The debate between synthetic and analytic mathematics propelled Italian authors to publish textbooks supporting their views [Giacardi and Scoth, 2014]. Rubini wrote the article, “Application of the Theory of Determinants: Note," with the hope of showing the reader the great benefits and progress resulting from analytic mathematics, as had been done by his professor, Fortunato Padula [Padula, 1839]. He frankly stated in the very first paragraph of the article,
[W]e propose to show with some examples how the algorithm of determinants can be useful in the presentation of the theory of equations, and how easily it leads to some formulas, which would be otherwise very difficult to deduce .... [Rubini, 1857, p. 179].
Here, "otherwise'' indirectly referred to the use of synthetic methods [Rubini, 1857, p. 179]. Predicting the theory of determinants to have even more powerful mathematical applications to algebra than functions, he declared that the introduction of determinants to algebra would cause the course of algebra to "change form more than it did with the use of functions." He thus made claim to the revolutionary impact that determinants would have on the state of algebraic mathematics [Rubini, 1857, p. 179]. Furthermore, Rubini concluded Section 5 of his article by declaring, "This way of proving formula (19) seems preferable to those ordinarily used in Algebra," explicitly reinforcing his purpose of showing the benefits that determinant theory introduced to mathematics [1857, p. 186].
Rubini may also have published his paper on determinants in order to expose the reader to various analytic ideas. His introductory paragraph ends with a declaration that he did not make "claim to the originality of the subject matter" he would present, leaving the reader curious as to why Rubini would bother to publish a work which did not introduce any new ideas or theorems about the topic [Rubini, 1857, p. 179]. The mathematical schism that resonated throughout Rubini's intellectual life provides a likely motivation [Mazzotti, 1998]. Rubini's article may not have had a global mathematical audience, but its publication was very beneficial to Italian mathematicians during this time period because it introduced them to various ideas about determinants from several analytic mathematicians, including William Spottiswoode (1825–1883), Brioschi, Cauchy, Lagrange, and Adrien-Marie Legendre (1752–1833), to which they may not have been exposed otherwise due to the schism. Rubini's compilation of others' work brought more ideas to the Kingdom of Two Sicilies on determinants and provided the reader with diverse perspectives on the applications of this new concept of determinants.
During the period of the Bourbon rule, Rubini may have published this work to insure the transmission of analytic methods and ideas among the mathematicians of Italy. When the Bourbons regained control of the Kingdom, they censored analytic mathematics by removing analytic mathematics teachers and replacing them with those aligned with the synthetic branch [Mazzotti, 1998], causing their students to be ignorant of the new analytic mathematics. It can be argued that without mathematicians like Rubini in the Kingdom of Two Sicilies, the analytic methods of mathematics would have reached many regions of Italy even later than they did, causing this territory to be left further behind in the mathematical progress being made throughout the rest of Europe.
Provided below is a brief outline of the material Rubini discussed in each section of his 'Note'.
Download the authors' English translation of Raffaele Rubini's article, "Application of the Theory of Determinants: Note."
In the next four webpages, we discuss, respectively, Rubini's notation throughout his article, Sections 1-4 and 8-10, Sections 11 and 13, and Section 12.