Analysis and Translation of Raffaele Rubini's 1857 'Application of the Theory of Determinants: Note' - Rubini's Notation

Salvatore J. Petrilli, Jr. (Adelphi University) and Nicole Smolenski (Adelphi University)

In Rubini's article, the reader will find the modern notation for determinants, with elements arranged in a table bounded by vertical lines enclosing the table. This notation for determinants first appeared in Arthur Cayley's (1821–1895) "Mémoire sur les hyperdéterminants" [1846], demonstrating Rubini's exposure to this relatively new notation [Cajori, 1993]. Throughout Rubini's article, he also utilized another modern notation, \(a_{r,s},\) to represent the element in the \(r\)th row and \(s\)th column. This notation was presented in Cauchy's 1815 work "Memoire sur les fonctions qui ne peuvent obtenir que deux valeurs egales et des signes contraires par suite des transpositions operees entre les variables qu'elles renferment," with Rubini most likely being exposed to the notation through Brioschi's [1854] textbook, La teorica dei determinanti e le sue applicazioni. This notation can be seen throughout Rubini's article (except for Sections 6 and 7), such as in the following expression:

\[P=\begin{vmatrix}a_{1,1} + h_{1,1} & a_{1,2} + h_{1,2} & \ldots & a_{1,n} + h_{1,n}\\a_{2,1} + h_{2,1} & a_{2,2} + h_{2,2} & \ldots & a_{2,n} + h_{2,n}\\\vdots & \vdots & \ddots & \vdots\\a_{n,1} + h_{n,1} & a_{n,2} + h_{n,2} & \ldots & a_{n,n} + h_{n,n}\end{vmatrix}\]

where \(a_{r,s}\) and \(h_{r,s}\) are real number elements from two different matrices being added together.

In Section 5, Rubini discussed solutions of general polynomials of the form:

\[f(x) = x^n + A_{1}x^{n-1} + A_{2}x^{n-2} + \ldots + A_{n-1}x + A_{n},\]

where the \(A_i\) are defined as

\(A_{1} = -(a_{1,1} + a_{2,2} + a_{3,3} + \ldots + a_{n,n})\)

\(A_{2} = +(a_{1,1}a_{2,2} + \ldots + a_{1,1}a_{n,n} + a_{2,2}a_{3,3} + \ldots+ a_{3,3}a_{n,n} + \ldots + a_{n-1,n-1}a_{n,n})\)

\(A_{3} = -(a_{1,1}a_{2,2}a_{3,3} + a_{1,1}a_{2,2}a_{4,4} + \ldots+ a_{1,1}a_{n-1,n-1}a_{n,n} + \ldots + a_{n-2,n-2}a_{n-1,n-1}a_{n,n})\)


\(A_{n} = (-1)^{n}a_{1,1}a_{2,2}a_{3,3} \ldots a_{n,n}\)

Rubini continued to use Cauchy's notation, as opposed to the more common notation for a root of an equations, \(a_r,\) except in Sections 6 and 7. In these sections, he continued to discuss the solutions of polynomials, but abruptly changed his element notation to the simpler \(a_r\) after for the first paragraph in Section 6. This notation is first presented in the following equation:

\[\begin{vmatrix}x - a_{1} & 0 & 0 & \ldots & 0\\0 & x - a_{2} & 0 & \ldots & 0\\\vdots & \vdots & \vdots & \ddots & \vdots\\0 & 0 & 0 & \ldots & x - a_{n} \end{vmatrix} = 0,\]

where \(a_r\) replaced Cauchy's notation of \(a_{r,r}\) to denote a root of an equation. However, Rubini provided no explanation of the significance of this new notation until the end of Section 6, where he stated that the \(a_r\) are solutions of equations. He could have made a smoother transition to using the \(a_r\) notation, had he started utilizing it at the very beginning of Section 6 or in Section 5 when he began presenting ideas that involved the roots of polynomials. Even though Rubini changed to the more conventional notation when describing the roots of an equation, this change in notation is rather confusing for the reader due to the lack of an explanation. He may have provided these two different notations, though, to expose readers to some of the notations that were present in the analytic repertoire, in his attempt to insure the circulation of these analytic ideas.