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

Figure 1. Seki Kōwa, Creative Commons Attribution-Share 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 Pierre-Simon 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. Augustin-Louis Cauchy, public domain.

In his 1812 paper on determinants, Augustin-Louis 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 skew-symmetric matrix of odd order equals zero [Eves, 1990], where a skew-symmetric 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.