etry with biology'' (Bookstein, 1982). The dual emphasis has remained
in theory, but applications have often failed on one or the other of these
For morphometrics to fulfill its promise of fusing geometry with
biology, there must be equal emphasis on the two components.
Morphometric techniques need to be designed and applied with biolo-
gy in mind. Quantitative results must be directly interpretable biolog-
ically. In addition, biologists have needs specific to the nature of life.
Biologists require methods for correctly analyzing change over time
due to growth or evolution, methods that provide for the completion of
partial specimens using available data, and methods for predicting, or
“retrodicting” in the case of paleontology, the geometry of hypothetical
forms based on evolutionary, ecological, functional, and/or develop-
mental considerations. We believe that morphometrics should provide
Although our focus is on studies in modern biology at the level of
the organism, we hope that those seeking tools for use in studies of the
cellular or molecular levels of scientific investigation will find value
and potential applications in this topic. We provide a molecular exam-
ple through a contribution from our colleague Tim Cole in Chapter 7.
We also hope that workers in alternate disciplines such as ecology,
geology, paleontology, meteorology, and geography may find value in
this topic. If our methods are truly useful, they should be applicable to
any discipline that requires a statistical tool for the study of geometric
relationships when coordinate data are given.
1.2 Foundations for the study of biological forms
We believe, as Bonner and Medawar did, that the success of D'Arcy
Thompson's masterful morphometric work, On Growth and Form , can
be explained by the uniqueness of the author's scholarship and abili-
ties. According to Medawar (1958), D'Arcy Thompson had not only the
makings, but the actual accomplishments of three scholars: classicist,
mathematician, and naturalist. The fact that these talents were inte-
grated within one person explains his achievement in combining math-
ematics and biology into a single work that is at the same time a beau-
tiful piece of literature. Neither of the authors of this volume are
experts in more than one field. For this reason, we joined forces in
authorship. The motivation for comparing biological objects and the
knowledge for interpreting those comparisons must come from biology,
but the tools for comparison come from statistics and mathematics.