The Beginning of Modern Mathematics

The late fifteenth century was an exciting time in Europe. Western civilization woke with a start after the slumbers of the medieval age. Johannes Gutenberg’s printing press arrived in 1450 and changed everything. Universities in Bologna, Oxford, Salamanca, Paris and elsewhere began to flourish. Leonardo da Vinci was in his prime and Christopher Columbus was discovering a new world.


Illustrations by Leonardo da Vinci in Pacioli’s De Divina Proportione.

Luca Pacioli

Luca Pacioli (1447-1517) was an Italian Renaissance mathematician and a contemporary and friend of Leonardo da Vinci (1452-1519). He is often called the “Father of Accounting”, as he was the first person to publish a description of double-entry book-keeping. At the age of 25, he became a Franciscan friar. In 1477 he was appointed to the chair of mathematics in Perugia. He wrote several textbooks on mathematics, the most noteworthy being his Summa de Arithmetica, Geometria, Proportioni et Proportionalita, which was published in Venice in 1494.

In 1497, Luca moved to Milan where he befriended Leonardo da Vinci. They lived there together for two years. Their close association and collaboration continued for many years, and Luca taught mathematics to Leonardo.

Pacioli’s Summa synthesized the mathematical knowledge of the time and was the first work on mathematics in the Italian language. It provided a comprehensive treatment of Renaissance mathematics, including arithmetic, geometry and algebra, and also contained an account of the book-keeping methods in use at that time. Pacioli presented his methods by providing examples, but also by giving general arguments and logical proofs of many statements. Thus, his book initiated a movement towards rigorous methods in mathematics, as in the tradition of the Greek geometers.

De Divina Proportione


Title page of Pacioli’s De Divina Proportione,

In 1509, De Divina Proportione On the Divine Proportion was published in Venice. This book discusses both mathematical and aesthetic proportions In particular, it describes the mathematics of the golden ratio and its applications in art and architecture. The illustrations for this work were drawn by Leonardo. They included depictions of the regular polyhedra in skeletal form, giving an illusion of solidity while providing visibility through to the backs of these geometric forms. Pacioli also discussed the use of perspective by artists such as Piero della Francesca. Indeed, much of the content of Pacioli’s book was taken directly from Francesca’s writing. And Pacioli was accused of plagiarism by the art historian Georgio Vasari.

Cubics and Quartics

In 1494 Luca Pacioli published his Summa de Arithmetica, presenting methods of solving linear and quadratic equations. He used the Italian word “cosa” (the thing) for an unknown quantity that we would denote by x today. He presented no method of solving general cubic equations of the form

x3 + a x2 + b x + c = 0

(we can take the coefficient of the highest power to be unity). Indeed, Pacioli stated that no general method of solving cubic equations was known and that probably no such method was possible. This acted as a spur to other mathematicians to find a solution of the cubic. Scipione del Ferro (1465-1526) managed to solve the depressed cubic some time later, and Tartaglia, Cardano, Ferrari and Bombelli completed the job of solving general cubics and quartics.

This period could be seen as the time when modern mathematics was born.

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