Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/180

We have next the addition of plane surfaces (Plangrössen). The equation

${\text{ABC}}+{\text{DEF}}+{\text{GHI}}={\text{JKL}}$

signifies that the plane ${\text{JKL}}$ passes through the point common to the planes ${\text{ABC, DEF,}}$ and ${\text{GHI,}}$ and that the projection by parallel lines of the triangle ${\text{JKL}}$ on any plane is equal to the sum of the projections of ${\text{ABC, DEF,}}$ and ${\text{GHI}}$ on the same plane, the areas being taken positively or negatively according to the cyclic order of the projected points. This makes the equation equivalent to four ordinary equations.

Finally, we have the addition of volumes, as in the equation

${\text{ABCD}}+{\text{EFGH}}={\text{IJKL}},$

where there is nothing peculiar, except that each term represents the six-fold volume of the tetrahedron, and is to be taken positively or negatively according to the relative position of the points.

We have also multiplications as follows: The line (Liniengrösse) ${\text{AB}}$ is regarded as the product of the points ${\text{A}}$ and ${\text{B}}.$ The Plangrösse ${\text{ABC,}}$ which represents the double area of the triangle, is regarded as the product of the three points ${\text{A, B,}}$ and ${\text{C}},$ or as the product of the line ${\text{AB}}$ and the point ${\text{C}},$ or of ${\text{BC}}$ and ${\text{A}},$ or indeed of ${\text{BA}}$ and ${\text{C}}.$ The volume ${\text{ABCD}},$ which represents six times the tetrahedron, is regarded as the product of the points ${\text{A, B, C,}}$ and ${\text{D}},$ or as the product of the point ${\text{A}}$ and the Plangrösse ${\text{BCD}},$ or as the product of the lines ${\text{AB}}$ and ${\text{BC}},$ etc, etc.

This does not exhaust the wealth of multiplicative relations which Grassmann has found in the very elements of geometry. The following products are called regressive, as distinguished from the progressive, which have been described. The product of the Plangrösen ${\text{ABC}}$ and ${\text{DEF}}$ is a part of the line in which the planes ${\text{ABC}}$ and ${\text{DEF}}$ intersect, which is equal in numerical value to the product of the double areas of the triangles ${\text{ABC}}$ and ${\text{DEF}}$ multiplied by the sine of the angle made by the planes. The product of the Liniengrösse ${\text{AB}}$ and the Plangrösse ${\text{CDE}}$ is the point of intersection of the line and the plane with a numerical coefficient representing the product of the length of the line and the double area of the triangle multiplied by the sine of the angle made by the line and the plane. The product of three Plangrössen is consequently the point common to the three planes with a certain numerical coefficient. In plane geometry we have a regressive product of two Liniengrössen, which gives the point of intersection of the lines with a certain numerical coefficient.

The fundamental operations relating to the pointy line, and plane are thus translated into analysis by multiplications. The immense flexibility and power of such an analysis will be appreciated by