The geometry of petrological sample space shows that a rock sample, defined by a particular composition or proportion of components, is uniquely described by a radial direction (= compositional axis) from the origin of a Cartesian reference frame where amounts of each component are measured along the axes. The original measured amounts determine the direction of the compositional axis defining a particular mixture of components. Normalization (i.e., expressing amounts as proportions of the whole) merely provides radial projection of the sample onto a proportions frame (three components = the ternary diagram) to permit a form of comparison of different compositions. Such comparisons become inconsistent when traditional parametric approaches are used because, as the "constant sum problem" has long revealed, numbers expressing proportions or percentages are not amenable to standard statistical methods. In addition, because of the disproportionate display of radial difference over the ternary diagram, it cannot be used as a basis for reliable calculations. Recognition of the fundamental radial character of compositional data reveals that they should, most logically, be analyzed on the sphere. Not only does such an approach bypass previously recognized problems in statistical treatment of compositional data, but also it opens the way for the application to petrological data of a whole class of established parametric and nonparametric statistical methods.

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