During the two decades before the start of the French Revolution in 1789, King Ferdinand (1751-1825) began to reform the educational system in the Kingdom of Naples by removing the Jesuits from their positions as educators [Goodwin, 1842]. Ferdinand's reaction to the revolution in France, however, was to abandon his reforms, reconcile with the Catholic Church, and impose a more repressive monarchy which, save for a six-month period in 1799, lasted until Napoleon Bonaparte's (1769-1821) conquest of the Kingdom of Naples in 1806 [Mazzotti, 1998, pp. 684-685; *Encyclopedia Britannica, *2008]. Although mathematics was for the most part neglected throughout Ferdinand's reign [Goodwin, 1842], during the 1780s, a mathematical schism developed between the synthetic and the analytic mathematicians [Mazzotti, 1998, p. 684].

Until Rene Descartes (1596-1650) and Pierre de Fermat (1601-1665) introduced analytic geometry, there was only one type of mathematics, synthetic. Synthetic mathematicians would solve geometric problems using the methods of Euclid and his contemporaries [Mazzotti, 1998]. However, after the introduction of analytic mathematics, these "synthetics" would accept solutions to problems that utilized algebraic tools as long as the mathematical reasoning could be explained by pure geometry [Mazzotti, 1998]. "Analytics" arrived at generalizations through the use of equations and variables, and their mathematics was generally more abstract [Otte and Panza 1997]. (Readers interested in a more in-depth historical and philosophical discussion of the analytic and synthetic debates should refer to the collection edited by Otte and Panza [1997].)

Even though this mathematical schism persisted, the educational systems in the Kingdom of Naples and, to some extent, the Kingdom of Sicily, were truly reformed during the reign of Joachim Murat (1767-1815), Napoleon's brother-in-law [Goodwin, 1842], which began in 1808 [*Encyclopedia Britannica, *2008]. Murat "decreed that a primary school, or school for reading and writing, should be established in every commune; [and] that secondary or classical schools should be founded in every province . . . " [Goodwin, 1842, p. 63]. French *licei,* now known as *scuola superiore* or high schools in Italy, were also introduced to the territory under Murat's rule and are believed to be one of the most important legacies of the Napoleonic domination of Italy [Giacardi and Scoth, 2014]. At the time *licei* were founded, they were on a higher academic level than colleges, providing students with other opportunities to learn from university professors besides at the kingdom's one university, the University of Naples [Giacardi and Scoth, 2014]. At both the colleges and the *licei,* "'pure and mixed' mathematics were taught" [Giacardi and Scoth, 2014]. Although Italy was a male-dominated society, "free schools for girls were founded in the capital and the provinces" [Goodwin, 1842, p. 63]. There were also private religious schools in the two kingdoms, run by Jesuits and other Catholic orders. These schools, especially, provided a stimulus for the numerous translations of French books published during the first half of the nineteenth century, mainly in Naples [Giacardi and Scoth, 2014]. The French educational reforms "laid the foundation for an education that was state controlled and secular and affirmed the importance of educating citizens who were responsible and aware of their place in society" [Giacardi and Scoth, 2014, p. 201].

Under the decree of the Congress of Vienna in 1815, the Italian peninsula was divided into seven states [Giacardi and Scoth, 2014]. One of these states was the Kingdom of Two Sicilies, which was formed by uniting the Kingdom of Naples and the Kingdom of Sicily [Giacardi and Scoth, 2014]. This new kingdom was once again returned to the Bourbons, under the rule of King Ferdinand [Giacardi and Scoth, 2014]. He continued the educational reforms implemented by the French until the revolts of 1821, after which education was once again "managed entirely by ecclesiastic and private entities, with religious orders being given the most important secondary schools and colleges" [Giacardi and Scoth, 2014, p. 204]. After several uprisings in 1848, it appeared that public education would once again receive its much-needed reforms; however, the counterrevolution of 1849 ended this reform project [Giacardi and Scoth, 2014].

Under Bourbon rule, the last known mathematical duel between Italian mathematicians occurred, essentially between the analytic and synthetic branches of mathematics [Mazzotti, 1998, pp. 680-683]. In Italy, it was a common occurrence for one mathematician to challenge another to a duel, in which each mathematician would provide challenging problems to his opponent and the mathematician who correctly answered the most questions at the end of a certain time period would be awarded a substantial prize. The best known mathematical duel was the one held between Nicolo Tartaglia and Antonio Fior in 1535. In 1839, a contest was arranged by Vincenzo Flauti (1782-1862), a synthetic mathematician who was the secretary of the Royal Academy of Sciences in Naples, with hopes that the contest would show the superiority of synthetic mathematics over analytic mathematics [Mazzotti, 1998, p. 680]. Flauti's student, Nicola Trudi (1811-1881), was the synthetic participant in the contest, and Fortunato Padula (1815-1881) the analytic contestant [Mazzotti, 1998, pp. 680-681]. Trudi was declared victor of the contest, winning the monetary prize and giving the synthetic method a victory as well [Mazzotti, 1998, p. 683].

The schism between mathematicians in the Kingdom of Two Sicilies reflected a broader intellectual and political schism [Mazzotti, 1998]. From the 1760s through the 1780s and again after Napoleanic forces took control of Naples, the citizens of the two kingdoms, Naples and Sicily, were exposed to the ideals and beliefs of the Enlightenment. Mathematical scholars, in particular, were exposed to the analytic mathematics that had been developed in France, beginning with Descartes' and Fermat's analytic geometry and including Joseph Louis Lagrange's (1736-1813) algebraic foundation of calculus. Those who would utilize the new analytical methods to solve mathematics problems tended to be receptive as well to the broader intellectual, political, and educational ideals of the French Republic and to side with Napoleanic forces, with some even fighting and dying for revolutionary ideals [Mazzotti, 1998]. During 1799-1815, as the government oscillated between the revolutionary leadership and Bourbon rule, every aspect of society, including schools and their administrators and teachers, would change depending on which side was in power [Mazzotti, 1998]. The return of the Bourbons to the Kingdom of Two Sicilies "generally implied a return to the past for the educational systems, a greater involvement of ecclesiastic authorities, and strict control over teachers and students, fundamentally dictated by the desire to quell any revolutionary spirit" [Giacardi and Scoth, 2014, p. 201]. In mathematics instruction, analytic teachers tended to be replaced by synthetic teachers, greatly reducing the exposure of scholars of mathematics to analytic thought [Mazzotti, 1998]. With the educational system of the Kingdom of Two Sicilies emphasizing the ancient, synthetic methods of mathematics, a few of its mathematicians decided they must try to bring the new and exciting algebraic concepts to the kingdom. One of them was a student of Padula, Raffaele Rubini (1817-1890).

The works of many mathematicians during this historically significant time period need to be examined, including those of Rubini, Brioschi, Padula, Trudi, leader of the Neapolitan synthetic school Nicola Fergola (1753-1824), and noted algebraist Alfredo Capelli (1855-1910).