Luca Pacioli - mathematician and geometrist

Friar who became known as ‘Father of Accounting’

Jacopo de' Barbari's portrait of Luca Pacioli with
an unknown geometry student attending

Luca Pacioli, the Franciscan friar and mathematician who would become known as the ‘Father of Accounting’, died on this day in 1517 in Sansepolcro, a town in eastern Tuscany in the province of Arezzo.

Taking advantage of the development of the printing press, Pacioli is thought to have published at least 10 mathematical textbooks, of which the best known is his Summa de arithmetica, geometria, Proportioni et proportionalita - usually known as simply Summa.

Published in Venice in 1494, it was a comprehensive treatise of every aspect of mathematical knowledge that had been explored to that time and the first book to include a description of the double-entry book-keeping system, widely used by Venetian merchants. The principals he outlined in Summa still influence business practices today.

Pacioli, who taught mathematics in several Italian cities and was appointed the first chair in mathematics at the University of Perugia, was a figure of influence in the arts world as well as commerce.

Having possibly been a student of Piero della Francesca, a pioneer of the use of geometry and perspective in painting and, like Pacioli, a native of Sansepolcro, he later became a close friend of Leonardo da Vinci.

Da Vinci’s drawings illustrated another of his notable works, Divina proportione, an exploration of mathematical and artistic proportion, focussing especially on the so-called golden ratio of proportionality and its application in architecture.

The title page of a 1523 edition of
Pacioli's Summa, his greatest work

Pacioli was around 70 years old when he died, having been born between 1446 and 1448. He received an education based around maths and commerce in Sansepolcro, before moving to Venice in around 1464, continuing his studies there and being appointed tutor to the three sons of a wealthy merchant, Antonio Rompiasi.

From Venice, he moved to Rome, where he became friends with the architect, artist and mathematician Leon Battista Alberti, before returning to Sansepolcro to enter the Franciscan Order in 1470. In the following years, he taught mathematics in Perugia, Florence, Venice, Milan, Pisa, Bologna and Rome. 

After the publication of Summa in 1494, he accepted an invitation from Duke Ludovico Sforza to work in Milan. There he met and taught Leonardo da Vinci, for a time living with the polymath as a house guest. Da Vinci acquired his knowledge of geometry and its applications in art and architecture from his association with Pacioli.

Pacioli might have remained in Milan longer had the city not been overrun by Louis XII of France at the start of the Second Italian War. Ludovico Sforza fled the city, as did Pacioli and da Vinci, first to Mantua, and then Venice.  

Although Pacioli left a significant intellectual legacy in the fields of accounting and mathematics, he was effectively accused of plagiarism by 16th-century artist and historian Giorgio Vasari.

Vasari was critical of the inclusion in Divina proportione - without credit - of the translated text of della Francesca’s book, originally written in Latin, De quinque corporibus regularibus, about the geometry of polyhedra - solid objects with polygonal faces, such as pyramids. Other scholars, though, argue that the edition of Divina proportione that Vasari read may have been appended to include della Francesca’s work after Pacioli’s death.

Pacioli’s final university teaching post is thought to have been in Rome during 1514 and 1515, before returning to Sansepolcro as his health began to decline.

Piero della Francesca's painting, The Resurrection,
which may have spared Sansepolcro a WW2 attack

Travel tip:

Sansepolcro, a town nestled in the Valtiberina valley at the foot of the Apennines, has a rich artistic and cultural heritage. As well as being the home town of Luca Pacioli,  it is known as the birthplace of Renaissance master Piero della Francesca, whose brilliant painting, The Resurrection, is housed in the Museo Civico di Sansepolcro. The presence of the painting probably saved the town from destruction in World War Two after Tony Clarke, an art-loving British Royal Horse Artillery officer who, halted a planned  Allied artillery attack. Another treasure, a 12th-century polychrome wooden crucifix known as the Volto Santo, can be seen in the town’s beautiful Romanesque duomo, the Cattedrale di San Giovanni Evangelista. Look out also for The Lamentation over the Dead Christ, by the Mannerist painter Rosso Fiorentino, in the church of San Lorenzo. Other museums include the Aboca Museum, dedicated to the history of medicinal herbs, featuring antique books, laboratory tools, and botanical collections, as well as the Bernardini-Fatti Museum of Antique Windows, the world’s first museum dedicated to stained glass windows. Sansepolcro’s medieval streets, fortified walls, and historic palazzos make it a delightful place to explore. 

One of Arezzo's notable sights is its sharply
sloping medieval main square, Piazza Grande

Travel tip:

Arezzo is one of the wealthiest cities in Tuscany. It is situated about 80km (50 miles) southeast of Florence, at the confluence of four valleys - Tiberina, Casentino, Valdarno and Valdichiana. Its medieval centre suffered massive damage during the Second World War yet the Basilica di San Francesco, with its beautiful frescoes by Piero della Francesca, the central Piazza Grande, with its sloping pavement in red brick, and the Medici Fortress, the duomo and a Roman amphitheatre survived, among other historic sights.  Arezzo’s original cathedral was built on the nearby Pionta Hill, over the burial place of Donatus of Arezzo, who was martyred in 363. In 1203 Pope Innocent III had the cathedral moved within the city's walls, to the current site in another elevated position a short walk from Piazza Grande.  The interior contains several notable artworks, including a relief by Donatello, entitled Baptism of Christ, and a cenotaph to Guido Tarlati, lord of Arezzo until 1327, said to be designed by Giotto, near to which is Piero della Francesca's Mary Magdalene.  The wooden choir of the Grand Chapel was designed by Giorgio Vasari, who was born in Arezzo. The city is home to an annual medieval festival called the Giostra del Saracino (Saracen Joust), in which "knights" on horseback representing different areas of the town charge at a wooden target attached to a carving of a Saracen king and score points according to accuracy. Arezzo had a starring role in Roberto Benigni's film Life Is Beautiful, which won three Academy Awards, as the place in which the main characters live before they are shipped off to a Nazi concentration camp.

Also on this day:

1918: The death in action of WW1 fighter pilot Francesco Baracca

1932: The birth of actress twin sisters Pier Angeli and Marisa Pavan

1951: The birth of Giro d’Italia-winning cyclist Francesco Moser

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Ulisse Dini – mathematician and politician

Patriotic professor was proud to serve the new Kingdom of Italy

Ulisse Dini went to Paris to further
his mathematical education

The mathematician Ulisse Dini, who wrote many books and papers based on his research and came up with original theories that advanced the knowledge in his field, died on this day in 1918 in Pisa.

Now regarded as one of the 19th century's most important contributors to the evolution of mathematics, Dini was also active as a politician and was elected to Pisa city council before becoming a member of the parliament of the new Kingdom of Italy. 

His political views were shaped by the changing landscape of Italy while he was growing up, as the country moved closer to unification, and he was always keen to help his local area and his country.

Originally intending to become a teacher, Dini, who was born in Pisa in 1845 when the city was part of the Grand Duchy of Tuscany, attended the Scuola Superiore in Pisa, a teachers’ college, where one of his teachers was the mathematician, Enrico Betti.

However, in 1865, Dini received a scholarship that enabled him to go to Paris to further his mathematical studies, where he was taught by the French mathematicians, Charles Hermite and Joseph Bertrand. He produced seven mathematical publications based on the research he was able to do while living in Paris.

The following year he was appointed by the University of Pisa to teach algebra and geodesy, the science of measuring and representing the geometry, gravity, and spatial orientation of the earth in temporally varying 3D. 

Dini succeeded Betti as professor for analysis and geometry at the university in 1871. He was rettore - rector - of Pisa University from 1888 until 1890, and of the Scuola Superiore in Pisa from 1908 until his death. One of his students at Pisa University was the Italian mathematician, Luigi Bianchi.

Enrico Betti was 
Dini's teacher

Professor Dini is particularly known for his contribution to the branch of mathematics known as real analysis, which studies the behaviours of real numbers, sequences and series of real numbers, and real functions. He collected his theories in his book, ‘Fondamenti per la teorica delle funzioni di variabili reali’, a collection of ideas that were entirely originated by himself. 

Despite his heavy academic workload, Dini had entered politics by the age of 25, when he was elected to the city council in Pisa.

When Dini was 13, the fight had begun to win back areas of the country from the Austrian occupying forces. Within a few years, forces representing the Kingdom of Sardinia had gained control of many areas in central Italy. Victor Emmanuel II of Sardinia later assumed the title of King of Italy, and Giuseppe Garibaldi and his 1000 volunteers set out to unify the country, starting by invading Sicily. With his forces, Garibaldi gradually moved northwards up the peninsula and this led to the official proclamation of the Kingdom of Italy in 1861.

Ulisse Dini is remembered by historians and writers as an upright, honest, and kind man, and for being devoted to the well-being of his city and country. He divided his time between teaching and research and public service, and in 1892 he became a senator in the Italian parliament. 

In the field of mathematical analysis, he is remembered for being the originator of Dini criterion and Implicit function theorem. He was considered one of the most important European mathematicians of the 19th century and he was elected as an honorary member of the London Mathematical Society.

After his death, Dini was buried in Pisa’s monumental cemetery, Campo Santo, on the northern edge of the Campo dei Miracoli. There is a statue commemorating his life in Via Ulisse Dini in the city’s historic centre. 

The 15th century Palazzo della Sapienza, built by
the Medici family, remains the heart of the university

Travel tip: 

The University of Pisa, founded in 1343, is one of the oldest in Italy and rivals Rome’s Sapienza University for the title of the country’s best. Various scholars place its origins in the 11th century, although those claims have never been verified. What is certain is that the papal seal “In Supremae dignitatis”, issued by Pope Clement VI on 3 September 1343, granted the Studium in Pisa the title of Studium Generale. At the outset in Pisa, lessons in Theology, Civil Law, Canon Law and Medicine were established. In its early years, the university struggled to survive at times, but investment from the Medici family in the 15th century saw the construction of the Palazzo della Sapienza, still the centre of the present-day university. Cosimo I de’ Medici instituted programmes to improve the quality of teaching and later Galileo Galilei, universally seen as the founder of modern science, was initially a student and then a teacher of Mathematics at the university. Today, Pisa has a student population of around 50,000, who give the city a vibrant cafe and bar scene.

The bronze statue of Ulisse 
Dini was finished in 1927

Travel tip: 

The bronze sculpture of Ulisse Dini in Via Ulisse Dini was created in 1927 by Leonardo Bistolfi, a sculptor from Casale Monferrato in Piedmont, whose work was praised for its likeness to the subject.  Via Ulisse Dini goes from Borgo Stretto to Piazza dei Cavalieri and Dini’s statue stands beside the Church of Santo Stefano dei Cavalieri. Once a stream flanked by wooden houses belonging to the city’s higher social classes, the street subsequently became a centre for forges and metalworkers’ workshops. Some of the stone arches lining the modern street date back to the 12th century.  Aside from the Campo dei Miracoli, the biggest attraction of which is the city’s world famous leaning bell tower, Pisa offers visitors a wealth of well-preserved Romanesque buildings, Gothic churches and Renaissance piazzas.

Also on this day:

312: The Battle of Ponte Milvio

1639: The death of composer Stefano Landi

1922: Mussolini’s Fascists march on Rome

1963: The birth of singer-songwriter Eros Ramazzotti

1973: The death of actor and illustrator Sergio Tòfano

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