Luca pacioli divine proportions. Accounting, Divine Proportion and Astrology

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 on religious themes, and his own style reflects this influence in all its aspects of 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 natural continuation background, as in "The Baptism" (now in the National Gallery, 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 showed remarkable mathematical abilities from his early youth, and attributes to him the writing of “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, since Piero della Francesca decided to prove that the technique of conveying perspective in painting is based entirely on mathematical and physical properties 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 contemporary artist David Hockney writes 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 a complete perspective analysis 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 “O 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:
The central point of the human body 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” ( 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 point of 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 Pacioli's works and Pierrot's ideas and mathematical constructions, which were not published in print, probably would not have achieved 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 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 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 (acquired in Venice Latin translation, 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 Measurement 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 right hand she has 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 a so-called rhombohedron (a geometric body with six faces, each of which is a rhombus, see Fig. 57), cut so that it can be inscribed in 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 at the time of his mother's arrest was already famous person, and the news of his mother’s trial caused him “unspeakable 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 a background, who had such ups and downs, such a turbulent life, managed 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 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 areas 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 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 due to thick line, indicating 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 work in Linz was the publication in 1619 of his second major work on 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 Galileo Galilei's father Vincenzo's Dialogues Concerning the 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 unaware of all the mathematical intricacies of parquet flooring, 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 interesting job about 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 Universe 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 a 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. Probably said it best English poet John Donne (1572–1631) in the pamphlet “Ignatius and his Conclave”: Kepler “made it his duty to see that nothing new happened in the heavens without his knowledge.”

Accounting is an integral element of the modern economic system. As historical practice shows, ideas about money and its circulation are inextricably linked with the existing economic structure. With the development of statehood, the need arose to systematize and streamline financial transactions. A huge contribution to solving this problem was made by Luca Pacioli, the “father” of accounting. Next, we will find out what the merit of this mathematician is.

Luca Pacioli: biography

He was born in 1445 in the Apennines, in the small town of Borgo Sansepolcro. While still a boy, he was sent to a local monastery to study with an artist. In 1464 Luca Pacioli moved to Venice. There he raised merchant sons. It was at that moment that his first acquaintance with financial activities occurred. In 1470, Luca Pacioli (photo of the mathematician is presented in the article) moved to Rome. There he finished compiling his textbook on commercial arithmetic. After Rome, the mathematician goes to Naples for three years. There he was engaged in trade, but, apparently, without success. In 1475-76, he took monastic vows and joined the monk. From 1477, Luca Pacioli taught for 10 years at the University of Perugia. During his career, his teaching abilities were repeatedly recognized with salary increases. While working at the university, he created the main work, one of the chapters of which was “Treatise on Records and Accounts.”

In 1488, the mathematician left the department and went to Rome. For the next five years he was on the staff of Pietro Valletari (bishop). In 1493, Pacioli moved to Venice. Here he prepared his book for printing. After resting for a year, Pacioli accepted a chair at the University of Milan, where he began teaching mathematics. Here he meets Leonardo da Vinci and becomes his friend. In 1499 they moved to Florence. There Pacioli taught mathematics for two years. After this he goes to Bologna. In this city, almost half was allocated for the maintenance of the university. The acceptance of a mathematician to such a profitable and prestigious position indicates his recognition.

A few years later, part of the book written by Luca Pacioli, “Treatise on Accounts and Records,” was published in Venice. The publication date of this work is 1504. By 1505, the mathematician had practically retired from teaching and moved to Florence. But in 1508 he again went to Venice. There he gave public lectures. However, his main occupation at that time was preparing for the publication of his translation of Euclid. In 1509, another book was published, written by Luca Pacioli, “On Divine Proportion.” In 1510, the mathematician returned to his hometown and became a prior in the local monastery. However, his life was burdened by numerous intrigues of envious people. This was the reason that four years later he left for Rome again. There he taught at the mathematical academy. Luca Pacioli returned to his hometown shortly before his death - in 1517.

Contribution of a mathematician to the development of methodology

To fully understand the significance of the book that Luca Pacioli wrote (Treatise on Accounts and Records), it is necessary to appreciate the principles he laid down in the system. Almost all experts say that the criteria proposed by the mathematician existed before him. For example, Luca Pacioli cannot be considered the author of the double entry. It existed before him. In this case, the question arises: what is the contribution of the mathematician in such a case? Unlike his contemporaries, Pacioli believed that everything important had already been invented earlier. He saw the main task of scientists in the most effective construction of a training course. Pacioli did not imagine scientific creativity beyond pedagogical process. Therefore, teaching became an integral element of his life.

The ideas that Luca Pacioli had completely determined his scientific approach to solving mathematical problems and related disciplines. This position was later determined quite accurately by Galileo. Luca Pacioli's knowledge of mathematics was closely connected with the study of the harmony of the world. At the same time, the correctness of geometric figures, as well as the convergence of the balance, became for him manifestations of this harmony. The scientist did not simply record those practices that existed previously, but gave them a scientific description. This is the main significance of the activities carried out by Luca Pacioli. The Treatise on Accounts and Records thus became the foundation for improving the balance sheet system.

The essence of the scientific approach

Reflection of facts at the time of their existence is the most accurate. But at the same time, such a technique does not contribute to the further development of practices, since the method of cognition is focused on the past, an accurate reproduction of what has already happened and is taking place. The approach used by Luca Pacioli made it possible to assess the situation not only at the stage of its development, but also in the future, as well as from the point of view of systemicity and integrity. In his work, the mathematician did not take into account much, made a number of mistakes, and described the more outdated Venetian system, rather than the progressive Florentine one. However, Luca Pacioli's Treatise showed that a scientific approach can also be applied when preparing financial statements. He was able to turn the formation of a balance sheet into one of the directions. This, in turn, led many people (Leibniz, Cardano and others) to become interested in accounting theory.

Implementation of a mathematical system

In his Treatise, Pacioli supplemented the existing methods with ideas about combinatorics. Balance sheets at that time used fractions due to the simultaneous use of several currencies. But during operations they were simply rounded up. However, the main contribution of the mathematician to the methodology is considered to be his introduction of the idea of ​​​​the integrity of the accounting system and the fact that the convergence of the balance acts as a sign of its harmony. The latter definition was considered at that time not only as an aesthetic, but also an engineering category. Assessing the trade balance from this position made it possible to present the enterprise as an integral system. The method that Luca Pacioli perfected - double entry - in his opinion, should have been used not only for a specific trading enterprise, but for any organization and for the entire economy as a whole. This allows us to conclude that the approach that the mathematician introduced predetermined not only the development of financial reporting, it became the foundation for the formation and subsequent implementation of economic thought.

Luca Pacioli: "Treatise on Accounts and Records" (summary)

First of all, it should be said that a mathematician’s financial balance is presented in the form of a strictly ordered sequence of operations. The most complete reflection of “procedurality” can be seen in the principle of maintaining three accounting books. The first - "Memorial" - reflects the chronological sequence of all cases. The sixth chapter of the Treatise describes the procedure for conducting it. Over time, the Memorial was replaced by primary documents. As a result, there was an inconsistency between the dates of the statement, the transaction and the registration of the fact.

The next book is "The Journal". It was intended exclusively for internal use. It recorded all the operations that were described in the Memorial, but at the same time took into account their economic meaning (loss, profit, and so on). It was intended for postings and was also compiled in chronological order. The third book was "The Main". It is described in Chapter 14 of the Treatise. It recorded transactions in a systematic rather than chronological order.

Clarity

This is the next principle that was described by Pacioli. Clarity meant providing users with clear and complete information about the business activities of the enterprise. All entries in books, in accordance with this principle, should be compiled in such a way that they provide for conceptual reconstruction. In other words, transactions must be recorded in such a way that it is subsequently possible to restore the participants in the act, objects, time and place of the fact. To achieve the greatest clarity, proficiency in the accounting language is necessary. The mathematician used the Venetian dialect when writing the book and used mathematical concepts everywhere. It was Pacioli who formed the prerequisites for the creation of an accounting language that was most understandable for the majority of Italian financiers.

Inseparability of the property of the owner and the enterprise

This principle was quite natural for that time. The fact is that many merchants then acted as the sole owners of the enterprise, managers and recipients of losses and profits from trading activities. In accordance with this, accounting is carried out in the interests of the owner of the company. However, in 1840, Hippolyte Vanier formulated a different approach. In accordance with it, accounting is conducted not in the interests of the owner, but of the company. This approach reflected the spread of share capital among the broad masses.

Credit and debit

One of Pacioli’s most important principles was dual recording. The mathematician took the position that each should be reflected in both debit and credit. This approach has the following goals:

In his work, Pacioli paid much attention to the first task. At the same time, the second and third remained undeveloped. This leads to the formation of a method that distorts the correctness of the turnover. The fact is that Pacioli was first and foremost a scientist, and then a financier, so he considered the double entry system within the limits of cause and effect. Presumably, the mathematician saw the cause in debit, and the effect in credit. This way of viewing the financial system has primarily found application in economics. The most succinct formulation of this principle was given by Yezersky: without expenditure there can be no income. Pacioli accepted the following as the main aspects of the dual notation:

1. The amount of debit turnover will always be identical to the amount of credit.
2. The value of debit balances will always be identical to the value of credit balances.

These principles subsequently became widespread in accounting systems.

Subject of reporting

Pacioli's role was to execute the purchase and sale agreement. Reducing all agreements to a document of this type was quite typical for that time. Undoubtedly, today's variety of forms of economic life cannot fit into the framework of the concept of purchase and sale (for example, offset, barter, and so on). However, in Pacioli's time this idea was very progressive. In addition, this approach made it possible to formulate an adequate definition of value for that period as not only a fair price, but also a consequence of cost and the market situation.

Principle of adequacy

Its essence is that all the expenses that an enterprise incurs are correlated over time with the income it receives. Pacioli's principle of adequacy presupposes rather than introduces it directly and explicitly. Only money received is considered income. At that time, the concepts of profitability and depreciation were just beginning to form. Taken together, all this contributed to the creation of ideas about both monetary and other forms of profit. In accordance with the new understanding of income, we can say that it is generated not only as a result of business transactions, but also as a result of the application of accounting methodology.

Maintaining a balance

Pacioli considered accounting to be something valuable in itself, and in connection with this, the value of reporting results acted as a relative concept. The results recorded in one or another book depend largely on the reporting method. This provision is consistent with the idea of ​​​​the most accurate recording of business transactions in the balance sheet, since all methods require a fairly accurate reflection of the facts, despite the fact that the conclusions can often be directly opposite. Pacioli understood this all too well. In this regard, as the main result of financial reporting, he saw its impact on decision-making in the field of economic management.

Honesty

This is the last principle that Pacioli proclaimed in his Treatise. The person who balances the balance must be absolutely honest. This should apply not only to the employer himself. An accountant must be primarily honest with God. In this regard, relying on it in almost every chapter for a mathematician is neither a tribute to tradition nor the fulfillment of a monastic duty, but most importantly Pacioli considered the deliberate distortion of accounting information not only a financial violation. For the mathematician, this was primarily a disorder of divine harmony, which he sought to comprehend through calculations.

Disadvantages of the job

It should be said that Pacioli's work acted primarily as a theoretical book. As such, it does not reflect many elements of the financial statements that existed at that time. These, in particular, include:

1. Maintaining additional and parallel books.
2. Industrial cost accounting.
3. Maintaining a balance for analytical purposes. At that time, reporting was already carried out not only to reconcile information and close books, but also acted as a management and control tool.
4. Maintaining nostro and loro accounts.
5. Basics of audit and procedure for checking balance.
6. Calculation methods relating to profit distribution.
7. The procedure for reserving funds and distributing results over adjacent periods.
8. Confirmation of reporting information using inventory methods.

The absence of these components primarily indicates Pacioli's lack of commercial experience. It is likely that he did not include the given details due to the fact that they simply did not fit into the holistic system he created.

Finally

Pacioli's work was one of the first to use the Italian language as a means of expressing scientific ideas. The principles and categories formed by the mathematician are still used today. Pacioli's main merit is not that he recorded them - after all, this would have been done anyway. His contribution is that it was thanks to his book that accounting was elevated to the status of a science.

One of the basic rules of wealthy people is to keep records of their income and expenses. Not a single company, not a single enterprise, not a single trade can do without accounting. Modern accounting is based on double entry - a method of maintaining accounts in which any movement of funds is recorded twice: on the left and on the right side of the account. And few people know that the author of this system is a monk of the mendicant order, the Florentine Luca Pacioli.

Magnificent century

We associate the Renaissance with Dante, Boccaccio, Michelangelo, Leonardo... But the flourishing of the arts and sciences in the states of the Apennine Peninsula of the 14th-16th centuries - Florence, Genoa, Venice, Naples - would have been impossible without the flourishing of the economy and trade. Great artists, architects, scientists, poets did not create on their own, but at the request of their sponsors - dukes, gonfaloniers and simply wealthy citizens. Cultural figures, especially favored by the powers that be, themselves patronized those in whom they saw talent.

One of the famous patrons was the ruler of the state of Urbino, Duke Federico de Montefeltro. In the 15th century, he maintained a rich court and patronized the sciences and art. Among the talents with which the Duke liked to surround himself, Leon Battista Alberti stood out - a scientist, writer, musician and an outstanding architect. He broke with the canons of medieval Gothic architecture and turned to the heritage of Ancient Rome.

It was he who noticed in 1464 19-year-old Luca Pacioli, who at that time served as an apprentice to famous artist Piero della Francesca. The young man was originally from the small town of Borgo San Sepolcro, located on the banks of the Tiber and then belonging to the Florentine Republic.

Alberti recommended the young man to the wealthy Venetian merchant Antonio de Rompianzi as a mathematics teacher for his children. Luke moves to Venice. Merchant offspring needed mathematics for their business - to trade competently and successfully. Therefore, Pacioli published in 1470 tutorial in commercial arithmetic.

In the same year, the young scientist leaves Venice and moves to Rome, to his patron, the architect Alberti. In the Eternal City, Luke sees numerous ruins of the capital of the ancient empire, a huge number of dilapidated crumbling medieval buildings and a few buildings of new great architects. Pacioli had the opportunity to clearly see that only those who had previously carried out the correct calculations build well.

At that time, this was not so obvious: when laying the foundations of buildings, they often made “construction sacrifices”, walling up babies or women alive in the foundation of the house. And they found explanations for this savage custom in Christian theology: “The Eternal Father placed his own son as the cornerstone of all creation in order to save the world from decay and through the death of an innocent person to stop the furious onslaught of hellish forces.”

The Renaissance, in addition to the flourishing of economics, science and art, was also characterized by the wildest superstitions and special cruelty. The historian and philosopher Alexei Losev wrote: “In the Renaissance, they told fortunes on corpses, conjured public women, made love potions, summoned demons, performed magical operations when laying buildings, and practiced physiognomy and palmistry.” The “witch hunt” did not take place in the Middle Ages, as is commonly believed, but rather in the Renaissance.

Photo: Mary Evans Picture Library/Global Look

In the world that surrounded Pacioli, in addition to cruelty, debauchery and superstition, greed also reigned. In contrast to the Middle Ages, steeped in Christian ascetic ideology, during the Renaissance people strived to live well. Myself the great Leonardo admitted: “I serve the one who pays the most,” “I almost don’t care what I do and what I get paid for.” Perhaps the desire to distance himself from all this and not be distracted from the research that interested him prompted Pacioli to adopt asceticism.

In 1472 he left Rome and joined the Franciscan friars. He put on sandals on his bare feet, pulled on a brown robe made of coarse wool and girded himself with a white rope with three knots, as a sign of three vows: obedience, chastity and poverty. It was as a monk of the Barefoot Order, as the Franciscans were then called, that “Brother Luke” wrote his books with advice to the rich on how to manage money and make life beautiful and harmonious. How much wealth hinders a scientist is debatable, but in our time the mathematician Grigory Perelman also refused a well-deserved million dollars for proving the Poincaré conjecture.

Pacioli devoted his life to science, not to his own enrichment. Money management was of purely scientific interest to him. Already a monk, from 1477 to 1480 he taught mathematics at the University of Perugia. He spent eight years in Zara (now Zadar in Croatia), and then changed his place of residence more than once. And he wrote main work of his life - the encyclopedic “Sum of Arithmetic, Geometry, Relations and Proportions.” In 1493, work on the book was completed.

Queen of Sciences

The monk dedicated his work to Guido Ubaldo de Montefeltro, the son of the former Duke of Urbino. The high patron helped publish the book in Venice in 1494. This is an extensive encyclopedic volume with 300 pages. 224 sheets are devoted to arithmetic and algebra, 76 sheets to geometry.

The book was written not in Latin, which was then used by learned men around the world, but in Italian that time. In the dedication, the author explains: “The correct understanding of difficult terms among Latinists has ceased due to the fact that good teachers have become 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 vernacular language, so that both educated and uneducated alike can enjoy these activities.

In the preface, Pacioli states his belief that mathematics deals with “a universal law applicable to all things.” He gives examples from astronomy, architecture, talks about numerous painters who developed the art of perspective, “which, if examined carefully, would be empty space without the use of mathematical calculations." The masters of art, whom he appreciates, “using calculations in their works with the help of a level and compass, brought them to extraordinary perfection.” Pacioli also mentions the importance of mathematics for music, cosmography, mechanical arts, warfare and, of course, trade.

In another of his works, “On Divine Proportion,” dedicated to the Duke of Milan Ludovico Sforza, Brother Luca wrote: “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.”

At the University of Milan, where Pacioli was invited to head the newly created department of mathematics, he met and then became friends with Leonardo da Vinci. The great master illustrated the works of Brother Luke. They separated in 1499, when Milan was occupied by the French.

In 1509, in Venice, Pacioli published an even more extensive work. Its very name is noteworthy: “Divine Proportion. A work very useful to every insightful and inquisitive mind, from which every student of philosophy, perspective, painting, sculpture, architecture, music or other mathematical subjects will extract the most pleasant, witty and surprising teaching and will entertain himself with various questions of the most secret science. As we can see, Pacioli considered mathematics the basis of almost all disciplines. In fact, he repeated the idea of ​​Pythagoras that the basis of the universe is number.

Debit and credit

In his Treatise on Accounts and Records, Pacioli described three conditions “necessary for anyone who wishes to conduct trade in a proper manner.” The first and most important thing is cash and valuables. The second is the ability to keep books correctly and count quickly. The third is to conduct your affairs in due order and as it should, “so that all information can be obtained without delay, both regarding debts and claims, for trade does not extend to anything else.”

It was here that Pacioli created his know-how, which served as the basis for modern accounting and is still used today. The principle of duality or double entry is formulated in the treatise as follows: “In the Journal there are two expressions: one - Per (on), the other A - (from), and each of them has its own special meaning: “on” always means a debtor (debtor) - one or many, “from” - always a trustee (creditor) - one or many. It is impossible to submit a single ordinary article to the Journal... without first providing it with both of these terms. At the beginning of every article it is placed “on” because first the debtor must be identified, and then, immediately after him, his trustee, that is, “from”. One is separated from the other by two small vertical lines».

Thus, when assets decrease (when a credit entry is made), liabilities also decrease (a debit entry is necessarily made) by the same amount. It would seem - a purely technical detail that does not affect the essence of the process. But in fact, this gives a huge advantage: the merchant could immediately see both his demands and debts. “Perhaps the beauty of accounting is the “harmony and harmony” of the accounts that make up the balance sheet of which they are parts,” Pacioli believed.

“After you have written down all the articles in the Journal, you should make a selection from it and transfer it to the third book - the large one, or the Main Book, which is usually kept with double the number of sheets compared to the Journal. It is customary to include an alphabetical index - a repertory, otherwise called a register, or index. The Florentines call it extract. In the register you will enter all debtors and trustees in alphabetical order, indicating the page numbers where they appear. main book must be marked with the same sign with which the Memorial and the Journal are marked.”

To be fair, it should be said that Pacioli was not the first to invent the principle of double entry: this method of keeping accounts was already used by the Incas of the Tawantinsuyu empire. However, the current accounting system is based on the principles laid down specifically by Brother Luke the mendicant.

Nowadays, people strive for specialization, clearly realizing that it is impossible to “embrace the immensity.” It is better to be a specialist in one area and achieve perfection in it. This was not the case during the Renaissance. Then they looked at the world as a single whole, trying to find harmony in it and the key to understanding the entire universe. From Luca Pacioli's point of view, mathematics was the key. Moreover, finding a completely utilitarian use for it, the scientist did not seek benefit for himself. He wanted to make the world a little better. And he succeeded.

A.P. Stakhov

Under the sign of the “Golden Section”:
Confession of the son of a student battalion student.
Chapter 4. Golden ratio in cultural history.
4.8. "Divine Proportion" by Luca Pacioli

Culture Ancient Greece and the culture of Rome and Byzantium - these are two powerful streams of spiritual values, the merging of which gave rise to the new ones, the titans of the Renaissance. Titan is the most accurate word in relation to such people as Leonardo da Vinci, Michelangelo, Nicolaus Copernicus, Albert Durer, Christopher Columbus, Amerigo Vespucci. This galaxy rightfully includes the mathematician Luca Pacioli.

He was born in 1445 in the provincial town of Borgo San Sepolcro, which translated from Italian does not sound too joyful: “City of the Holy Sepulcher.”

We do not know how old the future mathematician was when he was sent to study in the studio of the artist Piero della Francesco, whose fame resounded throughout Italy. This was the first meeting of the young talent with the great man. Piero della Francesco was an artist and mathematician, but only the second aspect of the teacher found an echo in the heart of the student. Young Luke, a mathematician from God, was in love with the world of numbers; number seemed to him to be some kind of universal key, simultaneously opening access to truth and beauty.

The second great man who met on Luca Pacioli's life path was Leon Battista Alberti - architect, scientist, writer, musician. The words of Albert will sink deeply into the consciousness of L. Pacioli:

“Beauty is a certain agreement and consonance of parts in that of which they are parts, corresponding to the strict number, limitation and placement that harmony requires, that is, the absolute and primary principle of nature.”

In love with the world of numbers, L. Pacioli will repeat after Pythagoras the idea that number underlies the universe.

In 1472, Luca Pacioli was tonsured into the Franciscan order, which gave him the opportunity to engage in science. Events showed that he made the right choice. In 1477 he received a professorship at the University of Perugia.

Luca Pacioli

The following portrait description of Luca Pacioli from that time has been preserved:

“A handsome, energetic young man: raised and fairly broad shoulders reveal innate physical strength, a powerful neck and developed jaw, an expressive face and eyes that radiate nobility and intelligence, emphasizing the strength of character. Such a professor could make people listen to themselves and respect their subject.”

Pacioli combines his pedagogical work with scientific work: he begins to write an encyclopedic work on mathematics. In 1494, this work was published under the title “Summa of Arithmetic, Geometry, Doctrine of Proportions and Relations.” All material in the book is divided into two parts, the first part is devoted to arithmetic and algebra, the second to geometry. One of the sections of the book is devoted to the application of mathematics in commercial affairs, and in this part his book is a continuation of the famous book by Fibonacci “Liber abaci” (1202). Essentially, this mathematical work by L. Pacioli, written at the end of the 15th century, summarizes the mathematical knowledge of the Italian Renaissance.

L. Pacioli's monumental printing work undoubtedly contributed to his fame. When in 1496 in Milan, the largest city and state in Italy, the department of mathematics was opened at the university, Luca Pacioli was invited to take it.

At this time, Milan was the center of science and art, outstanding scientists and artists lived and worked there - and one of them was Leonardo da Vinci, who became the third great man who met Luca Pacioli on his life path. Under the direct influence of Leonardo da Vinci, he begins to write his second great book, “De Divine Proportione” (“On Divine Proportion”).

The book by L. Pacioli, published in 1509, had a noticeable influence on his contemporaries. Pacioli's volume, published in quarto, was one of the first excellent examples of the art of printing in Italy. Historical meaning The book was that it was the first mathematical work entirely devoted to the “golden ratio”. The book is illustrated with 60 (!) magnificent drawings made by Leonardo da Vinci himself. The book consists of three parts: the first part outlines the properties of the golden ratio, the second part is devoted to regular polyhedra, the third - applications of the golden ratio in architecture.

L. Pacioli, appealing to Plato’s “Republic”, “Laws”, “Timaeus”, consistently deduces 12 (!) different properties of the golden ratio. Describing these properties, Pacioli uses very strong epithets: “exceptional”, “most excellent”, “wonderful”, “almost supernatural”, etc. Revealing this proportion as a universal relationship that expresses the perfection of beauty in both nature and art, he calls it “divine” and is inclined to consider it as a “tool of thinking,” “an aesthetic canon,” “as a principle of the world and nature.”

Title page of Luca Pacioli's book The Divine Proportion

This book is one of the first mathematical works in which the Christian doctrine of God as the creator of the Universe receives scientific justification. Pacioli calls the golden ratio “divine” and identifies a number of properties of the golden proportion, which, in his opinion, are inherent in God himself:

“The first is that there is only one, and it is impossible to give examples of proportions of a different kind or even slightly different from it. This uniqueness is in accordance with political and philosophical teachings. There is the highest quality of God himself. The second property is the property of the holy trinity, namely, just as in the deity one and the same essence is contained in three persons - the father, the son and the holy spirit, so the same proportion of this kind can take place only for three expressions, and for There are no greater or lesser expressions. The third property is that, since God cannot be defined in detail or explained in a word, our proportion cannot be expressed either by a number accessible to us or by any rational quantity and remains hidden and secret and therefore mathematicians called irrational. The fourth property is that, just as God never changes and represents everything in everything and everything in each of his parts, and our proportion for every continuous and definite quantity is the same, whether these parts are large or small, in no way cannot be changed or otherwise perceived by reason. To these properties can quite rightly be added a fifth property, which consists in the fact that, just as God called into existence heavenly virtue, otherwise called the fifth substance, and with its help four other simple bodies, namely, the four elements - earth, water , air and fire, and with their help brought into existence every thing in nature, so our sacred proportion, according to Plato in his Timaeus, gives formal existence to the sky itself, for it is attributed to the type of body called the dodecahedron, which cannot be built without our proportion."

Dodecahedron drawn by Leonardo da Vinci for the book “The Divine Proportion” by L. Pacioli

In 1510, Luca Pacioli turned 65 years old. He is tired, old. The library of the University of Bologna contains the manuscript of L. Pacioli's unpublished work “On Forces and Quantities”. In the preface we find the sad phrase: “the last days of my life are approaching.” He died in 1515 and was buried in his cemetery hometown San Sepolcoro.

After his death, the works of the great mathematician were consigned to oblivion for almost four centuries. And when at the end of the 19th century his works became world famous, grateful descendants, after 370 years of oblivion, erected a monument on his grave on which they wrote:

"Luke Pacioli, who was the friend and adviser of Leonardo da Vinci and Leon Battista Alberti, who first gave algebra the language and structure of science, who applied his great discovery to geometry, invented double-entry bookkeeping and gave in mathematical works the foundations and unchangeable norms for subsequent generations" .

A.P. Stakhov, Under the sign of the “Golden Section”: Confession of the son of a student student. Chapter 4. The golden ratio in the history of culture. 4.8. “Divine Proportion” by Luca Pacioli // “Academy of Trinitarianism”, M., El No. 77-6567, pub. 13547, 07/12/2006

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