**Stephano degli Angeli** was in fact named Francesco degli Angeli as a child but later changed his name when entering a religious order. He entered the Order of the Jesuati (or Jesuate or Gesuati) which had been founded by Giovanni Colombini from Siena in 1365 at Città di Castello. Since the Order had a major influence on Angeli's life and mathematics, we should say a little about it. The name of the Order comes from the fact that members said "Praise be to Jesus Christ" at all times but their special devotion to Saint Jerome explained the longer name "Apostolic Clerics of St Jerome". The Order received the approval of the Roman Catholic Church from Pope Urban V in 1367. They were devoted to charity work and caring for the sick, while every day they would beat themselves to inflict pain. The Order, whose rules were originally based on those of St Benedict, flourished spreading rapidly throughout Italy. Later the Rule of St Augustine was adopted and in 1606 priests were allowed into the Order.

What would Angeli have worn as a Jesuati? Certainly he wore a white habit with a dark leather belt and sandals on his feet. He would have had little in the way of worldly possessions. After being educated within the Order he then taught literature, philosophy and theology in the Faculty of the Gesuati Order at Ferrara from 1644. His health, however, was poor and he was given medical advice to transfer to Bologna, which is about 50 km south west, in 1645. It might seem strange medical advice to move to another town only 50 km away, but in fact Ferrara is situated on marshland whereas Bologna is situated at a height of 55 metres above sea level at the foot of the Apennine mountains. It was a good move in terms of Angeli's health, and also a good move for mathematics since in Bologna he came under the influence of Cavalieri. Cavalieri was also in the Order of the Jesuati so the two naturally came in contact. Cavalieri was teaching at the University of Bologna, one of the oldest and most famous universities in Europe, dating from the 11^{th} century. At the time that Angeli was there the university was situated in the Archiginnasio Palace which had been built by Pius IV in 1562 [1]:-

Cavalieri was soon able to appreciate Angeli's ability, and encouraged him in his studies and mathematical researches.

After leaving Bologna, Angeli continued his contacts with Cavalieri by correspondence, and was entrusted to publish Cavalieri's final work, *Exercitationes geometricae sex*, since by 1647 Cavalieri's health had deteriorated to such an extent that he was unable to carry out the work himself. Angeli also corresponded with a number of other mathematicians including Torricelli and Viviani. After Cavalieri's death, later in 1647, Angeli was offered his chair of mathematics at the University of Bologna but he was still too modest about his own mathematical achievements to accept the position. He moved to Rome where he devoted himself to both mathematics and religious studies.

Angeli was Rector of a Jesuati Order religious establishment in Rome from 1647 to 1652. Then he moved to a Jesuati monastery in Venice but, shortly after going to Venice, he was appointed as provincial definer for the Jesuati Order. In addition to this religious role he continued to be heavily involved in teaching and research in mathematics and he was appointed professor of mathematics at University of Padua on 2 January 1662. James Gregory studied with Angeli in Padua from 1664 to 1668 and learnt from him about series expansions of functions. In fact the Jesuati Order was suppressed in 1668 by Pope Clement IX who believed that abuses had crept in over the years, but Angeli continued in the priesthood while holding the chair of mathematics at Padua. He remained in this chair until his death in 1697.

Angeli's many mathematical works were on infinitesimals and he used them to study spirals, parabolas and hyperbolas. While in Venice he published *De infinitorum parabolis* (1654), *De infinitorum spiralium spatiorum mensura* (1660) which contains a generalisation of Archimedes' spiral, and *De infinitorum cochlearum mensuris ac centris gravitatis* (1661) which carries out Torricelli's intention of finding the centre of gravity of a solid body called a cochlea. The approach followed by Angeli in all these works is that of his teacher Cavalieri and of Torricelli, so when Guldin and Tacquet attacked these methods and defended the approach of the ancient Greeks, Angeli disputed with them over indivisibles. One has to see both sides in this argument for although Angeli's methods were much more powerful, they were less rigorous than the method of exhaustion adopted by Archimedes. The gist of the argument is contained in the introduction to *De infinitorum parabolis* [1]:-

To those who opposed Cavalieri's method by asserting that the continuous is not composed of indivisibles, Angeli replied, in agreement with his master, that the method in question does not depend on the composition of the continuous.

Angeli examined fluid statics based on Archimedes' principle and Torricelli's experiments. He published *Della gravita dell aria e fluidi* in 1671 while holding the chair at Padua. He also considered the motion bodies falling towards a rotating Earth. Of course Angeli held the chair at Padua which had been held earlier by Galileo and his work shows strong influences from his predecessor. For example Angeli often refers to Galileo in his writings on physics, showing clearly how he has been influenced, particularly in terms of ways of approaching problems via the experimental method. Also clearly influenced by Galileo is Angeli's writings on the two systems of Ptolemy and Copernicus which he writes in Galileo's dialogue style.

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

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