# Giovanni Francesco Fagnano dei Toschi

### Quick Info

Born
31 January 1715
Sinigaglia (now Senigallia, Italy)
Died
14 May 1797
Sinigaglia (now Senigallia, Italy)

Summary
Giovanni Fagnano was an Italian priest who worked on geometry and calculus.

### Biography

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 12th 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Fagnano continued his father's work on the triangle and wrote an unpublished treatise on the topic.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
$x^{n}\sin(x)$ and $x^{n}\cos(x)$
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.

### References (show)

1. A Natucci, Biography in Dictionary of Scientific Biography (New York 1970-1990). See THIS LINK.
2. R Ayoub, The lemniscate and Fagnano's contributions to elliptic integrals, Arch. Hist. Exact Sci. 29 (2) (1984), 131-149.
3. G Loria, Storia delle matematiche (Milan, 1950), 664.