**Giovanni Fagnano** was the son of Giulio Fagnano dei Toschi. Giovanni 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 12^{th} century had been Lamberto Scannabecchi who became Pope Honorius II in 1124.

Giovanni's father Giulio Fagnano held high office in Sinigaglia. He was appointed gonfaloniere in 1723 when Giovanni was eight years old. *Gonfaloniere* literally means "standard bearer" and it was a title of high civic magistrates in the medieval Italian city-states such as Sinigaglia.

Giovanni was one of many children in his family but the only one to follow his father's interest in mathematics. He entered the Church, being ordained priest, then appointed as canon of the cathedral in Sinigaglia in 1752. In 1755 Fagnano was appointed as archpriest, a very high rank to achieve.

Fagnano continued his father's work on the triangle and wrote an unpublished treatise on the topic. One theorem on the triangle which he discovered is worth quoting. He proved that given any triangle *T*, then the triangle whose vertices are the bases of the altitudes of *T* has these altitudes as the bisectors of its angles.

Fagnano also considered integration computing the integrals of

x^{n}sin(x) andx^{n}cos(x)

by parts. In addition he calculated the integral of tan(*x*) as -log cos(*x*) and of cot(*x*) as log sin(*x*).

Some of Fagnano's publications appear in the *Nova acta eruditorum* in 1774. However, he never achieved the international standing of his father although he did publish some work outside Italy.

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