Scholars in the Kingdom of Two Sicilies, which united the island Kingdom of Sicily and the mainland Kingdom of Naples (the southernmost regions in modernday Italy) from 1815 to 1860, were exposed to some of the works of French mathematicians after translations of these works were completed. However, until Francesco Brioschi (18241897) published his works of algebraic theory, beginning in 1854 with his Teoria dei determinanti (Theory of Determinants) [O'Connor and Robertson, 2006], these scholars had limited knowledge of algebra, specifically determinants. Their ignorance was due not only to the kingdom's geographic isolation, but also to an academic schism between two branches of mathematical thought, synthetic and analytic. This schism had consequences for several Italian mathematicians, including Raffaele Rubini (18171890), and led Rubini to publish his 1857 article, “Application of the Theory of Determinants: Note." The present article provides the first English translation of Rubini's article on determinants, along with some biographical information on this obscure mathematician. The analysis of Rubini's work will examine his purpose in publishing it and the ways in which he achieved his aims. Finally, we suggest how this article might be incorporated into a high school mathematics curriculum that follows the Common Core through the process of “Stealing Time."
Download the authors' English translation of Raffaele Rubini's article, "Application of the Theory of Determinants: Note."
During the two decades before the start of the French Revolution in 1789, King Ferdinand (17511825) 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 sixmonth period in 1799, lasted until Napoleon Bonaparte's (17691821) conquest of the Kingdom of Naples in 1806 [Mazzotti, 1998, pp. 684685; 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 (15961650) and Pierre de Fermat (16011665) 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 indepth 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 (17671815), Napoleon's brotherinlaw [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 maledominated 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 muchneeded 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. 680683]. 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 (17821862), 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 (18111881), was the synthetic participant in the contest, and Fortunato Padula (18151881) the analytic contestant [Mazzotti, 1998, pp. 680681]. 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 (17361813) 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 17991815, 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 (18171890).
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 (17531824), and noted algebraist Alfredo Capelli (18551910).
Figure 1. Seki Kōwa, Creative Commons AttributionShare Alike 3.0 Unported. 
Figure 2. Gottfried Wilhelm von Leibniz, Painting by Christoph Bernhard Francke, public domain. 
Although the term "determinant" was not introduced until the nineteenth century, the mathematical concept was discovered independently during the same time period on the continents of Asia and Europe by the Japanese mathematician Seki Kōwa (1642–1708) and one of the fathers of calculus, the German mathematician Gottfried Wilhelm von Leibniz (1646–1716). Kōwa had published his work in 1683, demonstrating to his readers through examples some general methods of calculating determinants [Eves, 1990; O'Connor and Robertson, 1996, 1997]. He was able to compute determinants of up to size 5 x 5 matrices and apply this knowledge to solve equations [O'Connor and Robertson, 1996]. Just ten years later in 1693, Leibniz shared in letters to Guillaume de l'Hôpital (1661–1704) his method of determining whether a set of simultaneous linear equations, in the general form \(a + bx + cy = 0,\) had a solution [Boyer and Merzbach, 1989]. He organized the system of equations as follows:
\[a_0 + a_1x + a_2y = 0\]\[b_0 + b_1x + b_2y = 0\]\[c_0 + c_1x + c_2y = 0\]
which enabled him to determine whether the system was consistent. Leibniz did so by utilizing a method known today as expansion by minors or the Laplace expansion, after PierreSimon Laplace's (1749–1827) work with determinants [Boyer and Merzbach, 1989; O'Connor and Robertson, 1996]. If
\[a_0 \cdot b_1 \cdot c_2 + a_1 \cdot b_2 \cdot c_0 + a_2 \cdot b_0 \cdot c_1 = a_0 \cdot b_2 \cdot c_1 + a_1 \cdot b_0 \cdot c_2 + a_2 \cdot b_1 \cdot c_0,\]
then a solution existed [Boyer and Merzbach, 1989; O'Connor and Robertson, 1996]. Leibniz was the first mathematician to introduce a type of notation which gave a designation to each element's position in computing determinants [Cajori, 1993]. Leibniz denoted the coefficients \(a,\) \(b,\) and \(c\) in the equations above as \(1,\) \(2,\) and \(3,\) respectively, and paired these digits with their positions \(0,\) \(1,\) and \(2.\) The system of three equations above would then be represented as [Cajori, 1993; O'Connor and Robertson, 1996]:
\[10 + 11x + 12y = 0\]\[20 + 21x + 22y = 0\]\[30 + 31x + 32y = 0\]
Although Leibniz's work was done a decade later than Kōwa's, Leibniz has generally been the mathematician credited by historians in the west as the originator of the theory of determinants [Eves, 1990].
In 1750, Gabriel Cramer (1704–1752) published his work, Introduction à l'analyse des lignes courbes algébriques, in which the rule that bears his name, Cramer's Rule, appeared [Boyer, 1989]. Although the method is known as Cramer's Rule, it was actually published earlier, in 1748, in Colin Maclaurin's (1698–1746) work, A Treatise of Algebra in Three Parts, which he had written in the 1730s [Katz, 2004]. This rule gives a specific method for how to solve systems of linear equations using determinants. Given the system of equations
\[ax + by + cz = m\]\[dx + ey + fz = n\]\[gx + hy + kz = p\]
Maclaurin and Cramer solved for the variables using a fraction formed from two determinants. For the equations above (taken from Boyer [1989] and Katz [2004]), these two mathematicians would have found the solution for the variable \(z\) to be
\[\dfrac{aep  ahn + dhm  dbp + gbn  gem}{aek  ahf + dhc  dbk + gbf  gec},\]
where the denominator consists of the "various products'' of the coefficients of the variables and the numerator of the "various products'' of the coefficients of \(x\) and \(y\) with the constants substituted for the column of \(z\) coefficients [Katz, 2004, p. 668]. They would solve for the other variables in a similar manner. Cramer was able to generalize the method for \(n\) linear equations with \(n\) unknowns and today we still utilize Cramer's Rule, but we know that these "various products" are in fact the determinant computed using the Laplace expansion [Katz, 2004].
Figure 3. Gabriel Cramer, public domain.  Figure 4. AugustinLouis Cauchy, public domain. 
In his 1812 paper on determinants, AugustinLouis Cauchy (1789–1857), the most prolific contributor to the theory of determinants, introduced the now common notation for matrix entries, \(a_{i,j},\) to describe an element's position based on what row \(i\) and column \(j\) it is in [Boyer, 1989]. According to Eves [1990], in this extensive paper, Cauchy was the first to prove the important property of determinants and matrix products, that \(\det\left(A\times B\right) = \det \left(A\right)\times\det\left(B\right),\) where \(A\) and \(B\) are two \(n\times n\) matrices. In this paper, in which he called determinants the "symmetric system," Cauchy also applied determinants to find the volume of the parallelepiped [Boyer, 1989]. Although Carl Friedrich Gauss (1777–1855) had introduced the term determinant in 1801, this word was not used in its modern sense until Cauchy's paper, "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," was published in 1815 [Miller, 2010]. The term determinant was accepted by the mathematics community largely because of Carl Gustav Jacob Jacobi (1804–1851) and his extensive work with determinants [Eves, 1990]. In 1827, Jacobi showed that the determinant of a skewsymmetric matrix of odd order equals zero [Eves, 1990], where a skewsymmetric matrix is a matrix whose transpose is also its negative. In his 1841 memoir, "De determinantibus functionalibus," Jacobi demonstrated certain properties of the functional determinants also known as Jacobians [Boyer, 1989].
The theory of determinants, as well as other mathematical concepts, reached the country of Italy through the publications of Brioschi, including his major work, La teorica dei determinanti e le sue applicazioni, published in 1854 [Francesco Brioschi, n.d.]. Brioschi's works influenced and enabled Rubini, an overlooked Italian mathematician, to publish mathematics papers exploring determinants as well.


Figure 5. Raffaele Rubini, public domain. 
When Raffaele Rubini died in 1890, noted algebraist Alfredo Capelli (18551910), who had won a chair in mathematics at the University of Naples in 1886, wrote an obituary (necrologio) of him, which appeared in the 18901891 Annuario scolastico della Regia Università di Napoli (Scholastic Yearbook of the University of Naples) [Capelli, 18901891]. Unless otherwise noted, information about Rubini in the present "Biography" came from this obituary.
Raffaele Rubini was born on October 20, 1817, to Settimio Rubini and Giuseppa Gargiulo in Brindisi, a city in southern Italy in the region of Apulia. While attending college, Rubini became a student of Professor Fortunato Padula, who exposed him to the analytic approach to mathematics. In 1844, he graduated with degrees in mathematics and architecture. He began teaching mathematics at the Collegio Militare della Nunziatella, a military school, and then in 1848 he taught at a liceo in Lecce. Due to a profound new sense of nationalism, a number of revolutions occurred in Europe in 1848, including in the Kingdom of Two Sicilies. Rubini's forthright support of the Revolution of 1848 had the adverse effect of his forced removal from his teaching position at Lecce the following year when the Bourbon king returned to power. Other analytic mathematics instructors who had supported the revolution were removed from their positions in 1849 as well. For the next ten years, Rubini taught several students privately and he also published some mathematical works, including an Italian translation of Jacobi's work, Sul Numero delle Tangenti Doppie (On the Number of Double Tangents), published in 1851 [Capelli, 18901891; Rubini Raffaele, n.d.].
Figure 6. The main building of the University of Naples Federico II, public domain.
Rubini began to teach formally again in 1859 as a professor of mechanics at the Regia Scuola di Marina in Naples. In 1861, he began to work at the University of Naples (now known as the University of Naples Federico II), where he continued teaching mechanics. But soon he decided to teach mathematics, including algebra courses, viewing these subjects as more conducive to his own studies. After suffering from a nervous breakdown in 1870, he returned to his native town of Brindisi to rest, but was able to maintain the title of professor at the university. He continued to work for the University of Naples until 1886, at which point he was declared an honorary professor. Rubini had several publications, which was a bit uncommon at the time for Italian mathematicians, including works on geometry, algebra (the theory of determinants), and calculus. Many of his publications were translated into Spanish by Professor Marquez y Villaroe at the University of Seville. Rubini died on May 13, 1890, in Brindisi, where a public monument was erected in his honor [Rubini Raffaele, n.d.]. There is also a street named after him in Brindisi that still exists today [Mathematica Italiana, 20082011].
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 AdrienMarie 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 14 and 810, Sections 11 and 13, and Section 12.
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:
\[f(x) = x^n + A_{1}x^{n1} + A_{2}x^{n2} + \ldots + A_{n1}x + A_{n},\]
where the \(A_i\) are defined as
\(A_{1} = (a_{1,1} + a_{2,2} + a_{3,3} + \ldots + a_{n,n})\)
\(A_{2} = +(a_{1,1}a_{2,2} + \ldots + a_{1,1}a_{n,n} + a_{2,2}a_{3,3} + \ldots+ a_{3,3}a_{n,n} + \ldots + a_{n1,n1}a_{n,n})\)
\(A_{3} = (a_{1,1}a_{2,2}a_{3,3} + a_{1,1}a_{2,2}a_{4,4} + \ldots+ a_{1,1}a_{n1,n1}a_{n,n} + \ldots + a_{n2,n2}a_{n1,n1}a_{n,n})\)
\(\vdots\)
\(A_{n} = (1)^{n}a_{1,1}a_{2,2}a_{3,3} \ldots a_{n,n}\)
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.
After delivering a short introductory paragraph, Rubini delved right into formulas for special determinants and relations between them. Intriguingly, he did this without providing the reader with any of the basic properties of determinants or any explanation of his notation. In order for his readers to understand this "Note", it seems as if Rubini assumed they would be knowledgeable about determinants or have access to Brioschi's textbook, La teorica dei determinanti e le sue applicazioni, published in 1854, which was the first Italian work about determinants available to mathematicians in the Kingdom of Two Sicilies. Rubini opened his first section with a reference to this influential work and provided his readers with formulas which are given on page 44 of Brioschi's work [1854]. Although the modernday reader may be perplexed upon viewing the first determinant Rubini supplied, an individual with knowledge of expansion by minors and properties of determinants would be able to understand the majority of the mathematics presented.
After reading a couple of sections of Rubini's article, the reader will quickly discover that the author established few connections between the formulas he provided. Most ideas do not lead to the successive idea. Furthermore, where connections could be made within the paper, Rubini chose to format his article in such a way that these ideas were presented disjointly. For instance, in Section 3 of his article, Rubini supplied formulas (8) and (9), respectively:
\[{\begin{vmatrix}1 & 1 & \ldots & 1\\ 1 & 1+x & \ldots & 1\\\vdots & \vdots & \ddots & \vdots\\1 & 1 & \ldots & 1+x \end{vmatrix}}_{n1}= x^{n1}\quad\quad\quad(8)\]
and
\[{\begin{vmatrix}1+x & 1 & \ldots & 1\\1 & 1+x & \ldots & 1\\\vdots & \vdots & \ddots & \vdots\\1 & 1 & \ldots & 1+x \end{vmatrix}} _{n} = nx^{n1} + x^{n},\quad\quad(9)\]
with the indices of \(n\) and \(n1\) indicating the number of \(x\)'s along the main diagonal, but with both matrices of the \(n\)th order. Formulas (8) and (9) are very similar; the difference is that the matrix of formula (9) has \(x\)'s in every position on the main diagonal, while the matrix of formula (8) has its first row and column composed only of ones. Rubini's notation may seem confusing at first, but after comparing the matrices of formulas (8) and (9), it can be understood that these matrices are of the same order. As he provided no concrete examples in his article, we provide such an example here. Using \(2\times 2\) matrices with \(x = 3\) and computing these determinants by the Laplace expansion, the first formula can be illustrated by the following example:
\[{\begin{vmatrix}1 & 1\\1 & 4 \end{vmatrix}}_{1} = (1)(4)  (1)(1) = 3= 3^1,\]
and the second formula by the example below:
\[{\begin{vmatrix}4 & 1\\1 & 4 \end{vmatrix}}_{2} = (4)(4)  (1)(1) = 15= 2\cdot{3^1} + 3^2.\]
Note that, according to the first formula, for a \(2\times 2\) matrix, the determinant will always be equal to the value of \(x.\)
Now, substituting \(x\) for \(1 + x\) and \(x  1\) for \(x\) in formulas (8) and (9), we obtain formulas (10) and (11):
\[{\begin{vmatrix}1 & 1 & 1 & \ldots & 1\\1 & x & 1 & \ldots & 1\\\vdots & \vdots & \vdots & \ddots & \vdots\\1 & 1 & \cdots & \cdots & x \end{vmatrix}}_{n1}= (x1)^{n1}={\begin{vmatrix}x & 1\\1 & 1\end{vmatrix}}^{n1}\quad\quad(10)\]
and
\[{\begin{vmatrix}x & 1 & 1 & \ldots & 1\\1 & x & 1 & \ldots & 1\\\vdots & \vdots & \vdots & \ddots & \vdots\\1 & 1 & 1 & \ldots & x \end{vmatrix}}_{n}= + n(x1)^{n1} + (x1)^{n}.\quad\quad(11)\]
Rubini then supplied the reader with three additional formulas, labeled (12), (13), and (14), which he never utilized further in his paper; he possibly wanted to show the reader the various results that the analytic theory of determinants permitted mathematicians to find.
Rubini's equation (15) is his formula (11) written completely in terms of determinants:
\[{\begin{vmatrix}x & 1 & 1 & \ldots & 1\\1 & x & 1 & \ldots & 1\\\vdots & \vdots & \vdots & \ddots & \vdots\\1 & 1 & 1 & \ldots & x \end{vmatrix}}_{n}={\begin{vmatrix}1 & 1 & 1 & \ldots & 1\\1 & x & 1 & \ldots & 1\\\vdots & \vdots & \vdots & \ddots & \vdots\\1 & 1 & 1 & \ldots & x \end{vmatrix}}_{n}+n\,{\begin{vmatrix}1 & 1 & 1 & \ldots & 1\\1 & x & 1 & \ldots & 1\\\vdots & \vdots & \vdots & \ddots & \vdots\\1 & 1 & 1 & \ldots & x \end{vmatrix}}_{n1}.\quad(15)\]
The reader would have been able to follow this section more easily had Rubini presented equation (15) sequentially after formulas (10) and (11). Furthermore, it would have been of great assistance had Rubini gave the order of these matrices, because it is not apparent that these matrices are of different orders upon looking at them for the first time. For clarity in our explanation, equation (15) will be denoted as \(A_n = A_n' + n\,A_{n  1}.\) The term \(A_n\) (which appears on the lefthand side of the equation) is an \(n\times n\) matrix with only \(x\)'s appearing along the main diagonal. The matrix \(A_n\) is a submatrix of the \((n+1) \times (n+1)\) matrix \(A_n'\) (which appears on the righthand side of the equation); \(A_n'\) contains an additional row and column of ones. Rubini obtained the term \(n\,A_{n  1}\) by substituting in formula (10); this is an \(n\times n\) matrix with \(n  1\) \(x\)'s on the main diagonal. The reader would have also been better able to see the beauty of this equation through a numerical example, like the following one with \(x = 3\):
\[{\begin{vmatrix}3 & 1\\1 & 3\end{vmatrix}}_{2} = {\begin{vmatrix}1 & 1 & 1\\1 & 3 & 1\\1 & 1 & 3\end{vmatrix}}_{2} +2{\begin{vmatrix}1 & 1\\1 & 3\end{vmatrix}}_{1}.\]
The reader can check that each side of the equation equals \( 8.\)
This equation can be written as:
\[{\begin{vmatrix}3 & 1\\1 & 3\end{vmatrix}}_{2}  2{\begin{vmatrix}1 & 1\\1 & 3\end{vmatrix}}_{1} = {\begin{vmatrix}1 & 1 & 1\\1 & 3 & 1\\1 & 1 & 3\end{vmatrix}}_{2}.\]
This version of the example illustrates that formula (15) provides the reader with a shortcut to the Laplace expansion for \(n\times n\) matrices of this form. It is unfortunate that part of this beauty is lost in Rubini's article due to the lack of structure and explanation of his formulas. Although several of the formulas he presented would have been clearer with some concrete examples, like the one supplied above, this lack may demonstrate Rubini's intention to impress rather than teach.
The article's lack of structure is also seen in Rubini's decision to merely present formulas (16) and (17), the only two formulas or equations of any kind provided in Section 4, and then to reproduce these formulas in slightly different form four sections later, at the beginning of Section 8. It makes one wonder why he decided to place these formulas in their own section and why he placed them so far in advance from the section in which he utilized them. He may have done this to emphasize the equations' importance, but it would have been easier for the reader to follow if he had presented these formulas for the first time in Section 8. The disorganized structure makes Rubini's work a bit difficult to follow, but also supports the conjecture that Rubini's work is a compilation of various mathematicians' previous works on determinants.
In Sections 11, 12, and 13, Rubini demonstrated powerful applications of determinants. In particular, he used determinants to obtain Lagrange's Four Squares Theorem and then derived other known results from it.
We summarize Rubini's derivation of Lagrange's Four Squares Theorem below. In Section 8, he introduced matrices \(P_{a+ih}\) and \(P_{a  ih}\) as square matrices with entries consisting of corresponding complex conjugates \(\left(i=\sqrt{1}\right).\) Therefore, the product, \(Q =P_{a+ih} \times P_{a  ih},\) is a matrix with complex entries, except for the elements on the main diagonal, which are real. In Section 11, Rubini calculated the determinant of \(P_{a+ih} \times P_{a  ih}\left(=Q\right)\) to be (54):
\[{\left{\begin{vmatrix}a_{1,1} & a_{1,2}\\a_{2,1} & a_{2,2}\end{vmatrix}\begin{vmatrix}h_{1,1} & h_{1,2}\\h_{2,1} & h_{2,2}\end{vmatrix}}\right}^2+{\left{\begin{vmatrix}h_{1,1} & a_{1,2}\\h_{2,1} & a_{2,2}\end{vmatrix}+\begin{vmatrix}a_{1,1} & h_{1,2}\\ a_{2,1} & h_{2,2}\end{vmatrix}}\right}^2\]
\[={(a_{1,1}a_{2,2}  a_{1,2}a_{2,1} + h_{1,2}h_{2,1}  h_{1,1}h_{2,2})}^2+{(h_{1,1}a_{2,2}  a_{1,2}h_{2,1} + a_{1,1}h_{2,2}  h_{1,2}a_{2,1})}^2,\]
and the determinant of \(Q\) to be (55):
\(({a^2}_{1,1} + {h^2}_{1,1} + {a^2}_{1,2} + {h^2}_{1,2})({a^2}_{2,1} + {h^2}_{2,1} + {a^2}_{2,2} + {h^2}_{2,2})\)
\( \ (a_{1,1}a_{2,1} + h_{1,1}h_{2,1} + a_{1,2}a_{2,2} + h_{1,2}h_{2,2})^2\)
\( \ (a_{1,1}h_{2,1}  h_{1,1}a_{2,1} + a_{1,2}h_{2,2}  h_{1,2}a_{2,2})^2.\)
Rubini then set these two determinants equal to one another and simplified to obtain the following equation (56):
\(({a^2}_{1,1} + {h^2}_{1,1} + {a^2}_{1,2} + {h^2}_{1,2})({a^2}_{2,1} + {h^2}_{2,1} + {a^2}_{2,2} + {h^2}_{2,2})\)
\(=(a_{1,1}a_{2,1} + h_{1,1}h_{2,1} + a_{1,2}a_{2,2} + h_{1,2}h_{2,2})^2\)
\(+ \ (a_{1,1}a_{2,2}  a_{1,2}a_{2,1} + h_{1,2}h_{2,1}  h_{1,1}h_{2,2})^2\)
\(+ \ (a_{1,1}h_{2,1}  h_{1,1}a_{2,1} + a_{1,2}h_{2,2}  h_{1,2}a_{2,2})^2\)
\(+ \ (h_{1,1}a_{2,2}  a_{1,2}h_{2,1} + a_{1,1}h_{2,2}  h_{1,2}a_{2,1})^2.\)
The equation above led Rubini to the known mathematical theorem that "every number is the sum of four squares," which he credited to Legendre [Rubini, 1857, p. 197]. Rubini used the power of determinants to demonstrate this theorem, where mathematicians such as Legendre and Lagrange used number theory. (See the article by Beintema and Khosravani [2003] for a historical discussion of the proof of this famous theorem.). Although several mathematicians made contributions to this theorem, it is usually attributed to Lagrange, who proved the theorem in 1770. The theorem is more accurately formulated as "every natural number is the sum of four squares" [O'Connor and Robertson, 1999]. Today the theorem is known as Lagrange's Four Square Theorem. Numerical examples include:
\[50 = 6^2 + 3^2 + 2^2 + 1^2\]
and
\[13 = 3^2 + 2^2 + 0^2 + 0^2,\]
and thus this theorem can be more generally stated as:
For all natural numbers \(x,\) there exist integers \(b, c, d, e \geq 0\) such that \[x = b^2 + c^2 + d^2 + e^2.\]
These representations are not necessarily unique; some numbers are produced by more than one set of four squares. For example, the number \(50\) can also be expressed as \[50=7^2+ 1^2+ 0^2+0^2.\]
In Section 13, Rubini set certain elements equal to each other, putting \(a_{1,1}\) = \(a_{2,2},\) \(h_{1,1}\) = \(h_{2,2},\) \(a_{1,2}\) = \(a_{2,1}\) and \(h_{1,2}\) = \(h_{2,1}.\) Then he applied these newly established equalities to two previously defined equations: (54), which appears above, and (59), which expresses the partial determinant of \(P_{a + ih} \times P_{a  ih}\) as \( A \Sigma\,A_2 =\)
\( ({a^2}_{1,1} + {h^2}_{1,1} + a_{1,2}a_{2,1} + h_{1,2}h_{2,1})\)
\(\times \ ({a^2}_{2,2} + {h^2}_{2,2} + a_{1,2}a_{2,1} + h_{1,2}h_{2,1})\)
\( \ [a_{1,2}(a_{1,1}+ a_{2,2}) + h_{1,2}(h_{1,1} + h_{2,2})]\)
\(\times \ [a_{2,1}(a_{1,1} + a_{2,2}) + h_{2,1}(h_{1,1} + h_{2,2})]\)
\( \ [a_{1,2}(h_{1,1}  h_{2,2})  h_{1,2}(a_{1,1}  a_{2,2})]\)
\(\times \ [a_{2,1}(h_{1,1}  h_{2,2}) + h_{2,1}(a_{1,1}  a_{2,2})]\)
\(+ \ (a_{1,2}h_{2,1}  h_{1,2}a_{2,1})^2. \)
With these substitutions, formula (54) could then be expressed as:
\[({a^2}_{1,1}  {h^2}_{1,1} + {h^2}_{1,2}  {a^2}_{1,2})^2 + \ (h_{1,1}a_{1,1} + a_{1,1}h_{1,1}  a_{1,2}h_{1,2}  h_{1,2}a_{1,2})^2,\]
which can be simplified to:
\[({a^2}_{1,1}  {h^2}_{1,1} + {h^2}_{1,2}  {a^2}_{1,2})^2 + \ [2(a_{1,1}h_{1,1}  a_{1,2}h_{1,2})]^2.\]
Formula (59), could be expressed as:
\(({a^2}_{1,1} + {h^2}_{1,1} + {a^2}_{1,2} + {h^2}_{1,2})^2  \ [a_{1,2}(2a_{1,1}) + h_{1,2}(2h_{1,1})][a_{1,2}(2a_{1,1}) + h_{1,2}(2h_{1,1})]\)
\( \ [a_{1,2}(h_{1,1}  h_{1,1})  h_{1,2}(a_{1,1}  a_{1,1})][a_{1,2}(h_{1,1}  h_{1,1})  h_{1,2}(a_{1,1}  a_{1,1})]\)
\(+ \ (a_{1,2}h_{1,2}  h_{1,2}a_{1,2}),\)
and simplified to:
\[({a^2}_{1,1} + {h^2}_{1,1} + {a^2}_{1,2} + {h^2}_{1,2})^2  \ [2(a_{1,1}a_{1,2} + h_{1,1}h_{1,2})]^2.\]
Upon setting these new equations equal to each other and solving for \(({a^2}_{1,1} + {h^2}_{1,1} + {a^2}_{1,2} + {h^2}_{1,2})^2,\) Rubini arrived at the following equation (61):
\(({a^2}_{1,1} + {h^2}_{1,1} + {a^2}_{1,2} + {h^2}_{1,2})^2=\)
\(({a^2}_{1,1}  {h^2}_{1,1} + {h^2}_{1,2}  {a^2}_{1,2})^2+\ [2(a_{1,1}a_{1,2} + h_{1,1}h_{1,2})]^2 + [2(a_{1,1}h_{1,1}  a_{1,2}h_{1,2})]^2.\)
This led Rubini to yet another known mathematical theorem that he derived through determinants, the theorem – researched by various mathematicians including Leonhard Euler (1707–1783) and Christian Goldbach (1690–1764) – that the "square of a number is always the sum of only three squares" [Lemmermeyer, n.d.; Rubini, 1857, p. 200]. The formula above can be generalized to \(b^2 = c^2 + d^2 + e^2.\) Every Pythagorean triple can be written in this form with \(0\) as one of the three squares. We can get a better sense of the validity of this theorem and see how the theorem works by examining a couple of numerical examples:
\[5^2 = 4^2 + 3^2 + 0^2= 16 + 9 + 0= 25\]
and
\[3^2 = 2^2 + 2^2 + 1^2= 4 + 4 + 1= 9.\]
Rubini's selection of these two theorems about sums of squares as part of his article further supports the conjecture that one of his purposes in publishing this work was to compile some of the work previously done by other analytic mathematicians and share it with mathematical scholars residing in the Kingdom of Two Sicilies. Also, Rubini may have chosen these two striking mathematical theorems to show the superiority of the analytic concepts of determinants and functions over those of synthetic mathematics. Although these theorems can be reached through complicated procedures utilizing number theory, Rubini arrived at the theorems rather easily by using determinants, demonstrating to the reader the superiority of this new mathematical concept.
All of the sections of Rubini's article contain purely algebraic applications of determinants, except for Section 12. Rubini's manipulation of Lagrange's Four Square Theorem led him to the geometric application of finding the distance from a point to a plane and distances from projections of that point to planes with related equations (the coefficients in the equation of the plane were permuted). By substituting \(h_{1,2} = h_{2,2} = 0,\) \(a_{1,1} = \alpha_1,\) \(h_{1,1} = \beta_1,\) \(a_{2,1} = \alpha_2 = x,\) \(h_{2,1} = \beta_2 = y,\) \(a_{1,2} = \gamma_1,\) and \(a_{2,2} = \gamma_2 = z,\) in formula (56), he determined that the terms on the left side of equation (58),
\[\left(\dfrac{\alpha_1x+ \beta_1y + \gamma_1z}{\sqrt{({\alpha^2}_1 + {\beta^2}_1+ {\gamma^2}_1)}}\right)^2 + \left(\dfrac{\alpha_1z  \gamma_1x}{\sqrt{({\alpha^2}_1 + {\beta^2}_1+ {\gamma^2}_1)}}\right)^2\]
\[+\left(\dfrac{\beta_1z  \gamma_1y}{\sqrt{({\alpha^2}_1 + {\beta^2}_1+ {\gamma^2}_1)}}\right)^2 + \left(\dfrac{\alpha_1y \beta_1x}{\sqrt{({\alpha^2}_1 + {\beta^2}_1+ {\gamma^2}_1)}}\right)^2\]
\[= x^2 + y^2 + z^2,\]
represent the distances squared, respectively,
with point \(M\) existing in space, point \(N\) in the \(xy\)plane, point \(N'\) in the \(xz\)plane, and point \(N''\) in the \(yz\)plane. These formulas use notation very similar to that of the formula used today to find the distance from a point to a plane.
Rubini then provided another geometric application in Section 13, illustrating the theorem that the "square of a number is always the sum of only three squares" [Rubini, 1857, p. 200]. Rubini presented an analytic way to determine that if one is given a vertex \(V\) of a rectangular parallelepiped, the squared length of the diagonal to the opposite vertex is equal to the sums of the squared lengths of the three perpendicular edges meeting at \(V.\) He showed how this result can be easily found through the application of previously defined determinant equations. This result is the extension of the Pythagorean Theorem to the threedimensional rectangular parallelepiped, with the diagonal squared equal to the sum of the squares of the figure's width, length, and height. This can be written as \(d^2 = x^2 + y^2 + z^2,\) with \(d\) the length of the diagonal of the rectangular parallelepiped, \(x\) its width, \(y\) its length, and \(z\) its height. With the return of solid geometry into the school curriculum, this theorem is taught in most high schools using a more geometric approach, but due to Rubini's mathematical style, he proved this geometric idea using an analytic approach.
Rubini did not provide any visual representation of the rectangular parallelepiped, perhaps because his analytic beliefs led him to view geometric diagrams as unnecessary. Although synthetics believed it was acceptable to use some analytic mathematics in geometry as long as all the techniques had a geometric interpretation, analytics were very persistent in using only algebraic methods to solve geometry problems [Mazzotti, 1998]. As a product of analytic mathematics, it makes sense for Rubini to have omitted a picture of a rectangular parallelepiped, even though it would have been helpful for those unfamiliar with this figure as well as for those who are visual learners or not strong in spatial reasoning. Today, mathematics teachers emphasize the importance of drawing diagrams or marking up given diagrams with the given information, to further students' understanding of a problem. However, the visual representation of the diagram for the analytics in earlier times could have led readers astray from the purer algebraic way of completing the problem.
Rubini may have selected these specific applications for his "Note" to show his readers the complexity of problems that can be solved using the analytic concept of determinants, demonstrating its superiority over synthetic methods. These geometric applications furthered his mission to show the reader the powerful tool determinants provide to mathematicians.
We provide two possible educational activities related to the topics in this article, which could be implemented individually or jointly. These two methods have not been tested; however, we encourage teachers to give them a try with whatever modifications they care to make.
After learning and proving the basic properties of determinants in a Linear Algebra class, Raffaele Rubini's article, "Application of the Theory of Determinants: Note," could be used to further students' understanding of determinant theory. In particular, his article could show students how to find the determinant of matrices of the form
\[M =\begin{bmatrix}a \pm b & c \pm d\\e \pm f & g \pm h\end{bmatrix}.\]
Most lessons in a typical sophomorelevel course do not include content at this level; the matrices that students see have only a single number or variable in each position. One possible lesson could involve the teacher handing out Section 1 from Rubini's article for students to read for homework, possibly together with some numerical examples and preferably following a class discussion of their ideas for how they might approach computing such a determinant. In the subsequent class, students could discuss the article and how Rubini computed a determinant of an \(n\times n\) matrix in this form.
Teachers can utilize the article in this way if an individual student is naturally curious about how he/she could find the determinant of a matrix of such a form or as an extra topic in which teachers utilize the concept from Columbia University's Teachers College of "Stealing Time'' in the mathematics classroom. Stealing time involves "avoiding unnecessary repetition and review'' and replaces that with an advanced topic [Weinberg and Ferrara, 2013, p. 86]. Matrices have shown great potential to fill stolen time [Weinberg and Ferrara, 2013]. Teachers College has demonstrated a variety of success stories of such a nature.
As another example, a teacher could give his/her students equations (10), (11), and (15) from Section 3 of Rubini's article, which give the reader an alternative method to use instead of Laplace expansion when matrices are of a certain form. (Download a sample handout, Rubini and Determinants Classroom Activity.) After students examine the various examples, they can work in groups to determine how they could find a determinant of an \(n\times n\) matrix in such a form and what the benefits are to using the method presented in Rubini's article as opposed to Laplace expansion.
Rubini's article could also be used as a Mathematics Common Core interdisciplinary activity with high school World History and Algebra II (or Precalculus) classes. During the unit in which students learn of the Unification of Italy in World History class, mathematics teachers could have their students do a small project on a restricted list of female and male Italian mathematicians during this historical time period, including Rubini. With female mathematicians such as Maria Gaetana Agnesi (1718–1799), Maria Gramegna (1887–1915), and Pia Nalli (1886–1964) included on the list, students will learn that females also made contributions to the field of mathematics. Excerpts of Mazzotti's [1998] article could be listed as a possible source for students who decide to do their project on mathematicians such as Brioschi, Padula, and Rubini. These students would learn of the schism between synthetic and analytic mathematics in the Italian mathematical community and incorporate it into their brief presentations of their projects.
The theory of determinants was a revolutionary discovery for mathematicians, enabling them to obtain even more mathematical results more easily. On the European continent, this theory advanced most rapidly in France and Germany. Between language differences and political upheaval, Italian mathematicians were generally slow to learn of advances such as those in the theory of determinants. The mathematicians of the Kingdom of Two Sicilies would remain largely ignorant of the theory of determinants until Brioschi published these ideas in Italian towards the middle of the nineteenth century. In addition to the geographic isolation of the kingdom, there was a schism between synthetic and analytic mathematical branches that reflected the intellectual and political divides between traditional Bourbon and revolutionary rulers. When the traditional side was in power, traditional, synthetic mathematics dominated the university, colleges, and other advanced schools. This tended to hinder the spread of algebraic theory, like that of determinants, among mathematical scholars of the Kingdom of the Two Sicilies. Even though Raffaele Rubini did not play a crucial role for mathematics in general, he helped kept the analytic branch of mathematics alive in the kingdom. His article on determinants enabled Italian mathematicians to learn of contributions to the theory of determinants by different analytical mathematicians. Although Rubini's article claims to present no original material on the theory of determinants, it is valuable to look at his work, not only due to this intellectual schism, but also because this article can be used by mathematics educators.
It is important to have a full sense of a mathematician's historical background before deciding whether or not an article of his or hers is valuable to read, especially in cases like this where the author very candidly declares that it does not provide any new ideas about the topic at hand. The historical events occurring in a mathematician's life can be very consequential in determining the mathematics he or she publishes, as well as how he or she presents the material to the reader. It would be interesting to examine other works published by Italian mathematicians who lived in the Kingdom of Two Sicilies during this time period to see how they presented the material according to the synthetic and analytic branches of mathematics, and taking into consideration who was ruling the kingdom at the time that the articles were published.
Download the authors' English translation of Raffaele Rubini's article, "Application of the Theory of Determinants: Note."
We are extremely grateful to Associate Dean Briziarelli for her assistance in the translation of the article, as well as to Professor Rosaria for a primary source providing additional biographical information about Rubini. We would also like to thank Professor R. Bradley, Professor E. De Freitas, and Professor L. Stemkoski for their helpful suggestions on this paper. All five are professors at Adelphi University.
The authors are also extremely grateful to the referees for their many helpful suggestions and corrections. In addition, the authors express deep gratitude to Janet Beery, Editor of Convergence, for graciously dedicating so much of her time in making additional suggestions for this paper.
Salvatore J. Petrilli, Jr., Ed.D., is an Associate Professor of Mathematics and Department Chair at Adelphi University. He has a B.S. in mathematics from Adelphi University and an M.A. in mathematics from Hofstra University. He received an Ed.D. in mathematics education from Teachers College, Columbia University, where his advisor was J. Philip Smith. His general research interests include history of mathematics, mathematics education, and applied statistics. However, the majority of his research has been devoted to the life and mathematical contributions of FrançoisJoseph Servois.
Nicole Smolenski is pursuing a graduate certificate in International Education from Florence University of the Arts. She will next be pursuing her JD and MA in Public Policy to become an Educational Policy lawyer. She taught for the past three years as a middle school mathematics teacher in the New York City public schools and taught an Italian elective for a year. She has a B.S. in Mathematics from Adelphi University and earned her M.A. in Mathematics Education at Teachers College, Columbia University. Her research interests are in Mathematics Education and its history.
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