### Purpose

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.

### Outline

Provided below is a brief outline of the material Rubini discussed in each section of his 'Note'.

- In Section 1, Rubini presented a formula to the reader and explained how he found the determinant of a matrix using this formula.
- In Section 2, he utilized the formula from Section 1 to obtain other equivalent formulas.
- In Section 3, Rubini provided the reader with a shortcut to Laplace expansion for \(n\times n\) matrices of a particular form.
- In Section 4, Rubini introduced the subtraction sign into the determinant formula and explained how this affected the way in which one would go about computing a determinant.
- In Sections 5–7, Rubini discussed solutions of polynomials of the form \(f(x) = x^n + A_{1}x^{n-1} + A_{2}x^{n-2} + \ldots + A_{n-1}x + A_{n}.\)
- In Section 8, Rubini returned to the idea of computing determinants with a subtraction sign in between the elements, but entered the realm of complex numbers. He showed how the introduction of complex numbers would affect the computations when calculating the determinant.
- In Sections 9 and 10, he demonstrated how his results from Section 8 could be used in different methods for calculating determinants. Additionally, in Section 9, he presented a method of finding the determinant of a skew-symmetric matrix.
- In Section 11, he derived Lagrange's Four Square Theorem using determinants.
- In Section 12, he demonstrated a geometric application of Lagrange's Four Square Theorem.
- In the last section, Section 13, Rubini presented the following theorem:
*"every number is the sum of four squares"* [Rubini, 1857, p. 200].

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.