, p. 3–30.

CASTRUCCI S. Luca Pacioli da 'l Borgo San Sepolcro. Alpignano: Tallone, 2003.

DAVIS M. D. Piero della Francesca’s mathematical treatises: The “Trattato d’abaco” and “Libellus de quinque corporibus regularibus”. Ravenna: Longo Editore, 1975.

FIELD J. V. Rediscovering the Archimedean polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Drer, Daniele Barbaro and Johannes Kepler. Archive for History of Exact Sciences, 50, 1997, p. 241–289.

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LUCAS DE BURGO. Summa de Arithmetica, Geometria, Proportione & Proportionalita. Venetia:

Paganino de Paganinis, 1494.

LUCAS DE BURGO. Divina Proportione. Venetia: Paganino de Paganinis, 1509.

MANCINI G. L’opera “De corporibus regularibus” di Pietro Franceschi detto Della Francesca usurpata da fra Luca Pacioli. Accademia dei Lincei, 1909.

MORISON S. Fra Luca Pacioli of Borgo San Sepolcro. New York, 1933.

PICUTTI E. Sui plagi matematici di frate Luca Pacioli. La Scienze, 246, 1989, p. 72–79.

PIERO DELLA FRANCESCA. Libellus de quinque corporibus regularibus. Eds. M. D. Emiliani e. a.

Florence: Giunti, 1995.

PITTARELLI G. Luca Pacioli usurp per se stesso qualche libro di Piero de’ Franceschi? Atti IV Congresso internazionale dei matematici, Roma, 6–11 April 1908, III. Rome, 1909, p. 436–440.

PORTOGHESI P. Luca Pacioli e la “Divina Proportione”. In: Civilt delle machine, 1957, p. 22–28.

REGIOMONTANUS. Commensorator. Ed. Blaschke W., Schoppe G. Wiesbaden: Verlag der Akademie der Wissenschaften und der Literatur in Mainz, 1956.

RICCI I. D. Luca Pacioli, l'uomo e lo scienziato. Sansepolcro, 1940.

ROSE P. L. The Italian renaissance of mathematics. Geneva: Librairie Droz, 1975.

SPEZIALI P. Luca Pacioli et son oeuvre. Sciences of the Renaissance, Paris, 1973, p. 93–106.

TAYLOR R. E. No royal road: Luca Pacioli and his times. Chapel Hill: Univ. of North Carolina Press, 1942.

“Beauty is a certain agreement and consonance of parts in that of which they are parts”

Leon Battista Alberti
(mathematician, painter, musician, poet, public figure, great architect of the Renaissance)

1.
Beauty and harmony of the world.
Man not only finds them in nature or intuitively generates them in his creativity. He tries to comprehend their innermost secret, as the basis of the universe, in order to understand them more subtly and recreate them more accurately.

When interest in this mystery unites great people, moreover, at a glorious time in a wonderful place, then their creative community itself already reveals beauty and harmony. Its fruits are amazing.

It is possible that this has happened more than once in history, but there is such a thing.

2.
During the Renaissance, in the rich Duchy of Milan, a meeting took place between two great people - the mathematician Luca Pacioli and the creator and inventor Leonardo da Vinci.

Luke had a deep sense of beauty. At the same time, he was “in love with numbers” and gravitated towards ONE area - mathematics, considering it a unique key to truth and beauty, becoming a luminary in it. He considered it his mission to give practitioners in various fields of activity useful techniques and tools of mathematics.

Leonardo had the most powerful creative intuition, imagination and ingenuity, finding the use of his rich talents in a VARIETY of areas of practice and art. He shone with his creativity and ingenuity, constantly striving to find new, original, large-scale solutions and discoveries. For this, Leonardo resorted to versatile and subtle observations of life and the possibilities of science, including mathematics.

The partnership between Luca and Leonardo did not last long, about 4 years, but left both with grateful memories for the rest of their lives.

3.
It was the glorious era of the Renaissance, the era of the most powerful large-scale human creative explosion, which had two sides to its coin.

On the one hand, the sciences and arts were actively developing, humanism was flourishing: man, his capabilities and talents were put at the forefront. The Renaissance gave birth to talented, erudite and specialized people who strived to live in wealth in the broadest sense of the word. At that time, major geographical discoveries were taking place (Columbus, Magellan, Vespucci, da Gamma), interest in the beauty of the human body increased, a new understanding of the cosmos (Copernicus), the universe and society (Machiavelli, etc.) arose, a transition to manufacturing and capitalism was taking place, the ideals of antiquity were revived with its glorification of the harmonious person.

On the other hand, spiritual asceticism was leveled, the same one that, an era earlier, created the highest treasuries moral culture(John Climacus, Ephraim the Syrian, Isaac the Syrian, Andrei Kritsky, etc.). The Renaissance did not interfere with other morals. Deceptions, conspiracies on corpses, spells, murders (especially poisoning), demonology were widespread in a society that did not pay due attention to the moral side of life.

This situation, and not only in that era, pushed intelligent people to find the right harmony in their lives. Is it the power and beauty of creativity? Or in the right balance between the desire of human creativity for power, going beyond the given limits and small, but important, moral restrictions, which should not be exceeded?

We will pay attention to this side of the heroes later, as part of the narrative.

4.
The Duchy of Milan, in which Luca and Leonardo met, at that time (late 15th century), was the most economically powerful in Italy (especially after the death of the Duke of Florence in 1492 Lorenzo Medici, nicknamed "The Magnificent"). At that time, Italy was a collection of separate, disparate, sometimes at war with each other, states. Milan, in those years, was an active center of the financial and economic life of Italy, fashion, a center of gunsmiths and artisans. Unlike Florence, where the emphasis was on the arts and textiles, natural sciences, mathematics and engineering flourished more in the Duchy of Milan.

Lodovico Sforza il Moro ruled this duchy virtually from 1480, first working as regent for his weak-willed nephew, not interested in state affairs, Gian Galeazzo, the son of his eldest murdered brother Galeazzo Maria Sforza.

Ludovico Sforza was a luxurious, ambitious ruler who wanted to turn Milan into the best state in Italy.

He made a lot of efforts to take power into his own hands after the death of his brother. He managed to remove his brother’s wife from her, Bona of Savoy, a prominent, kind, but not smart woman, and instead became regent for her minor son Gian Galeazzo.

Uncle led a cunning policy. Outwardly, and very luxuriously, all honors were given to the nominal Duke Gian, but all decisions of national importance were made by Lodovico. The uncle enjoyed great confidence from his nephew. He created a life of pleasure for the young Duke, took him away from education, gave freedom to his vices, deposing him morally and taking him away from business. When Gian Galleazzo was no longer needed, he soon died unexpectedly at the age of 25. There were rumors that his uncle had a hand in this, but his alibi was “cast”: he was not in Milan at the time of his death. One way or another, but since 1494, Lodovico Sforza il Moro became the legitimate seventh Duke of Milan.

Lodovico earned the nickname il Moro for two reasons. Moro meant Moor. That's what they called him because of his dark complexion. But this is not the main meaning. Moro also means mulberry tree as a sign of valor and prudence. The mulberry tree is the last to leaf and the first to bear fruit. Lodovico was proud of this nickname. The Moor's head and the alkali tree were depicted on his coat of arms. In addition, he had a servant - a real Moor.

Lodovico came from the young Sforza family (Sforza means "Strong" in Italian). His grandfather, the founder of the dynasty, a hired warrior (condottiere) from the age of 15 Muzio (full name Giacomuzzo Attondole) earned this epithet for his enormous physical strength: he unbent horseshoes with his hands. Lodovico's father Francesco Sforza was just as strong, bending iron bars with his fingers. Francesco married for the second time the illegitimate daughter of Filippo Visconti Maria Bianca, who had no male heirs. So the fading ancient Visconti family passed the baton to the young Sforza family as the rulers of Milan. This is a significant role for the valiant and talented Francesco Sforza.

Francesco, Lodovico's father, was a valiant, strong warrior and achieved the rank of general in military service. Later, during the period of his government, he achieved significant political and economic successes through the balance (that same harmony) of force and diplomatic methods of management. He also almost rebuilt the monumental architectural structure of the Castello Sforzesco (Sforza Castle), which became the residence of the Sforzes clan. Frescoes and paintings inside the castle were then done by Leonardo da Vinci. By the way, the Italian architects who built the Moscow Red Kremlin took the Castello Sforzesco as the basis for the project.

Lodovico, unlike his father, was born a sickly child (one of Francesco’s 8 legitimate children, there were even more illegitimate children). Francesco's children from Maria Bianchi did not take after him in valor and strength, but were similar to their mother, inheriting the characteristic features of Visconti: cunning, subtlety, grace, etc. Lodovico experienced quite strong religious feelings, and also showed respect, respect, and kind feelings for father and mother.

Lodovico was cunning, perspicacious, although somewhat straightforward, in state affairs. He understood things and was not indifferent to beautiful and intelligent women. Like many other influential people of that time, he had favorites, the mothers of his bastards (illegitimate children). Lodovico generously rewarded and patronized his women. For example, after parting with one of them - Cecilia Gallerani (her portrait can be seen in Leonardo da Vinci's painting "Lady with an Ermine" (1489-1490) - he married her to Count Bergamino and gave her one of the castles. Another favorite is Lodovico – Lucrezia Crivelli (depicted in da Vinci’s painting “The Beautiful Ferroniere” (1496) - was revered as one of the most beautiful, whose beauty Leonardo sincerely admired.

Lodovico was married (since 1490) to one of the most beautiful women of the Renaissance - the cheerful, energetic, intelligent and educated Beatrice d'Este, daughter of the ruler of Ferrara. Among other things, she was morally stable and did not cheat on her husband.

Sforza loved his wife very much, showed her respect, gave her tenderness, attention, and luxurious gifts. The spouses were close in worldview. Beatrice was a valuable and intelligent comrade-in-arms for him, and sometimes even an interpreter who helped in state affairs and decisions (for she paid attention to significant little things that Lodovico could not pay attention to).

Lodovico was 23 years older than his wife (his parents had a similar age proportion). She bore him two sons, boys, Massimiliano and Francesco. She was expecting the birth of her third, but at the very beginning of January 1497, having given birth to a stillborn baby, she died. She was only 21 years old.

Lodovico's grief knew no bounds. No words can describe the Duke's emotional loss and condition! Black drape on all the windows of the Castello, lying, for two weeks, in his chambers without the powers of Sforza. Every night he woke up, put on a dark cloak and came to his wife’s grave. While she was alive and well, he prayed to the Lord to grant him to die first, because his wife was so young! After her death, he prayed to the Higher Powers to be able to communicate with her spirit. Historians suggest that if Beatrice had remained alive, Lodovico would not have suffered the same fate that happened to him. But more on that later.

5.
Let's return to Pacioli and da Vinci.

In 1496, Luca Pacioli was invited to Milan, to the department of mathematics at the University of Pavia, by the Duke of Milan, Lodovico Sforza il Moro. He was then 51 years old. In the same city, 44-year-old Leonardo da Vinci served in the guild of engineers, who arrived in Milan much earlier, in 1482.

Why did Sforza invite the mathematician Luca Pacioli to his court?

In 1494, Luca Pacioli published in Venice, in the printing house of Paganino Paganini, his most famous work, on which he worked long years: Summa de arithmetica, geometria, proportioni et proportionalita “Body of knowledge in arithmetic, geometry, proportions and proportionality” (briefly “Sum”).

It was a real useful encyclopedia of applied mathematical knowledge on various topics. The book was dedicated (as was customary according to the canons of that time) to an influential person - the Duke of Umbria Guidobaldo Montefeltro, who at one time studied mathematics with Pacioli.

The Summa was written not in Latin (as was customary in those years for scientific publications), but in the native Italian language. It was the language of practitioners, traders, to whom the book was addressed (Pacioli in his youth lived with the Venetian merchant Rompiasi, taught his three children mathematics; in the early 70s, Luca himself did a little trading, but to no avail). In the “Summa” there was a part “Treatise on Accounts and Records”, devoted to the systematization of knowledge on accounting, double entry, financial statements. Luca Pacioli owes this part of the book the honorary title of “father of the founder of modern accounting,” which his descendants gave him. And writing it in Italian perpetuated the basic terms of accounting in it: debit, credit, balance, subconto.

“Summa” was very popular in Italy and abroad, and the author was also known as an excellent teacher. More on this talent of Pacioli later.

Leonardo da Vinci read this book before meeting Pacioli, but did not know the author. Moreover, before reading the Summa, Leonardo, who was fond of mathematics, had the idea of ​​writing his own work on geometry, but after reading it he realized that he could not write better, and it was not worth it.

I knew about this book and its author and Lodovico Sforza. He wanted to invite Luca to his place, finding a way to interest him: by giving him the chair of mathematics at the prestigious University of Pavia, the opportunity to engage in science, research, teaching, giving free time for writing books.

Luke gratefully accepted the Duke's offer.

6.
Lodovico had an excellent ability to attract talented and necessary people to his service, choosing the best and knowing how to interest them. Many famous people of that era served at his court (Bramanto, Fidelfo, Castaldi, Tsaroto, etc.). Sforza knew how to competently manage creative people. Another great man - Leonardo da Vinci - was not at all easy to manage: ambitious, wayward, freedom-loving. However, Lodovico found an approach to him, giving him interesting and varied orders and resolving creative conflicts that arose.

Leonardo worked for Sforza for almost 17 years, and would have worked longer if not for the height of the Italian wars.

An ambitious ruler and an ambitious creator seem to have found each other! Harmony?

The first Milan period of Leonardo da Vinci's work at the court of the Duke of Sforza was one of the most productive and best in the life of the great Leonardo and in terms of the quality of his creations (for example, “Madonna Litta”, “Madonna of the Rocks”, “Madonna in the Grotto”, “Vitruvian Man” , the grandiose “Last Supper”, projects of an ideal city, an aircraft, a light bridge, a colossal equestrian monument to Francesco Sforza and much more) and by the number of it creative manifestations(musician, poet, writer, architect and sculptor, land reclamation engineer, cook, chess player, organizer of court balls and celebrations, painter, inventor and innovator).

7.
Leonardo began attending amazing lectures on mathematics by Luca Pacioli, admiring his talent as a teacher and the breadth of his mathematical erudition. Leonardo did not make friends with every person; he liked extraordinary, large-scale and competent people, which was Pacioli. In Da Vinci’s notebook from those years there is an entry: “Learn to multiply roots from Maestro Luca.” Or in another: “learn about the measure of scales from brother Luke.”

Luka showed a high level of excellence in teaching mathematics. He knew the subject deeply and thoroughly and was an expert in it. Pacioli looked the part. This is how Albert Dupont described him: “A handsome, energetic young man; raised and rather broad shoulders reveal innate physical strength, a powerful neck and developed jaw, an expressive face and eyes that radiate nobility and intelligence emphasize strength of character. Such a teacher could make you listen to yourself and respect your subject.”

In addition, Pacioli was polite and pleasant in communication (a quality that helped him not only in teaching, but also in communicating with influential persons and friends, of whom he had many and with whom he enjoyed success and patronage).

Pacioli's approach to teaching was based on a deductive principle - from complex to simple: first he explained the most difficult example, then simple ones were solved much easier. Pacioli formulated this approach (learning principle): “He who has not tasted the bitter first does not deserve the sweet.”

Luca Pacioli had a strong character. In 1477, at the age of 32, he entered monasticism. For the time when the morals described above were in use, this was a feat. Entering monasticism (now under the name Fra Luca of Borgo), Pacioli made three main vows: obedience, chastity, and non-covetousness. In 1486 he also became a doctor of theology (theology). But Luke did not at all abandon his calling - mathematics, but, on the contrary, in its name, he became a wandering mathematician monk. Monasticism allowed Fra Luca to do what he loved, and through this, serve God with his talent, and pass on mathematical knowledge that was useful to interested people. He did what he loved, not caring how much he would earn from it. This showed the orientation of the Franciscan Order of Minorites: not to run away from life, but to live in it, to show their talents to please God, but also to accept useful renunciations in order to avoid unnecessary temptations. By the way, for the same reason many creative people came to this order. Another example in history is the composer Franz Liszt.

Luca Pacioli, as a mathematician, was paid well for his lectures and his salary was constantly increased. He was quite popular. Loyalty to his vows allowed him not to fall into the greed of earning money, but to enjoy the process of science and teaching and develop in them. He tried not to “stay” too long in one place: one of the ways to stay on his toes, avoiding familiarity, and also to expand the reach of his audience. So he worked as a mathematician in Perugia, Zara (Croatia), Rome, Naples, and Venice. Isn't this one of the examples of a truly harmonious Renaissance man?

As a parallel, we note that Leonardo da Vinci did not take monasticism and did not take vows, but observed the canons of correct life in the high secular society of Milan. At one time, through Cecilia Gallerani (a favorite of Sforza, a person of excellent spiritual qualities and intelligence, who was a close friend of Leonardo, wrote poetry and led readings in her literary club), he met representatives of the Milanese elite and learned how to behave.

Leonardo, being an outwardly sociable person, an excellent storyteller and fabulist, who knew how to start and maintain a conversation on any topic, doing it with ease and humor, was at the same time secretive and careful in his communication. He never wrote openly or talked about three important things: his personal life, the history of his inventions and what others should not know. He had this to say notebook, in which he kept records in encrypted form, many of which have not yet been deciphered. Leonardo kept the necessary distance from people.

As a finishing touch, he was a vegetarian and avoided excesses in food (consider him observing informal fasts).

Leonardo’s approach to income was not like Luca’s, but entrepreneurial: he knew how to offer, “sell” himself as a master (which he successfully did in 1482 in relation to Il Moro, having arrived from Florence to Milan), worked for those who paid the most, and in the specialty for which they pay more. It was very much in the spirit of the Renaissance. Creative people often worked not out of disinterested inspiration, but on well-paid orders. But there were also plenty of orders, different and interesting! Patronage was also highly respected.

8.
Leonardo da Vinci began to study mathematics with interest from Pacioli.

The great advantage of Leonardo himself is that he did not hesitate to learn new and necessary things at any age or status, and he did it easily, without hurting his pride.

But I had to study.

Leonardo did not have a systematic education (having studied in his early youth with the architect and painter Andrea del Verrocchio in Florence, and self-taught) and had many gaps in knowledge. His strong intuition, surpassing the capabilities of his era, required reliance on solid knowledge, which was not always the case.

For engineering work, as well as for pouring bronze into a wax sculpture of the colossal equestrian monument of Francesco Sforza (about 7 meters high), he needed knowledge of mathematics. Luca Pacioli became the person who helped him in the calculations of materials for the statue, as well as engineering design for the creation of water utilities.

And the Duke of Sforza was demanding of the people who worked for him. What they did had to be done with high quality, gracefully, luxuriously down to the smallest detail. Lodovico, and especially Beatrice, were very scrupulous about the quality of the work of the people who served them.

9.
In those Milanese years, Luca Pacioli had already begun to write his other monumental work entitled “De Divina Proportione” (On Divine Proportion). Many ideas were raised earlier when writing the Summa and were partially covered in it. The theme of Divine proportion, as a code of beauty and harmony, brought Luca and Leonardo even closer together.

In painting, which Leonardo considered the highest and primary of the arts (for it, like no other, allows one to immediately highlight all the beauty of the depicted object), he was interested, among others, in two main topics: the quality of the drawing lines (the technique of blurred lines, similar to , as the human eye perceives them) and a reflection of perspective and proportions. The second theme was close to Divine proportion.

Luca Pacioli studied at one time with such great masters of painting as the artist, mathematician and creator of the ideas of descriptive geometry Piero della Francesca (whom Luca enthusiastically called “The King of Painting”), mathematician, painter, writer, architect, architect Leon Battista Alberti (who, in addition to education, he helped young Luka in relations with many influential people and patrons). Pacioli studied painting, but did not become an artist. Knowledge of it helped him in a deeper understanding of geometry and, of course, beauty and harmony.

The third significant person in this area was Leonardo da Vinci for Pacioli. But it was no longer a partnership between teacher and student, as before, but two creative friends, full of ideas and plans.

While Pacioli gave lectures on mathematics in Pavia, wrote his work “On Divine Proportion”, translated Euclid’s “Elements”, Leonardo painted the “Last Supper”, monumental in beauty and harmony, in the refectory of the monastery of Santa Maria della Grazia, wrote several treatises in parallel , carried out Sforza's engineering tasks and prepared the colossal equestrian statue of Francesco for pouring bronze.

Leonardo and Luke had deep and interesting conversations on the topic of Divine proportion, in which insights of extraordinary power and beauty were born.

Leonardo, at the request of Pacioli, also completed 60 color drawings for the treatise in the stereometry of regular and semiregular polyhedra. He did it, as Luke wrote about it in his treatise, “with his divine left hand” (da Vinci knew how to write and draw with both hands, and from left to right, and back, and in tone with the mirror image; he performed especially creative work with his left).

Leonardo drew polyhedra without calculations or compasses, and at the same time beautifully, harmoniously and accurately. Luka then, until his death, carefully kept a copy of the drawings. Pacioli himself made models of regular polyhedra based on them.

Finished copies of the manuscript with drawings and models were presented to influential people in Milan (as was expected according to the rules of that time).

The voluminous handwritten treatise “De Divina Proportione” in 3 parts (on Divine proportion, on regular polyhedra, on architecture) was completed in December 1498 and dedicated to the Duke of Milan, Lodovico Sforza il Moro. It was printed in Venice, in the printing house of the same Paganino Paganini, only 11 years later, in 1509.

10.
In conclusion, a few words on the topic of Divine proportion itself, for this story began with words about the beauty and harmony of the world, as the mystery of the universe.

Luca Pacioli (or Fra Luca of Borgo) called the divine proportion what in the modern world is called the “golden ratio”. The last name was given to it in 1835 by the German mathematician Martin Ohm, brother of the famous physicist Georg Ohm. The topic has attracted many people in history, dating back to the times of Ancient Babylon and Egypt.

The “Golden Ratio” or “Divine Proportion” is understood as one of the secrets of the universe, a kind of universal and unique code of beauty and harmony. This is a combination of parts of a whole, perceived as the best (most beautiful) for a person’s aesthetic perception; when the smaller part relates to the larger as the larger part relates to the whole. It is described by the irrational number Phi (in honor of the ancient Greek architect Fideus) and is also called the number of God: 1.6180.... In percentage terms, conventionally, these are 62 and 38 percent.

The proportion of the “golden section” (or Divine proportion) is seen as universal, characteristic of most forms of natural objects (the proportions of the body and tail of a lizard, the human body (Vitruvius, da Vinci, Dürer, Zeising studied in more detail), a chicken egg, a snail’s spiral and a DNA molecule, the arrangement of leaves on a chicory branch, etc.), and the outstanding achievements of human creativity (in architecture and architecture, literature, painting, music, cinema, the geometry of beautiful polyhedra, etc.).

In his treatise “On Divine Proportion,” Luca Pacioli argued that this is the one and only proportion of beauty (as God is one and only) and there are no combinations better than it. That is why he spoke of her as Divine.

Luke proved the consequences of the theorem, revealing 13 properties of Divine proportion (the number 13 was chosen for a reason: 13 people sat at the table at the Last Supper).

He substantiated its use in architecture and architecture, spoke about it as the basis for the construction of regular geometric bodies (Plato’s 5 polyhedra, characterizing 5 cosmic elements: a pyramid (tetrahedron), consisting of 4 regular triangles - the element of fire, a cube (hexahedron), consisting of 6 squares - the element of earth, an octahedron consisting of 8 regular triangles - the element of air, an icosahedron consisting of 20 regular triangles - the element of water, a dodecahedron consisting of 12 regular pentagons - the element of ether or the Universe; and most of the 13 truncated polyhedra of Archimedes).

Pacioli turned as sources to the geometry of Euclid (book of the Elements), and to the works of Pythagoras, and to Plato’s Timaeus, and to the numbers and problems of Fibonacci, presented in his book of Abacus (counting board), and to Vitruvius, and to Alberti's works on architecture, which reveal the meaning and possibilities of Divine proportion.

Essentially, the work “De Divina Proportione” was an enthusiastic hymn to the “golden ratio”, written in the style of early Renaissance mathematics (somewhat complicated, sometimes mystical rather than logical). But it was an important encyclopedia of mathematical knowledge on beauty and harmony. A direction that would later be called “mathematics of aesthetics.” It was completed by Pacioli during a difficult historical period.

It was the height of the Italian wars, a troubled time, and people had no time for beauty and its universal codes. Any war (this is always its negative role) sometimes reduces people’s motives to primitive ones: to survive...

Only descendants, much later, appreciated this work of Fra Luca from Borgo.

11.
In 1499 Milan was captured by the French. Lodovico did not take into account the superior forces of the French king Louis XII. Sforza fled Milan, gathered an army of Swiss mercenaries and tried to recapture the city, but was defeated at Novara. The Swiss, for the right to their freedom, handed Lodovico over to the French. The Duke of Sforza was sent to prison in the ominous castle of Loches in the south of France and spent almost 8 years there. During the period of Sforza's defeat, Leonardo wrote in his diary: “The Duke lost his state, property, freedom, and not a single one of his affairs was completed by him.” Many of Leonardo’s own initiatives also turned out to be unfinished. The great colossal statue of Francesco Sforza, on which Leonardo worked for so long, was never cast in bronze (because it went into service), and its wax model was mutilated and destroyed by French riflemen.

The French king Louis XII treated Ludovico Sforza harshly and mercilessly, depriving him of everything he had and sending him to prison. As historians testify, one of the last words of this largely talented man, inscribed on the walls of his dark prison cell, was “Infelix sum” (“I am unhappy”; lat).

Sforza died in custody at the age of 55. Probably, being a gifted, perspicacious, and sometimes tough tactician, he was not so far-sighted and elegant in strategy. Being the initiator of the French coming to Italy to unite with them against Naples and Florence, he was defeated by them. Such mistakes are most often not forgiven by the powers that be.

12.
Luca and Leonardo successfully fled from Milan to Mantua, under the protection of the Marquise Isabella d'Este (married Gonzago), the elder sister of Lodovico's deceased wife Beatrice d'Este. She did not provide them with her permanent patronage, but offered them a short stay in Mantua. As a token of gratitude, Luca Pacioli, at the request of the Marchioness, wrote a treatise on chess for her in Latin (De Ludo Schacorum or Schifanoia"; On the Game of Chess or Banishing Boredom). Leonardo also proposed a number of entertaining tasks in it and completed all the drawings.

Isabella, Marchioness of Mantua, who loved to play chess, was presented with a treatise of 96 pages, with 114 entertaining chess problems, with drawings by Leonardo da Vinci (again, made by his “divine” left hand). The proportions of the chess pieces were executed by Leonardo according to the rules of the “golden section” (Divine proportion). Marquise Gonzago gratefully appreciated the gift.

Luca and Leonardo soon immigrated to Venice and then to Florence. Then their paths diverged and never crossed again, leaving only good, grateful memories of that Milan, the Sforza family, the mystery of Divine proportion, and each other.

*In the photo collage: against the background of the Castello Sforzesco (Sforzesco Castle) at the top left - Luca Pacioli, at the top right - Leonardo da Vinci, at the bottom left - five regular polyhedra of Plato, at the bottom right - the cover of the treatise “De Divina Proportione”.

**June 19, 2017 marked 500 years since the death of Luca Pacioli. He died and was buried in the same city where he was born - the Italian provincial Borgo San Sepolcro (city of the Holy Sepulchre).

During the Renaissance big interest Sculptors showed interest in the forms of regular polyhedra. architects, artists.
Leonardo da Vinci (1452 -1519), for example, was keen on the theory of polyhedra and often depicted them on his canvases. He illustrated Monk Luca Pacioli's book "On Divine Proportion" with regular and semi-regular polyhedra.
Pacioli was one of the largest European algebraists of the 15th century and, no less important, invented the principle of the so-called double entry, which is still used in all accounting systems without exception. So he can safely be called the “father of modern accounting.” However, the rather mysterious and controversial personality of Pacioli still causes fierce debate among historians of science.
In 1472 Pacioli under the name Fra Luca di Borgo San Sepolcro. In 1496 he was invited to give lectures in Milan, where he met Leonardo da Vinci. Leonardo, having read the Summa, abandoned work on his own book on geometry and began preparing illustrations for Pacioli’s new work, The Divine Proportion.
Leonardo da Vinci proposed an original way of spatially depicting a truncated icosahedron.
A reproduction of this beautiful image from the book illustrated by Leonardo by his contemporary, the Franciscan monk and mathematician Luca Pacioli (1445-1514) “The Divine Proportion” (“De Devina Proportione”), published in 1509, is shown in Fig. 1

Pacioli's book, for which Leonardo made 59 illustrations of various polyhedra, had a great influence on the development of geometry of that time, in particular, the stereometry of polyhedra.

It is no coincidence that Leonardo was involved in the study of the truncated icosahedron. Titan of the Renaissance, painter, sculptor, scientist and inventor Leonardo da Vinci (1452-1519) is a symbol of the inseparability of art and science, and therefore his interest in such beautiful things as convex polyhedra in general and the truncated icosahedron in particular is natural. Fig 2.

Leonardo precedes the engraving depicting a truncated icosahedron (Fig. 17) with the Latin inscription Ycocedron Abscisus (truncated icosahedron) Vacuus. The term Vacuus refers to the fact that the faces of the polyhedron are depicted as “empty” - not solid. Strictly speaking, edges are not depicted at all; they exist only in our imagination. But the edges of the polyhedron are not depicted geometric lines(which, as is known, have neither width nor thickness), but in rigid three-dimensional segments. Both of these features of this engraving form the basis of the method of spatial representation of polyhedra, invented by Leonardo to illustrate the book of Luca Pacioli and today called the method of rigid (or solid) edges. This technique allows the viewer, firstly, to accurately determine which of the edges belong to the front and which to the back faces of the polyhedron (which is practically impossible when depicting the edges with geometric lines), and, secondly, to look through the geometric body, to feel it in perspective, depth, which are lost when using the technique of solid edges. The technique developed by Leonardo is a brilliant example of geometric illustration, a new way of graphically representing scientific information. This technique was subsequently used many times by artists, sculptors and scientists.

G. Ya. Martynenko

Mathematics of Harmony: Renaissance (XIV–XVI centuries)
(to the 500th anniversary of Luca Pacioli’s book “On Divine Proportion”)

Divine proportion
Professor Fra Luca Bartolomeo de' Pacioli
The great dreamer of the wandering warehouse,
After wandering around and rubbing the calluses,
Got to Florence. Towards Leonardo

Da Vinci. Good God! That's the meeting.
Friends, hugging, almost strangled
Each other, but not to the point of injury.
Then they immediately got down to business.

Over a cup of wine, when it has cooled slightly,
Pacioli, drunk, told the sculptor,
What is in the wisdom of Euclid's Elements?

One proportion gave him strength.
Da Vinci beamed with an unusual smile:
“Look, my friend, she’s harmonious.”

Renaissance in the history of European culture is the era of transition from the Middle Ages to
new time, the era of a turn to living human thought, suppressed by asceticism
Middle Ages. This period is characterized by deep and fateful for Europe
processes: the agricultural revolution and the transition from craft to manufacture; great
geographical discoveries and the beginning of world trade. At this time feudal
fragmentation
inferior
centralized
authorities
And
are formed
modern
nation states. This era is associated with the beginning of printing, the “discovery”
antiquity, the flourishing of free thought, the emergence of Protestantism and the loss of the church
monopoly in spiritual life. At this time, natural science takes its first steps, blossoms
arts and literature, mathematics is developing rapidly.
The most common distinguishing feature of the Renaissance is the affirmation of the ideal
harmony of man and the integrity of the universe. Moreover, unlike the Middle Ages, these
categories, albeit not immediately, albeit evasively, began to be viewed as self-sufficient
essence, and not through the prism of the divine absolute. Related to this is the inherent culture
Renaissance secular and humanistic character and inclination towards cosmological and
natural philosophical vision of the world. An important role in this vision of the world was played by
mathematics, which liberated science and art from the shackles of medieval scholasticism and harsh
asceticism.
Unlike antiquity, Renaissance scientists did not shy away from purely practical
tasks. There were virtually no pure theoretical mathematicians. But even those who can
considered theorists, were engaged in astronomy, military affairs, anatomy, mechanics,
medicine, cartography, optics and other practical matters.

1. Ideas about harmony in the art of the Renaissance

During the Renaissance, the public authority of art sharply increases, but this is not
led to its opposition to science and craft, and was perceived as equal rights
various forms human activity in their unity. Compared to the Middle Ages
in art there is a sharp shift in emphasis. For a medieval man
the surrounding world is a mirror, drawings and statues in temples and manuscripts are also
mirrors; and even the encyclopedia of knowledge was then called “Speculum” (mirror). So what
reflected in these mirrors? According to medieval ideas, they reflected
perfection, some absolute, some boundless divine essence.

1.1. Art is like a mirror
During the Renaissance Holy Bible no longer considered a treasure trove
divine secrets, this is already a reflection of the real, real life and the existence of nature.
Leonardo da Vinci writes: “If you want to see whether your picture as a whole matches
an object copied from life, then take a mirror... On its surface things are similar
painting..." (Leonardo da Vinci, 1935, pp. 114–115). In other words, the painter must be
like a mirror to reflect the world around us, that is, as Leonardo da Vinci says
"You can't be a good painter unless you're a universal master."
in imitation by his art all the qualities of the forms produced by nature" (ibid.,
With. 88). Albrecht Durer also held similar views: “Our vision is like
mirror" (Dürer, 1957, p. 26). But we are talking here about visible arts. How can you be different?
types of art? That they have a mirror. And this is where the metaphors begin. Yes, George
Puttenham, in his book The Art of English Poetry (1859), writes: “The mind that has
imagination, like a mirror” (Gilbert, Kuhn, p. 182).
However, mysticism and the unquestioned authority of the church were not immediately supplanted
nature and reason. The Church retained its power over spiritual life for a long time.
life thinking people. At the same time, many “revivalists” saw a compromise in the fact that
theology is also poetry. So Petrarch wrote to his brother Gherardo: “Poetry is by no means
contradicts theology... We can say with some right that theology is the same as poetry, but
relating to God” (Gilbert, Kuhn, p. 186). According to Alberti and Leonardo da Vinci,
the artist must be a kind of priest, for piety and virtue
were considered then essential attributes artist. Art itself was considered
divine, and his role was primarily to inspire people with love and
worship of god.

1.2. Changes in the classification of arts
But in one area radical changes have occurred. It's about theology,
which began to increasingly penetrate the concepts and ideas characteristic of art.
There was a gradual erosion of theology against the background of the increasing role and prestige of art.
This was reflected in the fact that poetry, sculpture and painting began to refer to
categories of liberal arts. However, the secular trend in art made its way
the road extremely delicately, carefully, without “cavalry attacks”. It gained ground
thanks to the gradual intrusion into the sphere of the religious spirit of interest in science and
ancient heritage. Poets and artists understood that they had to tirelessly prove their
a place in the sun in the camp of liberal professions. And they did it through hard work,
perseverance, intelligence and “methodological mastery” characteristic of traditional
art that came from the Middle Ages. To raise their authority, artists and
poets worked tirelessly, because in the consciousness of man in the middle of the second millennium
the conviction was rooted that the more work put into creating a work, the
The more perfect it is, the more original and beautiful it is. Moreover, in disputes about what art -
painting or sculpture, painting or poetry is more important than the other, argument in favor
art that requires more labor, and visible, tangible labor, played
very important role.

1.3. The role of science and crafts

As they increased artistic skill, some figures
the arts of the Renaissance began to move away from strictly copying nature and tried
combine your artistic vision with the desire for ideal form and harmony
of your work. Nature has ceased to be just a “model”, a model for
copying, but turned into a source of hidden divine essence, which must
unravel.
But some artists and sculptors took a different path. They not only solved
but they also took it away. The basis of beauty is not so much the gift of God, but the choice of man,
which selects in nature the brightest options from the best and most beautiful forms.
“We must take the best features from many beautiful faces - such was the widespread
slogan" of the era (Gilbert, Kuhn, p. 205).
Another way was also popular. Based on a body of specialized knowledge in
areas of perspective, anatomy, mathematics, psychology, enhancing the senses,
Renaissance artists created a second “man-made” nature, but one that
corresponded to the plan of divine creation. In this case, the decisive role was played
mathematics. We will discuss its significance in the Renaissance in more detail below.



1.4. Attitude to harmony
If for a man of the Middle Ages harmony meant the maximum degree
following divine unity, then for Renaissance man harmony meant
complete correspondence of individual elements of a work of art to each other and
to the whole. In order to express the meaning of this correspondence, various
words and phrases: ratio, coordination, proportionality, agreement,
combination, consistency, proportionality, composition, arrangement, etc.
The concept of harmony ultimately finds for the Renaissance artist
embodiment in art of the project . This art is based on the study of many real
objects in order to create the perfect sample. This is how renaissance defines
project English art critic G. Vasari: “A project is like a form or idea of ​​all
things in nature; this is the most remarkable concept in its breadth, for not only on bodies
people and animals, but also on plants, buildings, sculpture and painting, the project shows
the relationship of the whole to the individual parts and each part to the other and to the whole... Of these
relationship, a certain concept and judgment arises” (Vasari, 1907, p. 205; cited in:
Gilbert, Kuhn, p. 207). And only after that the initial sketch or project
embodied in artistic reality.
The idea and practice of design goes back to the ideas of Vetruvius, who in his
projects were based on the proportions of the human body. Revivalists look at
the problem is broader. They take into account not only the proportions of the human body, but also any
proportions found in nature. But the great power of art often led artists away
from harmonic canons. For example, the American researcher J. writes about this.
Simon, discussing the work of Michelangelo, who often deviated in his work
on the proportions of the human body. Dürer thinks the same. In the third of his Four Books
about the proportions of the human body" he says that the artist has the power to deviate from the golden
middle towards big and small, thick and thin, young and old,
fat and thin, beautiful and ugly, hard and soft, but all this must be
subject to a consciously chosen method and art, which is firmly based on
nature and never repeats itself. For Dürer, a canon, a sample, a model, a project is not
dogma, but a guide to action for a free person with a “natural
a penchant for creativity.
So, in the art of the Renaissance there was a clear tendency towards the search for formal
regulators of the creative process. On the one hand, the criterion of truth becomes
a divine source, and on the other hand, mathematics begins to play a huge role. And this
The “combination” extended not only to art, but also to other areas
activities, primarily crafts and trade. So, a sailor who owns
mathematical skills, gained an advantage over his competitors thanks to
the ability to calculate the coordinates of a ship at sea, and a merchant proficient in accounting techniques
accounting, had a significantly greater chance of success in trading than his helpless in
math rivals. At the same time, traditional ideas argued that the universe
built by God according to a single plan, in which mathematics played an important role.
It is also noteworthy that during the Renaissance, harmonic representations
apply not only to nature and products of creative activity, but also to the entire
the range of human-nature interactions and human relationships. A striking example
such an expansive understanding of harmony is creativity Leona Battista
Alberti(1404 – 1472) - scientist, humanist, writer, one of the founders of the new
European architecture and leading theorist of Renaissance art.
Multi-talented and educated, he made a major contribution to the theory
art and architecture, literature and architecture, was interested in problems of ethics and
pedagogy, studied perspective theory, cartography and cryptography.
According to Alberti, harmony is the most important pattern of nature, the basis of world order.
A person included in the world order finds himself at the mercy of its laws - harmony and
perfection. The harmony of man and nature is determined by his ability to cognition
peace, to a rational existence striving for good.
Alberti created an original humanistic theory, going back to Plato and
Aristotle's concept of man based on the idea of ​​harmony. Alberti's ethics -
secular in nature - distinguished by its attention to the problem of man’s earthly existence, his
moral improvement.
The ideal person, according to Alberti, harmoniously combines the powers of reason and will,
creative activity and peace of mind. He is wise and guided in his actions
principles of measure, has a consciousness of his dignity. All this gives the image
created by Alberti, features of greatness.
Responsibility for moral improvement, which has both personal and
social significance, Alberti places on the people themselves. Choice between good and evil
depends on the free will of man. The humanist saw the main purpose of the individual in
creativity, which was understood broadly - from the work of a humble artisan to the heights of scientific
and artistic activities.
Alberti society thinks as a harmonious unity of all its layers, which
should facilitate the activities of rulers. Thinking through the conditions of achievement
social harmony, Alberti in his treatise “On Architecture” draws an ideal city,
beautiful in its rational layout and appearance of buildings, streets, squares. All
the human living environment is designed here so that it meets the needs of the individual,
families, society as a whole.
The embodiment of ideas about the ideal city in words or images was
one of the typical features of Renaissance culture in Italy. Projects of such cities
many paid tribute bright personalities this era. This is the architect Filaret, scientist and
artist Leonardo da Vinci, authors of social utopias of the 16th century. The latter reflected
dream
humanists
O
harmony
human
society,
O
external
conditions,
contributing to the stability and happiness of every person.

2. Mathematical studies
2.1. "Divine Proportion" by Luca Pacioli

In 1509, i.e. 500 years ago, on the advice of Leonardo da Vinci, Luca Pacioli published
book “On Divine Proportion” (“La Divina Proportione”) with the subtitle “Essay,
very useful to every insightful and inquisitive mind, from which each
student of philosophy, perspective, painting, sculpture, architecture, music or
other mathematical subjects, will learn the most pleasant, witty and amazing teaching
and will entertain himself with various questions of the most secret science.” The book explicitly stated
formulated law of the golden ratio. The book was elegant and knowledgeable
illustrated with images of polyhedra made by the great Leonardo. IN 2007
year, a Russian translation of “The Divine Proportion” appeared (Pacioli, 2007).
The Franciscan monk Luca Pacioli was a student of the artist Piero della Francesca,
who wrote two books, one of which was called “On Perspective in Painting.” This book
considered the forerunner of descriptive geometry. From the artist Pacioli received deep
knowledge of art and mathematics.
“La Divina Proportione” was a rapturous hymn to the golden proportion. Among
Monk Luca Pacioli did not fail to name many of the advantages of the golden ratio
“divine essence” as an expression of the divine trinity of god the son, god the father and god
holy spirit It was assumed that the small segment when dividing the segment into the extreme and middle
relation is the personification of the god of the son, the larger segment is the god of the father, and the entire segment is
god of the holy spirit.
The first part of “Divine Proportion” is devoted to the golden ratio, the second -
regular polyhedra, the third - architecture. Golden ratio and correct
Pacioli considers polyhedra in accordance with the XIV book of Euclid’s Elements.
Shortly before the publication of The Divine Proportion, Pacioli published an edited
Latin translation of “Beginnings” with its numerous commentaries.
Images of polyhedra on 59 tables made for his friend Leonardo da
Vinci, for whom Pacioli, for his part, calculated the amount of metal,
necessary for equestrian statue(Yushkevich, pp. 288–289). The book contains not only
five regular polyhedra (in full accordance with the Platonic solids), but also
polyhedra obtained from them by “cutting off” and “attaching” to each other. What
concerns the section devoted to architecture, then proportions are considered here
of the human body based on whole numbers in full accordance with Vetruvius' measurements.
“Divine proportion” for the mathematics of harmony is fundamental
meaning. It is interesting, however, that Pacioli considers divine proportion" with
cosmological positions in the Pythagorean-Platonic spirit, without tying it to
architecture, painting or any other art. This is evidenced by the fact that
Pacioli in his Treatise on Architecture, which forms the last part of the book, talks about gold
does not mention proportions. In other words, for Pacioli the golden ratio is, first of all,
Christianized mathematical-cosmic phenomenon.
Pacioli
glorious
Not
only
mathematical-harmonic
research.
His
Mathematical achievements in general are also of lasting importance.
In 1494, Pacioli published a mathematical work in Italian under the title
entitled "The Sum of Arithmetic, Geometry, Fractions, Proportions and Proportionality"
(Summa di arithmetica, geometrica, proportione et proportionalita). This essay outlines
rules and techniques for arithmetic operations on integers and fractions, problems on
compound interest, solving linear, quadratic and certain types of biquadratic
equations. Perhaps Pacioli's most significant innovation was his systematic
using syncopated algebraic notation - a kind of predecessor
subsequent symbolic calculus. Among the problems that have attracted the attention of mathematicians
subsequent generations, it is worth noting the problem of dividing the bet in an unfinished game.
Luka solved this problem incorrectly, but later it became the touchstone on which to hone
mathematical art. Ultimately, this task contributed to the emergence and
development of probability theory.

2.2. Symmetry theory and Leonardo da Vinci
There is a widespread opinion that the term golden ratio ( aurea sectio)
first used by Leonardo da Vinci. Is this really so, we cannot determine
managed. Perhaps Leonardo, exploring the structure of polygons and polyhedrons,
came across the golden ratio, known to him from Pacioli’s book. But for Leonardo, rather
In total, the golden proportion was only a manifestation of one of the types of symmetry. And the last one
he paid a lot of attention when designing his famous ensembles. Yes, Hermann Weil
(Weil, 2007, pp. 91–92, 100–101), notes that the simplest figures with
possible variants of rotational symmetry are regular polygons,
which are built in two-dimensional space. Leonardo da Vinci understood this well.
(Weil, 2007, pp. 91, 100). The number of such polygons is determined by the number of faces,
tending towards infinity. When the dimension of space increases to 3, the number
polyhedra are not infinite. There are only five of them. They are usually called Platonic solids.
This is a regular tetrahedron, cube, octahedron, as well as a dodecahedron, the faces of which are
twelve regular pentagons, and an icosahedron limited to twenty regular
triangles. Weil notes that “the existence of the first three polyhedra is
a very trivial geometric fact. But the discovery of the existence of the latter
two, was undoubtedly one of the most outstanding and beautiful discoveries made
throughout the history of mathematics” (Weil, 2007, p. 100). Difference between the two groups
polyhedra is that the cube and the octahedron have the same group
symmetry, because if you take the centers of the faces of a cube and “stretch” a polyhedron on them,
the result is an octahedron, and, conversely, the centers of the faces of the octahedron are the vertices of the cube. According to that
For the same reason, the dodecahedron and icosahedron have the same symmetry groups (Vinberg, 2001,
With. 19–20).
Weil also notes that Leonardo da Vinci was always concerned with the problem of choice
forms central building in architectural ensembles, as well as how to
to add chapels and niches to them without destroying the symmetry of the core of the ensemble.

2.3. Solving equations of the fourth and third degrees
Luca Pacioli finished the section on algebraic equations in the book “Summa”
remark that to solve cubic equations x 3 + b= ax And x 3 + ax= b
the art of algebra has not yet provided a method, just as it has not yet given a method for solving quadrature
circle. These words of Pacioli served as a starting point for Italian algebraists in
solving cubic equations. The discovery of this solution was a major mathematical
an achievement of the Renaissance that has retained its significance to this day. If
If we talk about the mathematics of harmony, then the solution of such an equation is related to
theory of equations generalizing the idea of ​​the golden section. It's primarily about
cubic equations of Padovan-Gazale and Alexey Stakhov (Gazale, 2002, p. 147; Stakhov,
2003, p. 10).
The first to solve one type of cubic equation x 3 + ax= b (a,b>0)
Professor of the University of Bologna Scipione del Ferro (1456–1526), ​​and after him
independently of him, a native of Brescia, Nicola Tartaglie (1500–1557), who decided and
other types of cubic equations. The Tartaglia formula was published by Giralomo Cardano.
(1501–1576) in his famous treatise "The Great Art" (1545). And although she
appears in the history of mathematics under the name Cardano, but the real author is
Tartaglia. By the way, other achievements of the inventive mind are associated with the name Cardano -
driveshaft and Cardano grille: maybe because someone invented it, and he
published?
It is interesting that Cardano’s formula was used by M. Ghazaleh (Ghazaleh, 2002, p. 158) when
calculation of the silver section proposed by the architect Padovan. For the equation
x 3 + ax= b Cardano's formula is:
3
2
3
2
 a
 b
b
 a
 b
b
3
3
x=
  +   +
−
  +   − .
 3 
 2 
2
 3 
 2 
2
Substituting into this expression a= −1 and b= 1, we can find a solution to the equation
3
p− p−1 = 0:
3
2
3
2
 −1
 1 
1
 −1
 1 
1
3
3
p=

 +   +
−

 +   −
=
 3 
 2 
2
 3 
 2 
2

23
1
23
1

3
3
=
+ −
− ≈
108
2
108
2
3
≈
,
0 461479103 + 5
,
0
3
−
,
0 461479103 − 5
,
0
≈
≈ 9
,
0 86991206 + 3
,
0 377226751 ≈ 3
,
1 24717957,
which, to within ten significant figures, is the same as the values ​​calculated by
successive iterations of the expression
3 1+ 3 1+ 3 1+ 3 1+ ... → p
As noted by Karl B. Boyer (1989, p. 282), and after him by Midhad Ghazaleh
(Ghazaleh, 2002, p. 160), year of publication of Cardano (1545) method of solving the cubic
equations marked the beginning of the modern era in mathematics. Let us add on our own that this
the date is also a harbinger of the development of the theory of high-order equations,
related to the golden ratio and Fibonacci numbers.
Cardano included in his book another discovery made by his student Lodovico
(Luigi) Ferrari: general solution to a fourth degree equation.
Italian mathematicians Del Ferro, Tartaglia and Ferrari solved the problem, with
which the best mathematicians in the world could not solve for several centuries. At the same time they
found that “strange” roots from negative numbers sometimes appeared in the solution.
After analyzing the situation, European mathematicians called these roots “imaginary numbers” and
developed rules for handling them that lead to the correct result. So in
mathematics included complex numbers for the first time.
The most important step The Frenchman François Viète (1540–1603) made a contribution to the new mathematics. He
finally formulated the symbolic metalanguage of arithmetic - literal algebra.
Another great great discovery of the 16th century - the invention of John Napier
logarithms, which greatly simplified complex calculations
And finally, at the very end of the 16th century, the Fleming Simon Stevin (1548–1620)
publishes the book “Tenth” about the rules for working with decimal fractions, after which the decimal
the system achieves a final victory in the field of fractional numbers. Stevin also
proclaimed complete equality of rational and irrational numbers, thereby deciding
one of the most pressing problems that puzzled the wise Greeks in ancient times
mathematicians and turned the vector of their research towards geometry.

2.4. Prospect theory
Euclid, in the “Optics” section of his “Principles,” formulated the rules for the first time
observational perspective, and also derived the laws of reflection of rays from flat, concave
and convex mirrors. The doctrine of perspective was later expounded in the treatise “Ten Books
about architecture" by the ancient Greek scientist and architect Vitruvius, who outlined
rules for constructing perspective, as well as drawing up architectural and construction plans
drawings containing the plan and facade of buildings.
During the Renaissance, a new stage in the development of the theory of perspective begins. Leon
Battista Alberti, in his treatises “On Painting” and “On Architecture,” outlined the mathematical
theory of proportions, based on the proportions of the human body. In promising
constructions, Alberti applied the method of constructing images located one after the other
friend of equal and parallel segments enclosed between two lines,
intersecting on the horizon line.
Leonardo da Vinci also made a great contribution to the theory of perspective. In "Treatise on
painting" he wrote that perspective belongs to the "mechanical sciences", which are not
should be neglected by any painter.
Leonardo
Yes
Vinci
divides
perspective
on
three
basic
parts:
1. Linear perspective, which takes into account the law of figures decreasing as they
distance from the observer.
2. Aerial and color perspective, which manifests itself in the color of objects,
depending on their distance to the observer.
3. Perspective on the clarity of the outline of objects depending on the structure
space and degree of illumination of its parts.
The first section of perspective theory subsequently developed into an exact science -
linear perspective, which later became an integral part of descriptive
geometry.
Outstanding German scientist, mathematician, engraver and artist Albrecht Durer
(1471–1528) in his work “Manual for measurements with compasses and rules”,
published in 1523, described graphic method constructing the perspective of objects from
using orthogonal projections, which received the name in educational literature
"Durer's method" Yushkevich notes that this work contains a huge
statistical material containing measurements various parts bodies of men and women
of different builds (Yushkevich, 1977, p. 324). It appears that these results were
the first serious step towards the establishment of anthropometry and rationalistic
aesthetics. Let us also note that Dürer’s achievements in this area are still waiting for a worthy
assessments.

3. The significance of mathematical-harmonic research in the Renaissance
During this period, mathematics for the first time went beyond the legacy left by
Greeks and mathematicians of the East.
1. Algebra and arithmetic received powerful development, finally breaking through
limits of geometry. For the first time, the concept of a real number was practically formed. All
"bad" numbers have become natural or, as Stephen wrote, "there are no absurd
irrational, irregular, inexpressible or dumb numbers” (Yushkevich, 1977, p. 325).
2. The range of ideas related to harmony has expanded significantly. Concept
harmony acquired an increasingly secular character, became more and more humanistic,
extending not only to nature, but also to individual person and human
society as a whole.
3. The concept of harmony for the creative person of the Renaissance finds
embodiment in art of the project, based on the study of many real objects with
with the goal of creating the perfect sample.
3. For the first time since Euclid, the conversation about the golden ratio was resumed,
Platonic solids and regular polyhedra.
4. In the works of Leonardo da Vinci, apparently, for the first time the question of various
types of symmetry of architectural structures.
5. A serious mathematical achievement of the era was the discovery of solution methods
equations of third and fourth degrees. On the one hand, this became the driving force for
development of algebra, and on the other hand, laid the foundations of the algebraic theory of harmony, in which
An important place is occupied by solving equations of high degrees.

Literature
Weil G. Symmetry. Translation from English M.: Publishing house LKI., 2007.
Vinberg E. B. Symmetry of polynomials. Series: Library “Mathematical Education”.
M.: MCMNO, 2001.
Ghazale M. Gnomon. From pharaohs to fractals. Translation from English Moscow-Izhevsk: Institute
Computer Research, 2002.
Gilbert K., Kuhn G. History of aesthetics. Translation from English M.: Publishing house of foreign literature,
1960.
Durer A. Diaries, letters, treatises. Art: M.-L.: 1957, vol. 2.
Yushkevich A. P. History of mathematics (edited by A.P. Yushkevich) in three volumes. Volume 1. C
ancient times until the beginning of modern times. M., Nauka, 1977.
Leonardo da Vinci. Selected works. M.-L..: Academia. 1935. T. 2.
Luca Pacioli. About divine proportion. Reprint ed. 1508 with translation appendix
A. I. Shchetnikova. M.: Russian Avant-Garde Foundation, 2007.
Luca Pacioli. Treatise on accounts and records. M.: Finance and Statistics, 1994.
Sokolov Ya. Luca Pacioli. Man and thinker. In the book: Pacioli Luca. Treatise on Accounts and
records. M.: Finance and Statistics, 1994.
Shchetnikov A. I. Luca Pacioli and his treatise “On Divine Proportion”. Mathematical
Education, No. 1 (41), 2007, pp. 33–44.
Boyer C., Merzbach U. A History of Mathematics. New York: John Wiley & Sons, 1989.
Vasari G. On Technique. Ed. G. B. Brown. London, 1907.