# Giulio Carlo Fagnano dei Toschi

### Born: 6 Dec 1682 in Sinigaglia (now Senigallia), Italy Died: 26 Sept 1766 in Sinigaglia (now Senigallia), Italy

Giulio Fagnano's father was Francesco Fagnano and his mother was Camilla Bartolini. Giulio was born into one of the leading families in Sinigaglia. The town of Sinigaglia, now known as Senigallia, is in central Italy and at the time of Giulio's birth was part of the Papal States. In fact the family went back very many generations in their association with Sinigaglia and one of the members of the family in the 12th century had been Lamberto Scannabecchi who became Pope Honorius II in 1124.

Fagnano was brought up to follow the family tradition of high office in Sinigaglia. He was appointed gonfaloniere in 1723. Gonfaloniere literally means "standard bearer" and it was a title of high civic magistrates in the medieval Italian city-states such as Sinigaglia. Such offices were not easy in these times and Fagnano was subjected to many false charges made against him by envious citizens who were maliciously trying to damage his reputation. Fagnano had many children, one of whom was Giovanni Fagnano who followed in his father's footsteps becoming interested in mathematics.

Giulio Fagnano was self educated in mathematics and treated the subject as a hobby. However, he achieved considerable international fame as a mathematician, and rightly so given the outstanding contributions which he made on a number of different topics.

Fagnano suggested new methods of solving equations of degree 2, 3 and 4. He improved Bombelli's work on complex numbers giving a famous formula

π/2 = √ log[(1 - i)/(1 + i)].

One of the topics for which Fagnano is best known is his work on triangles. Natucci in [1] writes:-

... he may well be considered the founder of the geometry of the triangle.

Considering triangles he looked at interesting problems such as:

Given ABC find P minimising PA2 + PB2 + PC2.

For a quadrilateral ABCD find P minimising AP + BP + CP + DP.

He also discovered that if X is the centre of gravity of the triangle ABC then XA2 + XB2 + XC2 = (AB2 + BC2 + CA2)/3.

In his study of the rectification of the lemniscate, Fagnano introduced ingenious analytic transformations that laid the foundation for the theory of elliptic integrals and his work was to lead to elliptic functions. Fagnano collected many of his published works, and a few unpublished ones, and produced the two volume treatise Produzioni matematiche in 1750. In 1751 Euler was asked to examine Produzioni matematiche and he found in this treatise relations between special types of elliptic integrals, that express the length of an arc of a lemniscate, which were quite unexpected to him. Generalising Fagnano's results, Euler went on to create a general theory of these integrals, in particular giving the famous addition formula for elliptic integrals. Fagnano had proved the duplication formula, a particular case of the addition formula, for the integrals.

In fact Fagnano had proved remarkable properties of the lemniscate, including the fact that its arcs may be divided in n equal parts using a ruler and compass construction, where n = 2×2m, 3×2m, or 5×2m. He also found the area of the lemniscate.

Fagnano made many other major contributions but his mathematical work was not without controversy. He was involved in priority disputes with Nicolaus(I) Bernoulli and, not surprisingly, the big dispute of the day which was between the supporters of Newton and those of Leibniz. Brook Taylor issued a challenge which Bernoulli and Fagnano both answered; the background and details of this controversy are studied in [3].

Fagnano received many honours. He had the title of count conferred on him by Louis XV in 1721, was elected to the Royal Society of London in 1723, and was made a marquis of Sant' Onofrio in 1745. In addition he was elected to the Berlin Academy of Sciences and was proposed for the Paris Académie des Sciences in 1766 but died before he could be elected.

Article by: J J O'Connor and E F Robertson

July 2000

MacTutor History of Mathematics
[http://www-history.mcs.st-andrews.ac.uk/Biographies/Fagnano_Giulio.html]