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An ant can carry many times its own weight, so an ant the size of a horse would be unstoppable. Except it would not be able to stand up, or breathe, and its legs would break under it while it was still trying. The reason is arithmetic you can do in a minute.
Films have been making giant insects for seventy years. A spider the size of a car. An ant the size of a horse. The joke everybody makes is that an ant can lift many times its own weight, so a horse-sized one would be able to throw a bus.
It could not. It could not stand up.
And the reason has nothing to do with biology being fussy. It is arithmetic, it was written down in 1638 by Galileo, and once you have seen it you cannot unsee it — you will start noticing it in the shape of every animal you look at.
When you imagine a giant ant, you imagine an ant, larger. Same proportions, same thin legs, same narrow waist, just scaled up like an image dragged bigger on a screen.
That is the assumption: that size is separate from shape, and you can change one without touching the other.
It is a completely natural thing to think, because it is true for a drawing. A drawing has no weight.
An animal does. And weight and strength do not grow at the same rate when you scale something up — not even close.
Take an animal and double every length: twice as tall, twice as wide, twice as long. Nothing else changes — same shape, same material. Watch the three columns come apart. Press for each step.
An engineer builds a scale model of a bridge at one tenth of full size, and it holds far more than its share of weight before breaking. What should they conclude about the real bridge?
Weight grows with volume, three lengths multiplied together. Strength grows with area, only two. So every time an animal gets bigger, its own weight gains on its strength — and it has to change shape to keep up.
Look at a gazelle's legs. They are twigs. Look at an elephant's. They are columns, held nearly straight underneath it, with the foot directly below the shoulder.
That is not a stylistic difference. The elephant cannot afford a bent leg, because a bent leg is held up by muscle rather than by bone stacked in a line, and its muscles are simply not in that league. So the elephant stands on its skeleton like a table stands on its legs, and it cannot gallop, cannot jump, and cannot get up quickly if it goes down.
A cat can leap several times its own height and land without giving it a thought. Scale that cat up to elephant size and the landing would break it. Not because it got clumsy — because the forces went up faster than the bones did.
Now run the arithmetic backwards, because this is the half that usually gets left out.
A body loses heat through its surface, and a body makes heat with its volume. Shrink an animal and volume falls away faster than surface does — so a small animal has an enormous amount of skin for the amount of body behind it, and heat pours out of it.
The shrew lives at the sharp end of this. It is among the smallest mammals there is, and it has to eat something close to its own body weight in food every day just to keep its temperature up. A shrew that cannot find food for a few hours dies. Not weakens — dies.
Meanwhile an elephant has so little surface for its bulk that the opposite problem arrives, and getting rid of heat becomes the difficulty. Which is what those enormous ears are for: thin, wide, full of blood vessels, flapped to move air across them. They are radiators, and the animal needed them because of the same arithmetic that gave it the legs.
Insects do not have lungs. They have a network of fine tubes running into the body from holes along their sides, and oxygen makes its own way inwards.
That works wonderfully at insect size and it is hopeless at any real size, because the tubes deliver oxygen across a surface while the body that needs it is a volume. The bigger the insect, the further the oxygen has to travel and the more of it is wanted at the end of the journey.
So there is a ceiling, and the fossils show us where it sits — and that it can move. In the Carboniferous, hundreds of millions of years ago, there were dragonfly relatives with wings spanning about seventy centimetres. The atmosphere then is estimated to have held far more oxygen than it does now, around thirty per cent against today's twenty-one. Richer air, bigger insects. The air thinned out; the giants went.
The horse-sized ant, then, would suffocate as well as collapse. It is a coin toss which one gets it first.
If the rule is true, big animals should have to spend a bigger share of themselves on skeleton than small ones do — because their bones have to be disproportionately thick just to stand still.
That is a testable prediction about real bodies, and it has been measured across a great many birds and mammals. Here is what it looks like.
The share of body weight made of bone, for animals from a mouse to an elephant. The rule predicts this line should climb. Read it and see.
| Animal, from lightest to heaviest | Skeleton share |
|---|---|
| Mouse (20 g) | 4.3 |
| Rat (300 g) | 5.5 |
| Human (70 kg) | 8.9 |
| Horse (500 kg) | 10.7 |
| Elephant (4 t) | 12.9 |
A blue whale is far heavier than any land animal has ever been, and its skeleton is not built like a scaled-up elephant's — its bones are relatively lighter and spongier. Why can it get away with that?
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Back to all episodesThe square-cube law was set out by Galileo Galilei in 'Discourses and Mathematical Demonstrations Relating to Two New Sciences' (1638), including his illustration of a bone scaled up and correspondingly thickened. Scaling every length by a factor k multiplies areas by k squared and volumes by k cubed; weight follows volume while the load-bearing strength of a bone or muscle follows cross-sectional area, so stress on the material rises in proportion to k. Skeletal scaling: Prange, Anderson & Rahn, 'Scaling of skeletal mass to body mass in birds and mammals', The American Naturalist 113:103-122 (1979). Mammalian skeletal mass scales with body mass raised to approximately 1.09, an exponent greater than 1, so the skeleton's proportion of body mass rises with size. NOTE ON THE CHART: the five plotted values are computed from that published equation at the stated body masses rather than being five independent measurements, and this is declared in the chart's own source line as well as here. Postural change with size — larger mammals stand with straighter, more columnar limbs, bringing bone into line with the load rather than relying on muscle to hold a bent joint — is a well-established finding in locomotor biomechanics, described in the scaling work of Biewener and others. Surface-area-to-volume and heat: heat loss scales with surface area while heat production scales with body mass, so small endotherms lose heat disproportionately fast and must eat correspondingly more for their size. Shrews are the standard example and must feed at intervals of a few hours or die; figures of roughly their own body weight to twice it in food per day are commonly reported for the smallest species. APPROXIMATE FIGURE, DECLARED: the exact multiple varies by species, ambient temperature and diet, and the episode says 'something close to its own body weight' rather than quoting a precise number. Elephant ears function in thermoregulation: they are thin, richly vascularised and large in surface area, and blood flow through them is regulated to shed heat. Well established in elephant physiology. Insect respiration is by a tracheal system of air-filled tubes opening at spiracles, with gas exchange substantially by diffusion. This imposes a size constraint, and the constraint is sensitive to atmospheric oxygen concentration. Carboniferous giant insects: Meganeura and related griffinflies had wingspans of roughly 70 centimetres. Geochemical models of Palaeozoic atmospheres (Berner and colleagues) estimate atmospheric oxygen in the late Carboniferous at around 30 to 35 per cent, against 21 per cent today, and the association between higher oxygen and larger insects is the leading explanation. DECLARED AS AN INTERPRETATION: oxygen is the leading account of the size ceiling shifting, not a proven sole cause — the absence of flying vertebrate predators at the time is also proposed as a contributor. Buoyancy and whale size: water supports a swimming animal's weight, removing the skeletal load that limits terrestrial body size, which is why the largest animals known are aquatic. Cetacean bone is comparatively light and cancellous rather than being scaled-up terrestrial limb bone. A stranded large whale is crushed by its own unsupported weight, which is why beachings are rapidly fatal. SIMPLIFICATION DECLARED: the doubling table holds shape and material constant and treats bone strength as proportional to cross-sectional area. Real bone failure involves bending and buckling as well as compression, and real animals change proportions as they scale, which is exactly the adaptation the episode goes on to describe. The direction and rough size of the effect are right; the table is an argument about proportions, not a stress calculation.