# data[].attributes.attachment_link.readable_text
The book contains geometric studies of capital letters, including M, which was the iconic logo of the Metropolitan Museum of Art in New York from 1971 to 2016.

There are 59 illustrations of polyhedra; some of which are solid, and some, like the one below, which are skeletal, making it easier to see edges and vertices.

The 1509 publication actually contains 3 works which were completed in 1498. The first is Divina Proportione; the second contains geometry related to architecture, including the golden ratio; and the third is an Italian translation of a work by Italian artist Piero della Francesca. There are two surviving manuscript versions, and it is possible to purchase facsimile editions of the manuscript version that is at Aboca Museum Edizioni.

For more information and images of Pacioli's Divina Proportione, visit the following pages:
Images in this article were taken by the author at the Linda Hall Library of Science, Engineering & Technology and are used with permission. The images may be downloaded and used for the purposes of research, teaching, and private study, provided the Linda Hall Library of Science, Engineering & Technology is credited as the source. For other uses, check out the LHL Image Rights and Reproductions policy.
References

Codices Illustres. http://www.codicesillustres.com/catalogue/de_divina_proportione/.

O'Connor, J. J., and E. F. Robertson. "Luca Pacioli." MacTutor History of Mathematics archive. http://www-history.mcs.st-and.ac.uk/Biographies/Pacioli.html.

O'Connor, J. J., and E. F. Robertson. "Piero della Francesca." MacTutor History of Mathematics archive. http://www-history.mcs.st-and.ac.uk/Biographies/Francesca.html.

Pedretti, Carlo. Leonardo da Vinci: The Complete Works. David & Charles Books, 2013.
Index to Mathematical Treasures

# data[].attributes.attachment_link.title
Mathematical Treasure: Luca Pacioli's Divina Proportione

# data[].attributes.attachment_link.description
The book contains geometric studies of capital letters, including M, which was the iconic logo of the Metropolitan Museum of Art in New York from 1971 to 2016. There are 59 illustrations of polyhedra; some of which are solid, and some, like the one below, which are skeletal, making it easier to see edges and vertices.

# data[].attributes.attachment_link.readable_text
The illegitimate son of a 25-year-old notary, Ser Piero, and a peasant girl, Caterina, Leonardo was born on April 15, 1452, in Vinci, Italy, just outside Florence. His father took custody of him shortly after his birth.

Growing up in his father's Vinci home, Leonardo had access to scholarly texts owned by family and friends. He was also exposed to Vinci's longstanding painting tradition, and when he was about 15 his father apprenticed him to the renowned workshop of Andrea del Verrochio in Florence. Even as an apprentice, Leonardo demonstrated his great talent. Indeed, his genius seems to appear in a number of pieces produced by the Verrocchio's workshop from the period 1470 to 1475. For example, one of Leonardo's first big breaks was to paint an angel in Verrochio's "Baptism of Christ," and Leonardo was so much better than his master's that Verrochio allegedly resolved never to paint again. Leonardo stayed in the Verrocchio workshop until 1477.

Seeking to make a living, and new challenges, he entered the service of the Duke of Milan in 1482, abandoning his first commission in Florence, "The Adoration of the Magi". He spent 17 years in Milan, leaving only after Duke Ludovico Sforza's fall from power in 1499. It was during these years that Leonardo reached new heights of scientific and artistic achievement.

The Duke kept Leonardo busy painting and sculpting and designing elaborate court festivals, but he also had Leonardo design weapons, buildings, and machinery. From 1485 to 1490, Leonardo produced studies on many subjects, including nature, flying machines, geometry, mechanics, municipal construction, canals and architecture (designing everything from churches to fortresses). His studies from this period contain designs for advanced weapons, including a tank and other war vehicles, various combat devices, and even submarines. Also during this period, Leonardo produced his first anatomical studies. His Milan workshop was abuzz with apprentices and students.

Unfortunately, Leonardo's interests were so broad, and he was so often compelled by new subjects, that he usually left projects unfinished. As a result, he only completing about six works in these 17 years, including "The Last Supper" and "The Virgin on the Rocks," leaving dozens of paintings and projects unfinished or unrealized (see "Big Horse" in sidebar). He spent most of his time studying science, either by going out into nature and observing things or by locking himself away in his workshop cutting up bodies or pondering universal truths.

Between 1490 and 1495 he developed his habit of recording his studies in meticulously illustrated notebooks. His work covered four main themes: painting, architecture, the elements of mechanics, and human anatomy. These studies and sketches were collected into various codices and manuscripts, which are now collected by museums and individuals (Bill Gates once paid $30 million for the Codex Leicester!).

Back to Milan - after Ludovico Sforza's fall from power in 1499 - Leonardo searched for a new patron. Over the next 16 years, Leonardo worked and traveled throughout Italy for a number of employers, including the infamous Cesare Borgia. He traveled for a year with Borgia's army as a military engineer and even met Niccolo Machiavelli, author of "The Prince." Leonardo designed a bridge to span the "golden horn" in Constantinople during this period and received a commission, with the help of Machiavelli, to paint the "Battle of Anghiari."

About 1503, Leonardo reportedly began work on the "Mona Lisa." From 1513 to 1516, he worked in Rome, maintaining a workshop and undertaking a variety of projects for the Pope. He continued his studies of human anatomy and physiology, but the Pope forbade him from dissecting cadavers, limiting his progress.

Following the death of his patron Giuliano de' Medici in March of 1516, he was offered the title of Premier Painter and Engineer and Architect of the King by Francis I in France. His last and perhaps most generous patron, Francis I provided Leonardo with a stipend and manor house near the royal chateau at Amboise.

Although suffering from a paralysis of the right hand, Leonardo (who wrote with his left-handed) was still able to draw and teach. He produced studies for the Virgin Mary from "The Virgin and Child with St. Anne", studies of cats, horses, dragons, St. George, anatomical studies, studies on the nature of water, drawings of the Deluge, and of various machines.

Leonardo died on May 2, 1519 in Cloux, France. Legend has it that King Francis was at his side when he died, cradling Leonardo's head in his arms.

# data[].attributes.attachment_link.readable_text
Summary

In 1482, Leonardo moved to Milan. He was now a master, but he seemed dissatisfied with his life in Florence. Science interested him just as much as painting, and he had begun making copious scientific speculations in his notebooks. Thus even though Milan was less artistically-centered than Florence, the city would provide him with more opportunities to apply his various interests.

But his primary motivation for moving was probably the opportunity to work for Milan's ruler, Ludovico Sforza: Milan held a highly strategic position in southwestern Europe, and was often a point of military contention; in a letter Leonardo wrote to Sforza before coming to the city, he offers his services as a military engineer. Although it is unclear whether Leonardo ever actually sent this letter, it proves Leonardo's strong interest in military engineering. During this time, Leonardo designed a number of military machines, including a primitive form of war tank and a wild chariot with scythes attached to its wheels. The aforementioned letter also alludes to an equestrian monument Sforza hoped to erect in honor of his father; Leonardo planned a gigantic horse to serve this purpose-one that would be a feat of engineering as well as art.

When Leonardo first arrived, however, his first major project was another painting, the Virgin of the Rocks.

Leonardo painted the work for the Confraternity of the Immaculate Conception in Milan; two brothers named de Predis painted the side panels. The contract with the Confraternity led to many legal problems, so much so that Leonardo eventually had to supervise the painting of a second version of the painting, in 1506.

Perhaps the most striking aspect about the Virgin of the Rocks is the Virgin's placement on rocks. A precipitous chasm separates the viewer from the holy scene, as if it were taking place on some unattainable plane of being. The rocky caves in the background, opening up onto a hazy landscape, contribute to the sense of otherworldliness. Leonardo often used hazy, rocky background, and this provides the extreme example. The Virgin herself is the most mature woman we have seen him paint at this point in his career. Her head is longer and more natural than the round, stylized faces of earlier paintings. An angel sits to her left, an infant John the Baptist toddles at her right, and the Christ Child rests at her feet. They form a triangle reminiscent of the Adoration of the Magi. The angel points at John the Baptist, and just above her hand hovers Mary's hand, open yet domineering. In the same vertical line, Christ lifts two pudgy fingers in benediction.

Even while he entrenched himself in this complex painting, Leonardo was beginning serious research into anatomy. In order to gain knowledge about the workings of the human body he learned to dissect cadavers; the resultant sketches and diagrams are some of the most meticulous studies of the human body made before the twentieth century. Not only did he fully record the known structure and functions of the body; he also made numerous revolutionary discoveries about anatomy, all of which contradicted contemporary medical beliefs. His interest in anatomy would persist throughout his life. As with many of his studies, he planned eventually to collect the results into a book- in this case the title was to be Of the Human Figure. However, like so many of his paintings, none of these books ever came to completion.

# data[].attributes.attachment_link.title
Leonardo da Vinci Study Guide: The Master Moves to Milan: 1482-1489 | SparkNotes

# data[].attributes.attachment_link.description
In 1482, Leonardo moved to Milan. He was now a master, but he seemed dissatisfied with his life in Florence. Science interested him just as much as painting, and he had begun making copious scientific speculations in his notebooks. Thus even though Milan was less artistically-centered than Florence, the city would provide him with more opportunities to apply his various interests.

# data[].attributes.attachment_link.readable_text
Leonardo da Vinci was a painter, engineer, architect, inventor, and student of all things scientific. His natural genius crossed so many disciplines that he epitomized the term " Renaissance man." Today he remains best known for two of his paintings, " Mona Lisa" and "The Last Supper." Largely self-educated, he filled dozens of secret notebooks with inventions, observations and theories about pursuits from aeronautics to human anatomy. His combination of intellect and imagination allowed him to create, at least on paper, such inventions as the bicycle, the helicopter and an airplane based on the physiology and flying ability of a bat.
WATCH: Behind the Da Vinci Code on HISTORY Vault
When Was Leonardo da Vinci Born?

Da Vinci was born in Anchiano, Tuscany (now Italy), in 1452, close to the town of Vinci that provided the surname we associate with him today. In his own time he was known just as Leonardo or as "Il Florentine," since he lived near Florence-and was famed as an artist, inventor and thinker.

Did you know? Leonardo da Vinci's father, an attorney and notary, and his peasant mother were never married to one another, and Leonardo was the only child they had together. With other partners, they had a total of 17 other children, da Vinci's half-siblings.

Da Vinci's parents weren't married, and his mother, Caterina, a peasant, wed another man while da Vinci was very young and began a new family. Beginning around age 5, he lived on the estate in Vinci that belonged to the family of his father, Ser Peiro, an attorney and notary. Da Vinci's uncle, who had a particular appreciation for nature that da Vinci grew to share, also helped raise him.

Early Career

Da Vinci received no formal education beyond basic reading, writing and math, but his father appreciated his artistic talent and apprenticed him at around age 15 to the noted sculptor and painter Andrea del Verrocchio of Florence. For about a decade, da Vinci refined his painting and sculpting techniques and trained in mechanical arts.

When he was 20, in 1472, the painters' guild of Florence offered da Vinci membership, but he remained with Verrocchio until he became an independent master in 1478. Around 1482, he began to paint his first commissioned work, The Adoration of the Magi, for Florence's San Donato, a Scopeto monastery.

However, da Vinci never completed that piece, because shortly thereafter he relocated to Milan to work for the ruling Sforza clan, serving as an engineer, painter, architect, designer of court festivals and, most notably, a sculptor.

The family asked da Vinci to create a magnificent 16-foot-tall equestrian statue, in bronze, to honor dynasty founder Francesco Sforza. Da Vinci worked on the project on and off for 12 years, and in 1493 a clay model was ready to display. Imminent war, however, meant repurposing the bronze earmarked for the sculpture into cannons, and the clay model was destroyed in the conflict after the ruling Sforza duke fell from power in 1499.

'The Last Supper'

Although relatively few of da Vinci's paintings and sculptures survive-in part because his total output was quite small-two of his extant works are among the world's most well-known and admired paintings.

The first is da Vinci's "The Last Supper," painted during his time in Milan, from about 1495 to 1498. A tempera and oil mural on plaster, "The Last Supper" was created for the refectory of the city's Monastery of Santa Maria delle Grazie. Also known as "The Cenacle," this work measures about 15 by 29 feet and is the artist's only surviving fresco. It depicts the Passover dinner during which Jesus Christ addresses the Apostles and says, "One of you shall betray me."

One of the painting's stellar features is each Apostle's distinct emotive expression and body language. Its composition, in which Jesus is centered among yet isolated from the Apostles, has influenced generations of painters.

'Mona Lisa'

When Milan was invaded by the French in 1499 and the Sforza family fled, da Vinci escaped as well, possibly first to Venice and then to Florence. There, he painted a series of portraits that included "La Gioconda," a 21-by-31-inch work that's best known today as "Mona Lisa." Painted between approximately 1503 and 1506, the woman depicted-especially because of her mysterious slight smile-has been the subject of speculation for centuries.

In the past she was often thought to be Mona Lisa Gherardini, a courtesan, but current scholarship indicates that she was Lisa del Giocondo, wife of Florentine merchant Francisco del Giocondo. Today, the portrait-the only da Vinci portrait from this period that survives-is housed at the Louvre Museum in Paris, France, where it attracts millions of visitors each year.

Around 1506, da Vinci returned to Milan, along with a group of his students and disciples, including young aristocrat Francesco Melzi, who would be Leonardo's closest companion until the artist's death. Ironically, the victor over the Duke Ludovico Sforza, Gian Giacomo Trivulzio, commissioned da Vinci to sculpt his grand equestrian-statue tomb. It, too, was never completed (this time because Trivulzio scaled back his plan). Da Vinci spent seven years in Milan, followed by three more in Rome after Milan once again became inhospitable because of political strife.

Inventions and Philosophy

Da Vinci's interests ranged far beyond fine art. He studied nature, mechanics, anatomy, physics, architecture, weaponry and more, often creating accurate, workable designs for machines like the bicycle, helicopter, submarine and military tank that would not come to fruition for centuries. He was, wrote Sigmund Freud, "like a man who awoke too early in the darkness, while the others were all still asleep."

Several themes could be said to unite da Vinci's eclectic interests. Most notably, he believed that sight was mankind's most important sense and that "saper vedere"("knowing how to see") was crucial to living all aspects of life fully. He saw science and art as complementary rather than distinct disciplines, and thought that ideas formulated in one realm could-and should-inform the other.

Probably because of his abundance of diverse interests, da Vinci failed to complete a significant number of his paintings and projects. He spent a great deal of time immersing himself in nature, testing scientific laws, dissecting bodies (human and animal) and thinking and writing about his observations.

Da Vinci's Notebooks

At some point in the early 1490s, da Vinci began filling notebooks related to four broad themes-painting, architecture, mechanics and human anatomy-creating thousands of pages of neatly drawn illustrations and densely penned commentary, some of which (thanks to left-handed "mirror script") was indecipherable to others.

The notebooks-often referred to as da Vinci's manuscripts and "codices"-are housed today in museum collections after having been scattered after his death. The Codex Atlanticus, for instance, includes a plan for a 65-foot mechanical bat, essentially a flying machine based on the physiology of the bat and on the principles of aeronautics and physics.

Other notebooks contained da Vinci's anatomical studies of the human skeleton, muscles, brain, and digestive and reproductive systems, which brought new understanding of the human body to a wider audience. However, because they weren't published in the 1500s, da Vinci's notebooks had little influence on scientific advancement in the Renaissance period.

How Did Leonardo da Vinci Die?

Da Vinci left Italy for good in 1516, when French ruler Francis I generously offered him the title of "Premier Painter and Engineer and Architect to the King," which afforded him the opportunity to paint and draw at his leisure while living in a country manor house, the Château of Cloux, near Amboise in France.

Although accompanied by Melzi, to whom he would leave his estate, the bitter tone in drafts of some of his correspondence from this period indicate that da Vinci's final years may not have been very happy ones. (Melzi would go on to marry and have a son, whose heirs, upon his death, sold da Vinci's estate.)

Da Vinci died at Cloux (now Clos-Lucé) in 1519 at age 67. He was buried nearby in the palace church of Saint-Florentin. The French Revolution nearly obliterated the church, and its remains were completely demolished in the early 1800s, making it impossible to identify da Vinci's exact gravesite.
READ MORE: Renaissance Art

# data[].attributes.attachment_link.description
Leonardo da Vinci was a painter, engineer, architect, inventor, and student of all things scientific. His natural genius crossed so many disciplines that he epitomized the term " Renaissance man." Today he remains best known for two of his paintings, " Mona Lisa" and "The Last Supper."

# data[].attributes.body
Nuland, Sherwin. Leonardo Da Vinci: A Life. United States, Penguin Publishing Group, 2005.

# data[].attributes.body
Leonardo was born in Anchiano, on April 15, 1452. At the time of his birth his parents were not married to each other. Later on, they each married different people.

Quote #1
"According to some, he was born in the nearby town of Anchiano and brought to Vinci only after a few years, or perhaps it was a few months." (3)

# data[].attributes.body
The name of Leonardo's mother was Caterina, she was a young peasan lady from Anchiano. After the birth of Leonardo, she got married to di Piero of Vinci. Very little is known about Caterina. She was not really in Leonardo's life but it is likely that she nursed and watched Leonardo the first couple years of his life.

Quote #2
"His mother was a girl of the town of Vinci or a neighboring community." (13)

# data[].attributes.body
Leonardo was an "illegitimate" son of Ser Piero. Ser Piero was a notary. Soon after the birth of Leonardo, his father married Albiera di Giovanni Amadori. Albiera grew to love Leonardo as her own. It was assumed that Leonardo spent most of his life in Vinci, with his father. Although, there is no way of actually knowing because he could've just been sent to his father a lot.

Quote #3
"ONE DAY in the early spring of 1452, a prosperous eighty-year-old landowner set down a few details of a recent notable event in his family: "A grandson of mine was born, son of Sir Piero my son, on April 15 Saturday at three o'clock in the night." (12)

# data[].attributes.attachment_link.title
Leonardo da Vinci | Biography, Art, Paintings, Mona Lisa, Drawings, Inventions, Achievements, & Facts

# data[].attributes.attachment_link.description
Leonardo da Vinci, the Renaissance genius, revolutionized art and science with his masterpieces like the Mona Lisa while pioneering advancements in anatomy, engineering, and invention.

# data[].attributes.body
Leonardo's education lasted till the age of 15. He learned basic math, reading, and writing. Not much is known about whether or not he had any formal education. He had little mastery of classical language and did not attempt to sudy it besides using Latin occasionally. His understaning of Greek, Latin, and Arabic came from reading it in Italian translation and sharing it with his colleagues. Because of this people called him "an unlettered man".

Quote #4
"Leonardo's education until the age of approximately fifteen was probably to a large extent in the hand of Alberia and her mother-in-law, Monna Lucia, who was fifty-nine when he was born." (21)

# data[].attributes.body
At around the age of 15-18, Leonardo and his family moved to Florence. His father introduced him to Andrea di Cione (Verrocchio), she was a leading artist within that city. From there, he became her apprentice. He learned and improved his painting and sculpting techniques.

# data[].attributes.body
Leonardo was fascinate with anatomy. He had the intention to enhance the art in anatomy, but during dissections he saw a fascinating and interesting side to it. He was the first artist to dissect a human body. He was also most likely the first person to accurately represent the internal human organ.

Quote #5
"Begun by Leonardo only with the intent of enhancing the art, in his probing hand these dissections gradually began to take on a more investigative aspect." (27)

# data[].attributes.body
Leonardo moved to Milan in 1482. He moved there primarily to be able to work with the ruler. Before he had moved to Milan, he wrote a letter to the ruler, Ludovico Sforza, for a job opportunity.

Quote #6
"Earlier that year, Leonardo sent the ruler of Milan, Ludovico Sforza, a letter that was essentially a job application." (31)

# data[].attributes.body
In Milan, Leonardo worked for Duke Ludovico. He took up the job to be a military engineer. He used his artistic abilities to design machines, weapons, and buildings. He had to study the state of the battle, carefully examine the weaponry used for war, etc.

# data[].attributes.body
In 1494, Ludovico Sforza, became the duke of Milan. There, he asked Leonardo to play the lute for him. Leonardo played an instrument for him that he made himself. The instrument was constructed from silver in the shape of a horse's head, this allowed the instrument to have a louder tone and be more deep.

Quote #7
"When Ludovico Sforza became due of Milan in 1494, he invited Leonardo most ceremoniously to come and play the lute before him." (38)

# data[].attributes.body
Leonardo had promised to make a statue for Ludovico. Sadly, the statue did not turn out good. He retried the horse again, but it did not work out again. He discontinued to work on this project. Later some time, the bronze that was used for casting the statue got sent to Ferrara. It was from Ludovico, to be made into canon. Leonardo carried on with his plan but he knew that it wasn't going to work. Eventually, the horse got destroyed from target practice and got decomposed.

# data[].attributes.body
Although Leonardo began his fascination with anatomy in Florence, Milan was where his fascination became more broad. His paintings became more precise and was more efficient. In the 1952 text, Leonardo da Vinci on the Human body, it has been expressed by Charles O'Malley and J.B de C. M. Saunders, that it contained errors with indications that most of his knowledge of anatomy had been from reading, observation and animal dissection, instead of actual human dissection.

# data[].attributes.body
The Last Supper is a memorable piece of art for the Christians.  The painting represents a significant part in the Bible. Leonardo wrote, "That figure is most praiseworthy which, by its actions, expresses the passions of the soul." Although we have never met these people in the picture, we know them.

# data[].attributes.body
Leonardo moved back to Florence. As recorded in a bank record, he withdrew 600 florins in April 24, 1500. After 17-ish, 18-ish years of living in Milan, he is returning to where he came from. He realized that things have changed since he lived there.

# data[].attributes.body
Leonardo was at the Santa Maria Nuova hospital to do dissection. He met a man that was a hundred years old, and according to him, in good health. After the man had passed while sitting in bed, Leonardo examined the anatomy. He discovered that the cause of the death was from failure in blood.

Quote #8
"Shortly thereafter, the ancient fellow suddenly died while sitting in bed, and Leonardo, who "examined the anatomy to ascertain the cause of the sweet death," discovered that it was caused by "a weakness through failure of blood and of the artery that feeds the heat and the lower members which I found to be very patched and shrunk and withered; and the result of this examination I wrote down very carefully." (71)

# data[].attributes.body
Leonardo begins many art but doesn't finish them. He receives many recognitions for his art, even the ones that are unfinished. Some of the praise that he got for his art are, Annunciation, Saint Jerome, and the Adoration of the Magi. Because of him not finishing some art but still receiving recognition, he was known as a man who, "began many things that he never completed."

# data[].attributes.body
While in Milan, Leonardo worked on architectural projects. The city's great cathedral was one of the most important one that he did. He worked on it between 1487 and 1490. One project that he worked on but later lost interest was a wooden model for the Duomo.

# data[].attributes.body
Luca Pacioli was a friend of Leonardo. He was a successful and recognizable mathematician. He was also a member of the Franciscan Order. Pacioli and Leonardo became close friends and ended up living together. The two of them later created a treatise, De divine proportione.

# data[].attributes.body
Michelangelo was a young artist. He dislikes Leonardo da Vinci. He had once asked Leonardo about how he made a drawing of a horse to be cast, but was not able to cast it so had to abandon it. The two were very different, they had different answer to the approaches of a human figure in action.

Quote #10
"This second Florentine sojourn provided the occasion for an abortive tournament of the titans between Leonardo and young Michelangelo, who by then had developed a considerable antipathy toward him." (73)

# data[].attributes.body
Leonardo had started painting the Mona Lisa at around 1503.  It is one of his most popular paintings. People really liked how realistic the Mona Lisa was. This piece of artwork also showed Leonardo's proficiency in anatomy and realism.

Quote #9
"The art historian Sydney Freedberg has said of the Mona Lisa that it is "an image in which a breathing instant had a composure for all time are held in suspension." (48)

# data[].attributes.body
He had made his will on the eve of Easter, 1519. In it, all of his notebooks were left to Melzi, and he arranged for the masses to be said to three different church, scattering his heritage. Leonardo da Vinci dies on May 2, 1519.