This scientist wrote a treatise on divine proportion. Luca Pacioli and his treatise “On Divine Proportion”

Accounting is an integral element of the modern economic system. As historical practice shows, ideas about money and its circulation have unbreakable connection 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. 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 outside the framework of the 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, this technique does not contribute further development practitioner, 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 areas. 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 everywhere used mathematical concepts. 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 whole system created by him.

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Lodovico Sforza il Moro ruled this duchy virtually from 1480, first working as regent for his weak-willed, uninterested government affairs, nephew - Gian Galeazzo, the son of his older murdered brother Galeazzo Maria Sforza.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Luke gratefully accepted the Duke's offer.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

But I had to study.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The French king Louis XII treated Ludovico Sforza harshly and mercilessly, depriving him of everything he had and sending him to prison. According to historians, one of the last words This man, talented in many ways, wrote “Infelix sum” (“I am unfortunate”; Lat.) on the walls of his dark prison cell.

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

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

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

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

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

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

G. Ya. Martynenko

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

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

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

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

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

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

1. Ideas about harmony in the art of the Renaissance

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

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

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

1.3. The role of science and crafts

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



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

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

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

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

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

23
1
23
1

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

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

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

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

A.P. Stakhov

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

The culture of Ancient Greece and the culture of Rome and Byzantium are two powerful streams of spiritual values, the merger 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 what he did 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. The historical significance of 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, in accordance with the 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: “they are approaching last days of my life". He died in 1515 and is buried in the cemetery of his hometown of 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

Sacred geometry. Energy codes of harmony Prokopenko Iolanta

Golden ratio. Divine proportion