A universal scaling relationship between body mass and proximal limb bone dimensions in quadrupedal terrestrial tetrapods

Imagine a fossil femur sitting on a lab bench. There is no muscle attached, no fat, no skin — just bone. The animal it belonged to has been dead for seventy million years. What could you possibly learn about how much it weighed? Campione and Evans found that one measurement of that bone — its circumference at the narrowest point along the shaft — predicts body mass with remarkable consistency across every living quadruped on Earth, from a mouse to an elephant to a crocodile. That's not a coincidence. That is a law of skeletal mechanics. Body mass is the keystone variable in paleobiology. Metabolic rate, growth rate, population density, home range — all of these scale with body size in predictable ways in living animals. Paleontologists use those relationships to reconstruct the biology of extinct ones. But you cannot put a dinosaur on a scale. So the question becomes: what do you use instead? Two approaches have dominated. Volumetric reconstructions build three-dimensional models of an animal and convert volume to mass using assumptions about tissue density. They are widely used, but they are also wildly sensitive to assumptions. For a single mounted Brachiosaurus brancai, different volumetric models produced estimates of roughly 38 tonnes and 74 tonnes — nearly a factor of two. The alternative is skeletal scaling: finding a statistical relationship between a measurable bone and known body mass in living animals, then applying that relationship to fossils. Skeletal scaling is repeatable, works on incomplete specimens, and returns explicit error bounds. The trouble is whether you can trust it. The best-known version of this approach is the Anderson method, derived from a sample of thirty-three extant terrestrial mammals. Anderson and colleagues related live mass to the combined minimum circumferences of the humerus and femur — the upper arm and upper leg bones, collectively called the stylopodia. Critics raised three pointed objections: Anderson's sample was taxonomically narrow, biased toward hoofed animals; different postures in different groups — a sprawling crocodile versus an upright elephant — impose different mechanical stresses on limbs and might break the relationship; and outliers among the largest animals could skew the regression. Could a single equation derived from a few dozen mammals really apply to a Triceratops? Campione and Evans set out to answer that directly. They built a dataset of two hundred mammal species and forty-seven non-avian reptile species — two hundred forty-seven taxa total — each represented by an individual skeleton with a known live weight. No estimated masses. Animals that had actually been weighed. For each specimen, they recorded the maximum length and the minimum diaphyseal circumference of both the humerus and femur. Circumferences were measured with thin paper tapes; lengths with calipers or fiberglass tape depending on size. All measurements were log-transformed to handle the extreme range of body sizes. They used standardized major axis regression — a technique that treats both variables as having measurement error, appropriate for comparing scaling relationships — alongside ordinary least squares regression for the predictive equations. The result is the kind of finding that looks almost too clean. Across all two hundred forty-seven species, the log-log regression of combined stylopodial circumference against body mass yields a slope of 2.78 with a coefficient of determination of 0.99. Mammals alone give a slope of 2.81. Reptiles alone give 2.79. Those numbers are statistically indistinguishable. On the log scale, body mass is a linear function of total stylopodial circumference — back-transformed, mass scales as a power of circumference, with essentially the same exponent whether you are measuring a shrew or a saltwater crocodile. The combined humerus-plus-femur circumference is the best single predictor of body mass among all the limb measurements they tested. The mean percent prediction error — the average deviation between predicted and actual mass — is 25.6 percent. A phylogenetically corrected version of the same regression, which accounts for the statistical non-independence of species that share recent common ancestors, gives almost identical results: a mean prediction error of about 25 percent. The signal survives removing shared evolutionary history. It is genuinely biological. Limb length tells a completely different story. Femur length alone has a mean prediction error of around 70 percent — nearly three times worse. Length-to-mass relationships differ substantially between clades and do not show the conserved pattern that circumference does. The bone's girth tracks mass. Its length tracks something else. Why? The mechanical intuition Campione and Evans offer is straightforward. Bone circumference is an index of compressive load capacity, and compressive load is dominated by body weight. Whether a limb is held sprawled out to the side or straight beneath the body, the weight of the animal must still be supported. Limb length, by contrast, is free to vary with locomotor style, posture, and ecological niche. The paper tests the circumference data against three classical scaling frameworks: geometric similarity, where shape scales proportionally as size increases; elastic similarity, the framework proposed by McMahon to describe structures resisting elastic deformation; and static stress similarity. No single model explains everything, but circumference consistently follows an allometric trajectory tied to mass while lengths vary across groups. The external girth of the stylopodial elements is most strongly related to the mass of the animal and only weakly influenced by whether the dominant forces on the limb are compressive or torsional. That is why a crocodile and an elephant fall on the same line. Campione and Evans then translate this directly into a tool for the fossil record. Their primary predictive equation, in plain terms: the base-ten logarithm of body mass equals 2.78 times the base-ten logarithm of the combined humeral and femoral circumferences, minus 1.16. Measure two bones, sum their narrowest girths, plug in the number, and you get a mass estimate with a prediction interval derived from the 25 percent mean error. That interval is honest — it reflects real biological variation — but it is also far tighter than what volumetric methods typically deliver. They apply this to a set of well-known non-avian dinosaurs using a phylogenetically corrected version of the equation. Iguanodon bernissartensis comes out at eight thousand six hundred eighty kilograms, with a prediction interval from six thousand five hundred ten to ten thousand eight hundred fifty. Triceratops horridus lands at seven thousand four hundred kilograms. Diplodocus longus at ten thousand nine hundred forty. Brachiosaurus brancai at thirty-five thousand seven hundred eighty kilograms — comfortably between the two wildly divergent volumetric estimates that started this whole comparison. These circumference-based estimates are also more consistent with Anderson's original method than with many volumetric reconstructions. The authors note that several volumetric models appear to underestimate mass, falling below even the lower bound of the circumference prediction interval. What this means practically is that paleontologists working on stem archosaurs, non-mammalian synapsids, or any extinct quadrupedal tetrapod now have a repeatable, specimen-based method with explicit uncertainty. Measuring circumference requires only the bone itself — no soft-tissue reconstruction, no assumptions about posture or density. The error bounds travel with every estimate. Because the relationship holds across the full phylogenetic breadth of living amniotes, the criticism that it cannot validly cross the mammal-reptile divide is, as Campione and Evans put it, conclusively rejected by the data. Return to that femur on the bench. Campione and Evans have shown that across hundreds of millions of years of evolution — through every shift in posture, every reinvention of locomotion, every transition from sprawling to upright — the minimum circumference of the major weight-bearing bones stayed tethered to body mass. That one structural constraint makes the fossil record legible in ways it otherwise would not be. Growth rates, metabolic regimes, the body-size changes that accompanied the origin of mammals from basal synapsids or the rise of large-bodied archosaurs — all of these become tractable questions the moment you can put a number, with honest error bounds, on how much an animal weighed. And all it takes is a measuring tape around a bone. This lecture was created by ennepō. Go to https://ennepo.ai to Discover, Create and Follow the latest research in your field. Read when you can. Listen when you want to.