Size
Why a Thing Cannot Simply Be Made Bigger
Twice as Big
Make a thing twice as big in every direction, and you have not made one quantity bigger. You have changed every quantity, each by a different amount. Every length doubles. Every area, from a footprint to a skin, grows four times, because an area is a length times a length. Every volume grows eight times, because it is a length times a length times a length.
Weight goes with volume, and almost everything that serves the weight goes with an area or a length. A bone or a cord holds by its cross-section, an area. A body gives off heat through its skin, an area, and takes in food and air through the linings of its gut and its lungs, more area. The grip of water’s surface acts along a line. So when a thing grows, its weight runs ahead of everything that has to hold it up, feed it and cool it. When it shrinks, its weight falls behind.
You have met this already, in a fall. The air pushes on a falling body’s face, an area, while its weight goes with its volume, so a small body falls slowly for its weight. By this arithmetic a mouse dropped down a deep shaft would land and walk away, and a horse would not. The rest of this page follows the same arithmetic down to the smallest warm animal and up to the largest, to the sizes where it decides what can exist.
Weight Against Strength
Shipbuilders knew it from practice long ago. A great galley needed heavier props and scaffolding for its launch than a small boat, or it could break its back under its own weight, while a small boat ran no such risk. The reason is the arithmetic of size. A beam, a rope or a bone holds by its cross-section, and the load it carries is a weight.
Make the same ball and cord three times as big, and the ball weighs 27 times as much, while the cord, three times as thick, is only 9 times as strong. A cord strong enough for more than twice its ball’s weight now breaks under it. Nothing about the brass or the cotton has changed. The size alone has made the thing a third as safe.
Turned around, the same rule makes small things strong for their size. A leafcutter ant carries pieces of leaf a few times its own weight back to its nest, and up to about nine times. It is not built of better muscle. Its muscles, like ours, pull in proportion to their cross-section, and at an ant’s size there is a great deal of cross-section for very little weight. Made the size of a person, the same ant would be some three hundred times as large and three hundred times weaker for its weight, and it could not hold its own body up.
Jumping is different, and the old boast that a flea as big as a person could leap over a tall building is wrong. A flea leaps about 20 centimeters, close to a hundred times its own length, while a person lifts their body about half a meter. But the energy a body can put into a jump grows with its muscle, and so with its bulk, and so does the weight it has to throw. Animals of one build should therefore jump to about the same height, whatever their size, and across a huge range they nearly do. Scaled up to your size, a flea would still jump about 20 centimeters.
Real animals do not simply scale up. By the rule alone, a giant would need monstrously thick bones. Measured leg bones say otherwise: from a mouse to a horse, they keep almost the same proportions. A large animal stands up straighter instead. A mouse runs crouched, its knees and ankles bent, while a horse stands on legs like columns, its weight carried down its bones, so its muscles need to pull far less for its weight. The stress in a galloping horse’s bones comes out about the same as in a ground squirrel’s, a quarter to a half of what would break them.
Above a few hundred kilograms there is no straighter to stand, and the largest land animals pay in other ways. An elephant’s bones are stouter, and it moves with care. Even at its top speed, about 25 kilometers an hour, one of its feet is always on the ground. That is not because it cannot run. At speed its body bounces the way a runner’s does, its hind legs working like springs. It never leaves the ground, though, and it cannot jump.
Held Up by the Water’s Skin
The surface of water pulls along every line that touches it, like the edge of a stretched sheet. That pull grows with the length of the line, and weight grows with the volume. Set the two against each other for water itself and they balance at about 2.7 millimeters, whatever the shape. Above that size weight wins, and a puddle lies flat. Below it the surface wins, and a small drop stays round.
A water strider lives on the small side of that line. It weighs about 10 milligrams and stands on six legs, the four long ones lying along the water at their ends. Each foot dents the surface, and the surface, pulling up along the edges of each dent, holds the insect up with room to spare: its weight is about a thirtieth of what the water could hold along its feet. The feet have to stay dry for this to work. They carry a pile of fine waxy hairs on which water beads up, and a leg that got wet would be pulled into the water instead of held up by it.
Each dent bends what the water mirrors, and it bends the sunlight passing through as well, away from its middle, so on the bed of a sunlit stream every foot casts a dark patch with a bright rim. Make the same insect five times as long and it weighs 125 times as much, while the line along its feet is only five times as long. The dents deepen, and at about that size the water lets go. To stand on water the same way, a person would need feet with a waterline about ten kilometers long.
The same rule makes water dangerous to the small. A film of water half a millimeter thick, about what clings to skin coming out of a bath, weighs about a kilogram on a person, a seventieth of their weight. On a mouse it would weigh a fifth of the mouse, and on a fly twice the fly. For an insect that falls in, a drop is a trap. An insect going for a drink, as one biologist put it, is in as much danger as a man leaning over a cliff for food.
Heat Leaks Through the Skin
A warm body keeps its warmth in its bulk and loses it through its skin. Three steel balls taken red-hot from a furnace show the rule plainly. Each holds warmth in proportion to its volume and loses heat in proportion to its surface. A ball twice as wide holds eight times the warmth and loses it through four times the skin, so it stays hot about twice as long.
A warm-blooded animal is one of those balls with a fire inside. To stay at its own temperature it has to make warmth as fast as heat leaks out through its skin, and a small animal has far more skin for each gram it keeps warm. Each kilogram of a 2-gram shrew comes with nearly a square meter of skin. Each kilogram of you comes with about a thirtieth of that, and a kilogram of blue whale with some four hundred times less than the shrew’s.
So the smallest mammals live at a furious pace. The Etruscan shrew, at about two grams the lightest mammal there is, has a heart that has been measured at 1,511 beats a minute, and it breathes up to nine hundred times a minute. It eats more than its own weight in insects every day. When food runs short it does what a steel ball cannot: it lets its fire die down, dropping into a torpor at about 12 degrees Celsius, and warms up again when it can eat. Without food it starves in about two days. Many hummingbirds do the same on cold nights, and one in the Andes was found at 3.3 degrees.
A small body’s whole life runs fast. The bigger a mammal, the more slowly its heart beats and the longer it lives, so that most mammals get somewhere near a billion heartbeats, give or take a few times, whatever their size. People, at about three billion, are an exception, and so are bats. A bat of about seven grams, ringed in a Siberian cave, was caught again 41 years later.
The Smallest Warm Body
Push the shrew’s arithmetic further down and it stops working. Three separate lines of warm-blooded animals stop at about the same size. The Etruscan shrew is the lightest mammal, at about two grams. The bumblebee bat of Thailand and Myanmar is the shortest, three centimeters long and about as heavy. The bee hummingbird of Cuba, the smallest bird, weighs about the same. Nothing warm-blooded is known to be smaller.
What sets the floor is a race between losing heat and refueling. Smaller still, a body would lose heat faster than any heart, lungs and gut could keep it supplied, and the shrew’s heart and lungs already move oxygen close to the most a body can. Exactly where the line falls is still argued. That it is set by heat is supported by water, which draws heat out of a body far faster than air: in water the floor is far higher, and the smallest seals and porpoises weigh tens of kilograms.
Insects go far smaller because nearly all of them stay at the temperature of the air around them, and pay nothing to keep warm. Some large ones warm up to fly. A bumblebee shivers its flight muscles to above 30 degrees before it takes off and keeps them there in the air, inside a coat of fur. That works only for a large insect, and it burns sugar fast. A tiny one could not stay warm for a moment.
Two Thirds or Three Quarters
If its skin alone set how fast an animal burns, the heat a resting animal gives off would grow as its skin does. Twice the body would give off about 1.6 times the heat: the two-thirds power of the weight. Put side by side, thirteen sets of careful measurements, from ring doves to steers, showed the heat growing a little faster than that, closer to a three-quarters power. That rule, Kleiber’s law, has been taught ever since, and the gap between the two numbers has kept biologists arguing for ninety years.
The measurements now cover some six hundred kinds of mammal, and the answer depends on which ones are counted. Fitted with one straight line, they give a power between two thirds and three quarters. Corrected for differences in body temperature, they come out close to two thirds. And the line is not straight: its slope rises from well under two thirds among the smallest mammals to well over three quarters among the largest. Across all of life there is no single line at all, and bacteria and single cells follow rules of their own.
The explanation is not settled. The best known derives three quarters from the branching of the blood vessels and airways that fill a body. It is elegant, its assumptions have been challenged one by one, and it predicts a curve bent the wrong way. What would settle the question is a theory that predicts the bend itself, and measurements at the top: no great whale has ever had its resting rate measured, and the largest animal whose resting rate has been measured directly is an orca of five tonnes. The direction is not in doubt: gram for gram, a small body burns far faster than a large one. What is open is the exact number, and how much of it the skin explains.
The Wait for Oxygen
Every cell needs oxygen, and oxygen gets into a body the way the bath gets into the jelly below: by diffusion, molecules wandering at random from where there are many to where there are few. Diffusion has an awkward rule. The time it takes grows with the square of the distance, so twice as far takes four times as long.
For oxygen in living tissue, crossing a bacterium takes a quarter of a thousandth of a second. A tenth of a millimeter takes about two and a half seconds, a millimeter about four minutes, and a centimeter about seven hours. A working cell cannot wait minutes. Worked out for a ball of living tissue, diffusion alone keeps it supplied only to a depth of somewhere between a third of a millimeter and two millimeters, depending on how hard the tissue works.
Every larger animal gets round the wait. A flatworm stays flat, so no cell is far from the outside, and a land flatworm can grow a meter long that way. A jellyfish is mostly jelly that needs little oxygen. Most animals pump: a heart drives blood to within a hair’s breadth of every cell, and lungs or gills fold a vast lining for air or water into a small space. Some salamanders a foot long have no lungs at all and breathe through their skin, but they have hearts. Insects carry air itself to every part of the body in a branching system of fine tubes, and oxygen moves through air about ten thousand times faster than through water.
Those tubes may set a ceiling of their own. About 300 million years ago, when most estimates put the oxygen in the air well above today’s 21 percent, perhaps near 30, giant relatives of dragonflies had wings 70 centimeters across. For 150 million years after, the largest insects grew and shrank as the oxygen is thought to have risen and fallen. In beetles alive today, the larger the beetle, the larger the share of its body its tubes take up. But a recent study found that the finest tubes, the ones that reach into each flight muscle, take up about one percent of it or less at every size measured, a share that would still be small at the size of the fossil giants. Whether oxygen, birds or the strength of a shell set the limit is still open.
The Largest
At the other end of the scale the problems turn over. Under the African sun an elephant’s bulk makes warmth faster than its skin can easily shed it, and it has no sweat glands in the skin of its body. It keeps cool in other ways. Water seeps out through its skin twenty to seventy times faster than through a person’s dry skin, and evaporates. The fine cracks in its hide hold five to ten times more water than smooth skin after a bath or a wallow in mud. And its ears, counting both sides of both, make up about a fifth of its whole skin: thin, full of blood vessels it can open, and fanned through the air.
On land, the largest animals ever were the long-necked dinosaurs, some of 50 to 70 tonnes or more, several times the largest land mammals. Why they could grow so much bigger is still being worked out; legs, heat, food and how the young are born all play a part. In water the weight is carried by the water, and the largest animal known to have lived is alive today. A blue whale can weigh well over a hundred tonnes, and some cut up and weighed piece by piece have come to 160 tonnes and more, before counting the blood they lost. What stops it growing bigger is not its bones but its food. A filter feeder’s size is limited by how much krill the sea can offer.
There is no such operation as making something bigger. There is only building something else, for a size at which other forces are in charge. Even the rule this page has followed assumes the same shape and the same material, and life keeps neither: the large animal stands straighter, the small one goes torpid, the insect breathes through pipes. Where the arithmetic says a design cannot work at a size, life has usually found another one. Where no design works, as for a warm body below about two grams, the arithmetic has drawn a line that life has not crossed.
