On Being the Right Size

In 1638, under house arrest and nearly blind, Galileo published a discussion of why a large animal cannot have the same shape as a small one. He drew two bones side by side: a normal one, and one scaled up in length and thickened grotesquely to bear the weight it would have to bear. The second looks absurd, and it is correct.

The reasoning is a single piece of arithmetic that governs more of the physical world than almost anything else you can learn in an afternoon.

Double every linear dimension of an object. Its surface area, which scales with length squared, goes up by four. Its volume, and therefore its mass, scales with length cubed and goes up by eight. The ratio between them has changed, and it changes with every increase in size, without limit. That is the square-cube law, and it is the entire argument.

Now attach that to biology. The strength of a bone or a muscle depends on its cross-sectional area, which grows as the square. The weight it must support grows as the cube. So an animal scaled up uniformly becomes steadily weaker relative to its own weight, and at some size it cannot stand. This is why an elephant's legs are pillars and a gazelle's are sticks, and why the giants of folklore, built to human proportions at three times the height, would break their own femurs standing up.

The same law explains why there are no dog-sized insects, and here two mechanisms compound. An insect breathes through tracheae: a branching network of tubes carrying air directly to the tissues, moved largely by diffusion. Diffusion is quick over a millimetre and hopeless over a few centimetres, because the time it takes scales with the square of the distance. Scale up the insect and oxygen cannot reach the interior. Then add the exoskeleton, which must thicken to carry eight times the mass, which adds mass, which requires more shell. The exception proves the rule: Carboniferous dragonflies reached seventy-centimetre wingspans in an atmosphere with substantially more oxygen than ours, which relaxed exactly the constraint that binds.

Scaling also reverses which forces matter. For a human, gravity dominates and surface tension is a curiosity; for an insect, surface tension is a serious force and gravity is manageable. This is why a wet insect is in real trouble (the water clinging to it is an appreciable fraction of its own weight) and why a creature can walk on a pond. Below a certain size, air behaves less like a thin medium to be pushed through and more like something viscous to be swum in.

Which gives J. B. S. Haldane the best-known passage in the literature, from his 1926 essay "On Being the Right Size": you can drop a mouse down a thousand-yard mine shaft and, on arriving at the bottom, it gets a slight shock and walks away. A rat is killed, a man is broken, a horse splashes. The drag that slows a falling body depends on its cross-sectional area; the momentum to be dissipated on impact depends on its mass. Small things reach a low terminal velocity and arrive gently.

A related unit artefact is worth clearing up, because it is repeated constantly. A flea jumps a hundred times its own height, which sounds like superhuman strength. Jump height depends on the energy a muscle can deliver per unit of mass, and muscle delivers roughly the same energy per kilogram across the animal kingdom, so absolute jump height is broadly similar for a flea and a person. A flea jumps a few tens of centimetres. Expressing it as body lengths makes small animals look miraculous and is telling you about the denominator. The genuinely remarkable thing about fleas is elastic energy storage (a protein spring released faster than muscle could contract) which is a story about power rather than strength.

Metabolism has its own scaling law, and it is one of the few numerical regularities in biology that holds across the whole of life. Max Kleiber found in the 1930s that metabolic rate scales with body mass to the power of about three-quarters, and it has since been shown to hold across more than twenty orders of magnitude, from bacteria to blue whales. Larger animals use less energy per kilogram than small ones, which is why a mouse must eat a substantial fraction of its body weight every day while an elephant eats a few per cent, and why small mammals live fast and die young. Why the exponent is three-quarters rather than the two-thirds that pure surface-area scaling would suggest is still argued over, with the leading explanation involving the geometry of branching supply networks.

And the law does not stop at biology. It is why scale-model testing flatters every structure: at one-twentieth scale a model's weight has fallen by a factor of eight thousand while its cross-sections have fallen by four hundred, making it twenty times stronger relative to its own weight. Engineers therefore test using dimensionless similarity parameters rather than raw load ratios. It is why large animals struggle to shed heat, since heat is generated by volume and lost through surface. It is why ants and bacteria live in a world governed by adhesion rather than by weight.

Galileo's conclusion holds for all of it. You cannot make something bigger and keep the same design. Above a certain size you have to build a different thing.