Bonaventura Cavalieri

Bonaventura Cavalieri (1598-1647) was an Italian mathematician who worked on indivisibles. Both he and his student Stefano Degli Angeli (1623-1697) were members of the order of the jesuats (with an "a"). Cavalieri published two books on indivisibles, and Degli Angeli published nine books on indivisibles. Their work was attacked by jesuit (with an "i") mathematicians Guldin, Tacquet, and others. The jesuat order fell out of favor with the pope who suppressed it in 1668 and transferred its assets to the war effort against the Ottomans, who prevailed a year later (upon hearing the news, the pope fell ill and died a few months later).

Archimedes was good at calculating areas but the art seems to have been put mostly in gestation until the 17th century, even though there were remarkable achievements in the meantime in areas such as power series developments of trig functions. However, there were some intermediate achievements by Ibn al-Haytham (c. 965–1040), Thābit ibn Qurra (c. 826–901), Al-Karaji (c. 953–1029), and Nicole Oresme (c. 1320–1382).

Before Cavalieri, Kepler used infinitesimals to calculate volumes (notably of wine barrels). But Cavalieri used a different method, that of indivisibles, not infinitesimals. Indivisibles have zero width, i.e., they are codimension one entities.

Many jesuits were accomplished mathematicians. An early example is Grégoire de Saint-Vincent, a jesuit and a pioneer of what will later become the calculus, but there were many others.

However, a number of influential mathematicians in the jesuit order opposed indivisibles, including Guldin, Tacquet, and Bettini, and wrote tracts attacking them.

Guldin even made an allusion in one of his texts suggesting that there may be a doctrinal issue behind his opposition to indivisibles, though he did not state this explicitly.

In the context of the jesuit order, a distinction must be made between jesuit mathematicians and jesuit superiors such as the generals of the order. These included Mutio Vitelleschi, Jakob Bidermann, and Sforza Pallavicino.

The generals issued a number of bans against indivisibles.

These bans explicitly applied both to physical and mathematical indivisibles.

The reasons for the bans were also mentioned explicitly. They included the perceived conflict between indivisibles and the catholic canon of transubstantiation, which endorsed an Aristotelian doctrine of hylemorphism (or hylomorphism) as the substatum of a theoretical explanation of the Eucharist. Atomism, closely associated with indivisibles, was viewed as a rival, and heretical, doctrine.

Galileo's fate was closely intertwined with that of his disciple Cavalieri. Galileo was similarly attacked for allegedly being contrary to canonical interpretations of the Eucharist. Several jesuits are on record attacking Galileo with the weapon of Eucharistic theology.

Cavalieri was a catholic friar, but he belonged to the "wrong" order: that of the jesuats (not jesuits). The jesuat order was an older order than the jesuits. Cavalieri is on record complaining that his difficulties in finding employment are due to the domination of the field by the jesuits, and his being a humble jesuat.

Cavalieri and his students Degli Angeli (similarly a jesuat) and Torricelli did some brilliant work and published a number of books on indivisibles. One would have expected this to lead to many students and a flowering of the new mathematics in Italy. Instead, the jesuat order was suppressed by a papal bull (Alexander erroneously refers to this as a 'brief') in 1668, following decades of speculation about a looming suppression. This cut short Degli Angeli's productivity: he published nothing on indivisibles after the ban, though he remained active at the university until the 1690s, when his interactions with the young Riccati are on record.

James Gregory visited Degli Angeli in the 1660s (not long before the ban). We have a record of his correspondence where Gregory expressed puzzlement concerning the fact that his books were suddenly suppressed in Italy.

There are some detailed studies of the persecution of Italian atomists by Rome in the 17th century.

In the 18th century, Agnesi dared to start picking up the pieces and re-launch the calculus from scratch in Italy, repairing some of the damage done earlier.

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