View source for Giulio Carlo de' Toschi di Fagnano

{{Short description|Italian mathematician (1682–1766)}}
[[File:Giulio Fagnano.jpg|thumb|Portrait of Giulio Fagnano (1682–1766)]]
'''Giulio Carlo, Count [[Fagnano family branch|Fagnano]], Marquis de Toschi''' (26 September 1682 — 18 May 1766)<ref name="CE"/><!-- MacTutor says 6 Dec – 26 Sept --> was an Italian [[mathematician]]. He was probably the first to direct attention to the theory of [[elliptic integral]]s. Fagnano was born in [[Senigallia]] (at the time spelled "Sinigaglia"), and also died there.<ref name="CE"/>

==Life==
Giulio Fagnano was born to Francesco Fagnano and Camilla Bartolini in [[Senigallia]] (at the time spelled "Sinigaglia") in 1682.<ref name="MacTutor"/>

Fagnano had twelve children.<ref name="CE"/> One, [[Giovanni Fagnano]], was also well-known as a mathematician. Another of Fagnano's children became a Benedictine nun.<ref name="CE"/>

In 1721, Fagnano was made a [[count]] by [[Louis XV]];<ref name="MacTutor"/> in 1723, he was appointed ''[[gonfaloniere]]'' of Senigallia<ref name="MacTutor"/> and elected to the [[Royal Society of London]];<ref name="MacTutor"/> in 1745 he was made a [[marquis]] of Sant' Onofrio.{{huh|date=December 2022}}<ref name="MacTutor"/>

==Mathematical work==
[[Image:Fagnano - Produzioni matematiche, 1750 - 1506824.jpg|thumb|''Produzioni matematiche'', 1750]]
Fagnano made his higher studies at the [[Collegio Clementino]] in [[Rome]], and there won great distinction — except in mathematics, to which his aversion was extreme.<ref name="CE"/> Only after his college course did he take up the study of mathematics; but then, without help from any teacher, he mastered mathematics from its foundations.<ref name="CE"/> Most of his important researches were published in the ''Giornale de' Letterati d'Italia''.<ref name="CE"/>

Fagnano is best known for investigations on the length and division of arcs of certain curves, especially the [[Lemniscate of Bernoulli|lemniscate]] (cf. [[Lemniscate elliptic functions]]); this seems also to have been in his own estimation his most important work, since he had the figure of the lemniscate with the inscription "Multifariam divisa atque dimensa Deo veritatis gloria" engraved on the title-page of his ''Produzioni Matematiche'',<ref name="Fagnano1750"/> which he published in two volumes (Pesaro, 1750), and dedicated to [[Pope Benedict XIV]]. The same figure and words "Deo veritatis gloria" also appear on his tomb.

Failing to [[arc length|rectify]] the [[ellipse]] or [[hyperbola]], Fagnano attempted to determine [[Arc (geometry)|arc]]s whose difference is rectifiable.
The word "rectifiable" meant at that time that the length can be found explicitly, which is different from its modern meaning. He also pointed out the remarkable analogy existing between the [[integral]]s which represent the arc of a [[circle]] and the arc of a lemniscate. He also proved the formula

:<math>\pi = 2i\log{1-i\over 1+i}</math>

where <math>i</math> stands for <math>\sqrt{-1}</math>.

Some mathematicians objected to his methods of analysis founded on the [[infinitesimal calculus]]. The most prominent of these were [[Vincenzo Viviani|Viviani]], [[Philippe de La Hire|De la Hire]] and [[Michel Rolle|Rolle]].
[[File:Acta Eruditorum - III geometria, 1763 – BEIC 13450778.jpg|thumb|Illustratiom from ''Illustratio theorematis actis lipsiensibus...'' published in [[Acta Eruditorum]], 1762]]

==References==
*{{Rouse History of Mathematics}}
<references>
<ref name="CE">{{Catholic Encyclopedia|wstitle=Giulio Carlo de' Toschi di Fagnano}}</ref>

<ref name="Fagnano1750">{{cite book |author=Giulio Fagnano |title=Produzioni Matematiche |volume=2 |publisher=Stamperia Gavelliana |location=Pesaro |year=1750 |url=https://books.google.com/books?id=EXvkqp6-xKEC&pg=PR3}}</ref>

<ref name="MacTutor">{{MacTutor Biography|id=Fagnano_Giulio}}</ref>
</references>

==External links==
{{Commons}}
* {{DBI |title= FAGNANO, Giulio Carlo |url=https://www.treccani.it/enciclopedia/giulio-carlo-fagnano_(Dizionario-Biografico)|last= Baldini|first= Ugo|volume= 44}}

{{Authority control}}

{{Italy-mathematician-stub}}
{{DEFAULTSORT:Fagnano, Giulio Carlo}}
[[Category:1682 births]]
[[Category:1766 deaths]]
[[Category:People from Senigallia]]
[[Category:18th-century Italian mathematicians]]
[[Category:17th-century Italian mathematicians]]
[[Category:Fellows of the Royal Society]]
[[Category:Scientists from the Papal States]]