Of all the articles on cephalometrics this journal has published over the last half-century, the one most cited across the scientific literature is the 1979 lecture “The inappropriateness of conventional cephalometrics” by Robert Moyers and me. But the durable salience of this article is perplexing, as its critique was misdirected (it should have been aimed at the craniometrics of the early twentieth century, not merely the roentgenographic extension used in the orthodontic clinic) and its proposed remedies have all failed to establish themselves as methods of any broad utility. When problems highlighted by Moyers and me have been resolved at all, the innovations that resolved them owe to tools very different from those suggested in our article and imported from fields quite a bit farther from biometrics than we expected back in 1979. One of these tools was the creation de novo of a new abstract mathematical construction, statistical shape space, in the 1980s and 1990s; another was a flexible and intuitive new graphic, the thin-plate spline, for meaningfully and suggestively visualizing a wide variety of biological findings in these spaces. On the other hand, many of the complaints Moyers and I enunciated back in 1979, especially those stemming from the disarticulation of morphometrics from the explanatory styles and purposes of clinical medicine, remain unanswered even today. The present essay, a retrospective historical meditation, reviews the context of the 1979 publication, its major themes, and its relevance today.
This essay is dedicated to the memory of Robert E. Moyers on the 100th anniversary of the American Journal of Orthodontics and Dentofacial Orthopedics.
Highlights
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“The inappropriateness of conventional cephalometrics” was published in 1979.
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The article, by Moyers and Bookstein, remains widely cited today.
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This essay reviews the context of the 1979 article, its themes, and its relevance.
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It is dedicated to the late Robert E. Moyers on the 100th anniversary of the AJO-DO.
The inappropriateness of conventional cephalometrics,” a lecture by Robert E. Moyers (1919–1996) and me that this journal published in 1979, remains its most cited article on cephalometrics of the last 50 years. Its title, though intentionally provocative, was not an exaggeration, and its argument, viewed from 37 years on, can still be viewed as a daring piece of intellectual criticism. From Moyers’ point of view it must have felt like a pitiless intellectual self-portrait, an explicit and mostly destructive critique of the biostatistical methodology embraced by the clinical profession (orthodontics) that he had served as an academic chairman for 15 years and then as the director of an even more interdisciplinary research center (craniofacial growth and development) for 15 years more (McNamara ).
The citation history of this article differs strikingly from the early-peaking, long-tailing shape that constitutes the commonest form of citation history for peer-reviewed biomedical articles in primary research journals. As Figure 1 shows, “Inappropriateness” (as I will be referring to this paper in these pages, for brevity) has been cited steadily, though irregularly, throughout the whole time span since its appearance. The year of peak citation count, in fact, was 2002, nearly a quarter of a century after its publication and 6 years after the death of its senior author, and it was cited as many times in 2014 as it was in 1982 or 1983. The article is actually the senior author’s most cited journal publication (though of course there are far more citations to his celebrated textbook of orthodontics) and also serves as my own most cited paper on any aspect of skeletal or craniofacial biology. Those who were present at its original presentation (the Cordwainers Lecture, Institute of Dental Surgery, University of London, May 17, 1978) surely numbered dozens more.
In addition to this odd stability of citation frequency, or perhaps in spite of it, “Inappropriateness” is unusual for 2 other reasons. As I already hinted, it objects to the methodology of nearly everything else that its senior author had been publishing recently, including the long-awaited and widely distributed Atlas of Craniofacial Growth that had appeared just a few years earlier after an enormous effort of compilation; and every methodological resolution that it prophesied for the problems and infelicities it diagnosed proved just about as inappropriate as the conventional methods that were pilloried in its own pages.
Whereas I was the brash young applied mathematician (less than two years past my own Ph.D.) slashing through the lore of an established field (clinical orthodontics) in which I had neither training nor professional stake, Moyers had committed decades of his career to publications based in just the quantitative language regarding which he had already turned skeptical, a language I seemed to be trying to demolish. But the proposed upgrades of method I could offer Moyers’ readers at the time “Inappropriateness” was written did not last even 5 years. Already by 1984, for instance, biorthogonal grids, the core technique of my doctoral dissertation, had been replaced by the shape coordinates (see below) that proved far more suitable for statistical summaries such as variance and covariance. So my co-author had far more status at risk than I had, and yet his contributions to the radical part of the argument proved far more cogent than mine. Of the systematic charges that “Inappropriateness” laid against conventional cephalometrics—I will review them below under the paper’s own rubrics of “fabrication,” “camouflage,” “confusion,” and “subtraction”—some remain unresolved to this day, while those that have been resolved owe their resolution not to the techniques named as promising in the same lecture—representations of curvature, medial axes, and biorthogonal grids—but to innovations arising from other branches of biometrics and medical image analysis entirely: the class of methods that are nowadays generally referred to as geometric morphometrics (GMM).
That was not what I expected the fate of this paper to be. I had not properly understood its own academic context, which was not the setting of craniofacial biology into which Moyers had inducted me when he appointed me assistant research scientist in his Center for Human Growth and Development at the University of Michigan in 1977, but the much older context of anthropometry and in particular craniometry ∗
∗ In this essay I will use the word “craniometry” in the orthodontist’s sense—the measurement of the cephalogram, the conventionally positioned lateral or anteroposterior roentgenogram. “Craniometry” or “craniometrics,” after the German or French, will instead mean the measurement of the solid skull as an object in the laboratory or, more recently, its virtualized equivalent as a 3-dimensional surface image. One hundred years ago the word “craniometry” already appeared in the standard unabridged dictionaries of educated English. For instance, on page 1331 of the 8600-page Century Dictionary of 1914, under this keyword, there is a thousand-word entry mentioning and displaying dozens of points and lines that might have served as a précis of Rudolf Martin’s textbook of the same year. At this time the word “cephalometrics” apparently did not even exist as a technical term in English. The closest my Century Dictionary comes is the 4-word entry on page 891 under “cephalometry”: “measurement of the head.” This follows the entry for “cephalometer,” which is “an instrument formerly used for measuring the fetal head during parturition.” As used in this essay, then, the word “cephalometrics” is a neologism postdating the development of the standardized roentgenography on which it is based (the Broadbent-Bolton imaging instrument of 1937).
that had been under relatively unsupervised development since at least the turn of the twentieth century. That “there is no theory of cephalometrics,” as “Inappropriateness” cogently argues, is because there was not and never had been any proper methodology of craniometrics either. The channels of “misinformation” that “Inappropriateness” unearthed were entirely exogenous in their origin: the combination of problems pervading all of twentieth-century craniometry with the newer ambiguities of a purportedly standardized medical imaging modality, to wit, cephalography. The solution to the problems highlighted by Moyers and me awaited a proper biometrical theory of the way that patterns of geometry in medical images could be turned into valid inferences about the underlying biological processes; but this theory did not exist until very recently, when “Inappropriateness” was already more than a third of a century old.
A brief iconoclastic history
A substantial minority of the problems that Moyers and I complained about were an intrinsic component of the classical stance in anthropometrics that had been in place since before he was born. I knew none of this back in 1979—it has come to my attention only since I’ve been professor of anthropology at the University of Vienna. As I have already noted in 2 recent books, the incoherence of craniometry as a scientific methodology was already clear in Rudolf Martin’s great textbook Lehrbuch der Anthropologie (first edition, 1 volume, 1914; second edition, 3 volumes, 1928, posthumous). Martin seems to have been concerned that an earlier generation of anthropometrists tabulated averages or ranges of physical measurements such as lengths, ratios, and angles without any principled basis for the list of measures thus presented. For instance, the Croatian anthropologist Karl (Dragutin) Gorjanović-Kramberger, in his great treatise of 1906 on the newly identified Krapina Neanderthals, couched his rhetoric of group comparisons as “tabular overviews” (tabellarische Übersichte) of seemingly arbitrary lists (see page 119 or page 167) each of which “instructs us” (belehrt uns) about what group a particular reconstructed skull “doubtless belongs to” (zweifellos angehört) .
Martin’s solution was a clever one: to focus on the actual formulas for the entries in these tables. (Our current approach emphasizes rather the information in the locations of the points that drive those formulas, but information in this narrowly statistical sense would not be invented for another 30 years.) As those formulas all involved measurements keyed to the locations of specific points, he called them Meßpunkte, “measuring points”; today, in English and in most other European languages, we call them “landmarks.” Martin systematized the images of these points in multiple views and wrote out all of their operational definitions at length. By the time of the 1928 edition there were 68 of these points. What remained was to list all the distances that anybody had ever reported among pairs of these points, all the lines set by pairs of points or tangents to curves, and all the ratios of 2 distances, heights of triangles, and 3-point or 4-point angles that any of his colleagues had ever used. The list of these indices ran to dozens of pages, many augmented by little tables of cutpoints for words like “brachycephalic” or averages by racial/ethnic group.
The missing “theory” of cephalometrics (or craniometrics) that “Inappropriateness” was referring to was reified in the chaos of Martin’s lists. Such a disorder persevered right into Moyers’ own work: the roster of measurements entailed in the Riolo et al. Atlas of 1974 was the exact equivalent of Martin’s approach when the points were restricted to the 45 highlighted there—the measurements tabulated were an equally arbitrary selection of all the possible choices (990 distinct distances among pairs of the 45 points, 42,570 distances from points to lines, 85,140 angles) potentially pertinent to analysis of the same data resource of point locations. The development of morphometrics from the late 1970s on can be summarized as a technology for literally reversing the logic Martin was instantiating: not guessing first at distances or ratios, then tabulating them for purposes of group comparison, but instead letting the group comparisons themselves (or the multivariate equivalents, such as shape regressions or principal components) drive the specification of variables or composites. The biorthogonal method highlighted in the constructive section of “Inappropriateness” was a first attempt at this reversal, but it did not support any parallel statistical method, and so was quickly replaced by the shape space methods that my colleagues and I would develop a few years later.
At just about the same time as Martin’s first edition, D’Arcy Thompson was announcing his “method of Cartesian transformations.” (My biorthogonal method actually incorporated a variant of this.) In this approach, which can be traced back centuries, any single shape comparison could be plausibly rendered as the deformation of a piece of graph paper. (Around 1960, by relaxing the constraint of continuity, the Boston orthodontist Coenraad Moorrees would convert this into an explicit diagnostic technique, his “mesh method.” ) But in the absence of appropriate statistical tools for samples of these grids, the visualization served mainly as a charming diversion, always “promising” but ultimately no help for any form of quantitative persuasion, calibration, or theory-testing. There is a review of the general history of Thompson’s suggestion in Bookstein.
I noted above that “Inappropriateness” was this journal’s most-cited discussion of cephalometrics of the last half-century. If I disable the date filter a new winner emerges, Cecil Steiner’s “Cephalometrics for you and me” of 1953. “The cephalometer is here to stay,” he opined, but is not yet in general use by clinical orthodontists, who must steel themselves to “master its mysteries.” He continued: “Hundreds of measurements and combinations of measurements [ratios] can be made from cephalometric x-rays. Many of these have value. … In our office, the following measurements and assessments are made. I now shall attempt to justify them.” In other words, Steiner conceded the main point “Inappropriateness” would make 26 years later: there would be no intersubjective basis for the quantifications that underlie cephalometric clinical judgments in any of the conventional systems—no logical basis for the proffered number of different features of form worth measuring, nor for the specific formulations of the distances, angles, or ratios that are put forward in order to measure them.
Steiner began with the problem of telling left-side landmarks from right, an issue overlooked by Moyers and Bookstein because the Atlas used averaged bilateral landmarks only. After that, he was particularly concerned to locate porion; we shall see presently that the team assembled by Horst Seidler in Vienna in 1998 ended up instead abandoning the very idea of landmark-based registration (whether via the Frankfort horizontal plane, which required porion for its alignment, or any other rule). The justification of the Frankfort, though, is because it was traditional for anthropologists in that its reference points were on the outer surface of the skull, where they were visually accessible. This is an unconvincing basis for a methodology like the Broadbent-Bolton that is intended for x-ray images instead. While the internal midsagittal points could not be located by the conventional anthropologist without hemisecting an actual skull, some bilateral landmarks could be located from the lateral cephalogram with great precision indeed. Steiner went on to highlight particular angles that he measured, among them the angle ANB, without ever admitting the possibility that what he was actually recommending was a scientific method—that its ambit might go beyond the task of clinical description so as to also embrace the assignment of making and verifying truth claims that could then be brought to bear upon clinical observations of an individual case.
Where Steiner was interested in representing the motion of the mandible over the opening-closing cycle, including a representation of a condylar “axis” that is explicitly criticized in “Inappropriateness,” Moyers and his colleagues in the Atlas work were more concerned with changes over time. For this they exploited a statistical method quite a bit older than Martin’s work: the method of growth charts that Edith Boyd reviewed from its origins in the 1700s. The tradition of referring growth to such charts, one distance or other extensive measure per page, was well-established by midcentury (see, for example, the systematic survey of California schoolchildren’s growth by Tuddenham and Snyder ). But, in what is an irony for our present purposes, the statistics for a proper analysis of groups of such curves did not mature until well into the 1990s (see, for instance, the book by Ramsay and Silverman ) and remains a topic of active research even today.
Similarly, the methods for combination of multiple measurements (what is now called “multivariate analysis”) that were in the hands of the anthropologists by the 1950s (for a bit of history, see my article ) did not acknowledge the possibility that measurements of landmark configurations would have to be treated differently. The problem was well-known since the work of Beni Solow in the 1960s —correlations of measurements overlapping in their lists of reference points were not distributed around zero the way the usual textbook formulas presume they should be—but no solution would be at hand until the mid-1990s (the principal components of shape with respect to Procrustes distance; see below).
If craniometrics can be blamed for about half the dilemmas that “Inappropriateness” was pinpointing, the other half can be attributed to the particular geometric problems of getting to measurements from roentgenography, the projection imagery that likewise predated my own life by at least half a century. Martin could describe his canonical views ( norma lateralis, norma frontalis, etc—what Melvin Moss somewhere sardonically refers to as “the six ladies of the cephalometric laboratory”) by direct inspection of the solid objects being measured. But the orthodontist could not see these objects, and every rule for their standardization came up against the same repeated objection—that the form-variations one was interested in discussing were entangled with the very conventions according to which those skulls were being reduced to flat images and then discrete points in the course of measurement.
The basic intuition driving the cephalostat—that general variations of a form as complicated as the human skull, even a nominally symmetric skull, can be captured by a set of just a few views at fixed angles to one another—is mathematically unfounded and geometrically unsound. The best critical language had not yet been developed (this theme will recur many times in my pages today)—it was first crystallized in accessible textbook form at the surprisingly late date of 1990 by the Dutch physicist Jan Koenderink. Koenderink’s book, as its title (Solid Shape) implies, deals with a variety of formalisms for comprehending and apprehending shape features of smoothly curving surfaces in three dimensions by looking at them from various directions carefully calculated so as to highlight a range of explicit features of those same surfaces. A surprising number of these features—call them “silhouettes,” the name they are assigned in the portrait literature—are manifest in the edges of these surfaces as they appear in projected images. (Please do not confuse these images with sections of the original solid; they are representations of surface geometry only.) The apparent edges of these projected images can have a variety of features, such as extremes of curvature (eg, gonion) or self-intersections (eg, articulare, pterygomaxillary fissure). Usually these and their analogues over the whole form are explicit functions of the coordinate system used for projecting the surface down onto a plane. Because they are coordinate-dependent, in turn, their locations are not functions of the biological form per se, but also depend on the choice of coordinate system, which is, of course, a human artifact.
A prime example, already considered by the authors of “Inappropriateness,” is orbitale, “the lowest point of the orbital rim.” Martin says of this point, “Es dient nicht als Meßpunkt” (p. 556)—it does not serve as a landmark; its sole function is to help align what he called the Ohraugen-Ebene, the ear-eye plane (nowadays the Frankfort horizontal plane). Orbitale is not a landmark because it is not actually on a curve—“the orbital rim” is not a well-defined curve on the actual cranial form, nor does it lie on any silhouette of the projected surface. Its location on the lateral cephalogram is entirely an aesthetic judgment corresponding to the schema of Koenderink’s Figure 418 when the book is viewed upside-down.
Such points are never among the Meßpunkte accepted by Martin in his craniometric discussions of a century ago. Rather, nearly all of Martin’s landmarks are intersections of curves, sometimes on the external surface of the skull and otherwise on the intersection of the solid form with its (nominal) midsagittal plane. This applies even when the curve in question is not itself explicit on the skull. Thus porion (Martin, page 512) is defined as a point on the upper border (Oberrand) of the porus acusticus externus, but that opening does not actually have an upper border any more than the bell of a tuba does (if you ignore the actual edge where the metal stops).
Inasmuch as neither orbitale nor porion is a valid craniometric point, the “line” they delineate, the Frankfort horizontal, cannot be a component of any biologically valid quantification. The problem is not their ambiguity as points sampled upon curves, but rather the nonexistence of the curves on which they purportedly lie: these are, in truth, curves neither on the solid skull nor on its radiographic image. Findings dependent on the Frankfort horizontal, therefore, cannot even be considered to be multilocal, analogous to angles among 3 or 4 points. Rather, the dependence of any measure on the Frankfort is entirely an artifact of where the observer chooses to place the nonphysical points porion and orbitale that purport to summarize small regions of the projected image. Interestingly, it was precisely this incoherence of the classical notion of the Frankfort horizontal plane that induced Horst Seidler, Professor of Anthropology at the University of Vienna, to assemble the conference in 1998 at which contemporary geometric morphometric ideas were first introduced to the European craniometric community. The conference instigated the 1999 paper by Bookstein et al and so was responsible for the entire “Vienna school of morphometrics” emphasizing not landmark points but semilandmarks upon curves and surfaces. (Ironically, the impossibility of validly operationalizing the Frankfort horizontal makes hardly any difference for the analysis of the lateral cephalogram, the main concern of “Inappropriateness” as it is of orthodontists today. It is the anteroposterior image that demands standardization in this or some other completely arbitrary way.)
The tension between geometric reality and cephalometric convention was already clear in the Riolo-Moyers Atlas of 1974. Figure 2 is a template of the 45 purported landmark points entailed in their study. But some points (#35, #28, #11) lie on no image structures at all, while others (eg, #40, orbitale; also #42, #43) lie on purported curves that do not actually exist on the solid skull. (The curving arc through #40 and #41, for instance, is actually the nonplanar curving locus in space where the surface tangent plane to a sheet of cortical bone passes through the origin of the x-ray beam, a location that has no meaning whatever as far as the biomechanics, growth, or physiology of the head is concerned.) Several points, including #36, #31, and #32, represent overlays of curves on the cephalogram that do not actually correspond to loci in space, but rather lines tangent to the solid skull at 2 separate locations that also happen to lie perpendicular to the nominal midsagittal plane. (In other words, the “landmark” as located combines the anteroposterior coordinate of one surface patch with the craniocaudal coordinate of another at some distance—quite a perplexing biological construction, to say the least.) Many points lie on single curves in ambiguous positions lacking robust operational definitions (examples include #37 glabella, #9 A-point, #4 B-point, #33 condylion, and all 4 of the points on the mandibular symphysis). Omitting the teeth, the number of points that could conceivably qualify as craniometrically valid is at most 7 (#10, #29, #30, #34, #38, #39, #45). So Moyers’ own reference publication is fully vulnerable to the sharp criticisms he would be publishing less than 5 years later. Finally, notice that porion does not appear in this figure at all—indeed, there is no curve for the auditory meatus, as whatever position its “border” might actually be imagined to occupy is obliterated in the image by the steel rods used to restrict the subject’s head position.
Fabrication, camouflage, confusion, and subtraction
“Fabrication, camouflage, confusion, and subtraction” are Moyers’ terms for 4 of the systematic ways in which the conventional cephalometrics of his day would “misinform”—would mismanage the data even after the images had been (inappropriately) reduced to landmark point locations already. Most of these problems are familiar to today’s student from later critiques of diverse morphometric methods.
“Fabrication” was Moyers’ term for the generation of “growth curves” as series of positions of a landmark point within a coordinate system registered on others in some fashion. The critique boils down to the observation that such traces are not susceptible to proper biometrical analysis, as they confound changes in the region of the landmark with changes and frank registration artifacts at a distance. The fallacy is declared to be a troublesome consequence of the fact that divergences between neighboring landmarks are “much less than their common translation.” The formal resolution of this problem was not published until just last year, but the 1979 critique was certainly valid as far as it went.
By “camouflage,” the 1979 authors meant the concealment of this registration dependence among the superpositions that would generally convey the findings of a conventional cephalometric analysis. Those superpositions would typically fix a landmark point (sella, perhaps) and some line through it in the direction of a second landmark point—nasion, perhaps. Figure 7 of “Inappropriateness” already announced the principle of “tensor biometrics” that would be the core of my lengthy innovation of 1984, the role of weighted averages of landmarks in breaking through the obvious asymmetries of this formulation to unearth simple summaries of change whenever such summaries were in fact available. “Inappropriateness” already acknowledged that for the proper description of a uniform transformation (a simple shear or directional stretch), a registration needs to be along the principal strains of the landmark change, directions that usually do not lie along explicit interlandmark segments. In other words, the appropriate registration is a function of the change to be described, not vice versa.
The Moyers rubric of “confusion” covered 3 major rhetorical difficulties besetting the conventional methodologies: description of rotations, “controlling for size” (the quotation marks are in the original article), and reporting changes in angle. As for the preceding 2 categories, while these criticisms were all correct, the corresponding resolutions were widely scattered in time and in discipline of application.
Reporting changes in angle turned out to be the formal dual of the problem of optimal reporting of a uniform transformation already noted under the heading of “camouflage.” This pairing, the explicit topic of a table on pages 220–221 of my treatise of 1991, originally appeared in this Journal as one component of the growth analyses in an article on craniofacial growth invariants.
The issue of “controlling for size” was resolved in several distinct steps. First came the introduction of a standardized size scalar, centroid size (square root of the sum of the principal moments of the suite of landmarks around their centroid). The extension of this concept to drive regressions of shape on size (the implementation of allometry in the context of landmark data) was the essence of the notion of shape regressions that was introduced at the NATO conference on morphometrics run by Les Marcus in 1993 (see the augmented proceedings volume ). Finally, the idea of form space, the extension of shape space to incorporate size explicitly, in the form of the natural logarithm of centroid size, was the subject of the theorems first presented in an article exploiting them in one classic application (the allometric shape trajectories of humans and great apes).
Rotations of small features of form were dealt with by the method of edgels published for a mathematical audience in a highly technical exposition and implemented in the 2-dimensional module ew2 of W. D. K. Green’s program package Edgewarp. In contrast, issues of rotation in the large remain a site of active innovation as of this writing. The regionalized rotation associated with the bending of a rigid body is dealt with as an adaptation of textbook formulas in Bookstein, while the general case remains unformalized.
Finally, Moyers’ fourth category of misinformation, “subtraction,” would today be regarded as a misnomer. The crux of the critique as published had nothing much to do with the arithmetical operation of that name. Its concern would better be described as the relation of any scheme of landmarks or curves to the actual osteology of bone, depending as it does on coordinated processes of deposition and resorption all over multiple surfaces. In other words, the objection was to the conceit that comparison of measurements on static structures could lead to biologically appropriate insights into actual processes of regulated biological change over time in osteal tissue. Already by 1979 we knew that analyses of growth as quantified via locations of metal implants were incommensurate with cephalometric analyses of the same growing animals. During growth, “form moves through bone”: Moyers himself had made this point in his Handbook of Facial Growth appearing just a few years before “Inappropriateness.” Recently a seminal article by Oxnard and O’Higgins has broadened the concern here from bone per se to the soft tissues adjacent, particularly the muscle insertions that we often rely on to infer selective effects of function on dried skeletal material. Other applications, such as the way the dental enamel comes to ride on top of the dentin, are even more recent.
In summary, Moyers’ pessimism about the state of cephalometrics circa 1979—its “inappropriateness” for most scientific purposes—was a natural outgrowth of methodological stasis across the fields for which he had been responsible at the University of Michigan: first orthodontics, then craniofacial growth and development more generally. By the 1970s, it was clear to many of us, not just Moyers and me, that something was deeply wrong with the way that sets of points, either on a skull or on an x-ray of a skull, were converted into lists of decimal numbers and then into pattern statements referring to those lists. There seemed to be no way that those numbers or pattern claims could be used either for consilient scientific research or for objectively rational clinical decision-making. In other words, cephalometrics was not yet a numerical method, but only a sort of subjective guide to form, along the lines of the guides to artistic anatomy that had concerned analogous representations of the face by portraitists for hundreds of years. This failure, which characterized Moyers’ own Atlas as much as it did anybody else’s work, was one of the main drivers of the “Inappropriateness” critique.
When I joined his research group in 1977, as a freshly minted Ph.D. in the measurement of biological shape, it was reasonable to hope that developments would come quickly. Some did—shape coordinates were published by 1986, thin-plate splines by 1989, analysis of asymmetry and analysis of semilandmarks on curves by 1991. The issues of “fabrication, camouflage, and confusion” in particular that concerned the 1979 authors were, for the most part, resolved by the mid-1990s, and the new methods had been published widely in the literatures of applied statistics and image-processing, with specific reference to some pioneering medical (but noncephalometric) applications such as the shape features of the human brain. But reversion of the new methods back into craniometrics and particularly into orthodontics was considerably delayed past that date, partially owing to lack of clinical demand but more, in my view, to our failure to deal with the more fundamental set of problems, the problems of the cephalogram as a source of data in the first place. As a result, from the date of publication of “Inappropriateness” (1979) to the date of that Vienna conference 19 years later, there seems to have been no obvious progress within the literature of cephalometrics itself: no resolution of any of the issues focused on by Moyers and me. Yet by the time of that Vienna meeting, the adaptive radiation that became modern geometric morphometrics was about to become explosive. What a shame that Moyers did not live to see any of this.
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