Using polar coordinates to measure variability in samples of Phenacolemur: a method of approach.

Polar coordinates can be used to quantify shapes on which homologous points cannot be readily identified such as lophs, edges, circumferences, outlines of areas, etc., This allows one to describe quantitatively characters, often diagnostic, that at present are not usually quantified and hence do not figure in the statistical analysis of variability that ought to accompany morphological descriptions in taxonomic work. Computer-driven plotters, using this data, provide an efficient means of graphing it, with great accuracy, in a variety of ways that can be used as an aid in qualitatively assessing variability in a sample or as a basis for further quantitative work. Principal component analysis of the data is essential to reduce the great quantity of measurements to a more manageable amount. It is also a valuable aid in interpreting the information. In the example using the Phenacolemur sample, it was shown with the data obtained from the use of polar coordinates in measuring that: (1) Discriminant functions can be used to distinguish between teeth that are very similar in shape and size, such as the Phenacolemur M1 and M2 and with which it is difficult to make such a distinction visually in an objective fashion. (2) It can be shown quantitatively that size is the only valid distinguishing criterion in splitting the AMNH Phenacolemur sample into three species and that the variability in shape is not significant.