Presented to the Missouri Section of the Mathematical Association of America, November 26, 1927.
In recent years many references have been made to the problem of the extension of the formulas of spherical trigonometry to space of higher dimensions. In particular, Karl Pearson, emphasized the importance of such an extension for the sake of its application to the theory of multiple correlation. E. V. Huntington revoiced the sentiment of Pearson in urging the importance of mathematics in modern statistics. Later, Dunham Jackson gave a trigonometric representation of correlation. The problem was considered in more detail by James McMahon in the paper, Hyperspherical goniometry and its application to correlation theory for n variables. In this article he generalized the formulas of spherical trigonometry for the hypersphere in n dimensions and then applied these formulas to correlation for n variables. McMahon considered only Euclidean spaces.
It is the object of the present paper to show that the desired trigonometric formulas are essentially contained in well known relations connecting certain invariants of space. Some of the more important of these date back to Grassmann. Formulas which may be regarded as the extension of the Grassmann formulas to curved or Riemannian space were established by Maschke.