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

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

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,

  • of a point \(M(x,y,z)\) from the plane \( \ \alpha_1x' + \beta_1y' + \gamma_1z' = 0,\)
  • of a point \(N'(x,0,z)\) from the plane \( \ - \ \gamma_1x' + \beta_1y' + \alpha_1z' = 0,\)
  • of a point \(N''(0,y,z)\) from the plane \(\ \alpha_1x' - \gamma_1y' + \beta_1z' = 0,\) and
  • of a point \(N(x,y,0)\) from the plane \(\ - \beta_1x' + \alpha_1y' + \gamma_1z' = 0,\)

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 three-dimensional 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.