The Same Law, Written Twice

Somewhere around 1513, in a small notebook, Leonardo writes down a rule about trees. All the branches of a tree, at any height, added together, are equal in thickness to the trunk below them. Cut the tree across at any level, sum the cross-sections of everything you have severed, and you get the same number every time.

Then, immediately afterwards, in the same passage, he writes it again about water. All the branches of a watercourse, at every stage of its course, are equal to the body of the main stream.

He is not noticing two things. He is stating one law and then pointing at a second place it holds. That habit—and it is a habit, it recurs everywhere in the notebooks—is the thing worth understanding about him, and it is more interesting than the usual story about a man who was good at a lot of subjects.

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One precision, because the popular version overreaches. Leonardo’s rule conserves cross-sectional area—the square of the radii. The law that governs blood vessels and the branching airways of the lung, Murray’s law, conserves the cube. They are relatives, not the same law, and merging them is a common error. But you do not need to merge them, because Leonardo’s own pairing is better: he wrote it for trees and for rivers, in adjacent sentences, and that pairing is his.

The Hypothesis Underneath

Why would anyone expect a tree and a river to obey the same rule? Because he thought they were the same kind of object, and he said so.

In a passage opening a treatise on water, he sets out what he actually believed about how the world is organised. The ancients, he says, called man a world in miniature, and the name is well given, because man’s body resembles the body of the earth. Rock does the work of bone. Soil does the work of flesh. The water in springs and rivers does the work of blood. And the six-hourly rise and fall of the tide does the work of breathing.

“as man has in him bones the supports and framework of his flesh”Leonardo da Vinci, Notebooks (Richter, §929)

It is tempting to read that as decorative Renaissance analogy, and it is not. It is a working hypothesis, and it does work. If the same mechanics run at every scale and in every material, then dissecting a heart and diverting a river are the same investigation conducted on different apparatus, and an observation made in one place can be tested in the other. That is exactly how he proceeded, and it is why the notebooks look chaotic and are not.

It is also, incidentally, wrong—there is no general principle guaranteeing that geological and biological structures share laws. But it is wrong in the productive way that good hypotheses are wrong. It told him where to look, and often enough there was something there.

The Winter He Spent Inside People

In the winter of 1510 and 1511 he produced what is now called Anatomical Manuscript A: eighteen sheets, over 240 individual drawings, and more than 13,000 words of notes in his mirror-writing, all working from human material. The Royal Collection’s estimate is that he may have dissected as many as twenty bodies in that period. He was working in association with Marcantonio della Torre, a professor of anatomy at Pavia—though it is worth saying that this collaboration rests largely on a passage added to the second edition of Vasari’s Lives in 1568, and some scholars are sceptical.

What he got first, and got right: the double-S curvature of the spine, which nobody had drawn correctly. The sacrum as five fused vertebrae. The first drawings of the human appendix. The first scientific depiction of a fetus in the womb. And the maxillary sinus, a hundred and fifty years before the anatomist whose name it now carries. A structure in the heart is still called Leonardo’s band.

There is a smaller detail I find more revealing than any of these. His skull studies of 1489 are the first anatomical drawings to use architectural section views—the convention a builder uses to show a wall cut through. He did not invent a way of drawing a skull. He took an instrument from another discipline and applied it to the head, because the problem was the same problem: how do you show the inside of a solid thing on a flat sheet.

And he got things wrong, in ways that are equally instructive. The famous fetus drawing gives the womb a cotyledonary placenta—the lobed, button-covered kind found in cattle. He had dissected a cow and, apparently having access to only one human pregnancy, filled the gap with the animal. He also believed the umbilical cord carried the fetus’s urine out. When his observation ran out, he substituted analogy, which is exactly the failure mode the macrocosm hypothesis sets you up for.

The Glass Heart

The strongest case that he was doing science rather than very good draughtsmanship is the aortic valve, and it deserves telling properly.

The valve at the exit of the heart has three leaflets, and behind them the vessel wall bulges out into three pouches. The question that occupied him around 1512 was how the leaflets close, cleanly, seventy times a minute, for a lifetime, without anything pulling them shut.

So he built a model. He took a mould from the aortic root of a bull’s heart, cast it in glass, and pumped water through it with grass seeds suspended in the flow so that he could see where the water went. It is a flow-visualisation experiment, done in about 1512, with the correct instinct that you cannot understand a fluid by looking at the vessel.

What he saw, and drew, were vortices forming in the pouches behind the leaflets. And he concluded that these eddies are what push the leaflets closed—that the valve shuts itself using the fluid passing through it.

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The vindication came in stages. In 1968 a laboratory model with pulsatile flow and dye reproduced the trapped-vortex mechanism. In 2014, four-dimensional flow MRI imaged it in living human beings, and the resulting paper’s title says plainly that it confirms Leonardo’s prediction. Its authors note that his depiction was strikingly precise in the size and position of the vortices relative to the aortic root.

And—this matters—he also got a piece of it wrong. He proposed further layered vortices continuing up the aorta beyond the root. Those do not occur; flow there stays laminar in a healthy vessel. He was right about the mechanism and wrong about its extent, which is what a real result looks like.

Water, and the Word He Needed

He spent more of his life on moving water than on painting, and he is probably the first person to use the word turbolenza in something close to its modern technical sense. Not turbulence as disorder—turbulence as a *regime*, a describable way for a fluid to behave.

He watched large eddies give rise to smaller ones, and he distinguished between the motion of the water and the motion of the form the water makes—which is the distinction on which the whole subject depends. In 2021 a team ran a modern particle simulation of the flow in one of his drawings of water falling into a pool and found it reproduced what he had drawn.

There is an overclaim to head off. You will read that he anticipated Kolmogorov’s theory of the turbulent cascade. He did not, and no peer-reviewed work establishes it. He saw eddies spawning eddies and drew them beautifully. He had no scaling law, and inserting one into his notebooks retrospectively does him no favours—what he actually did is impressive enough.

It culminates in the Deluge drawings—around eleven sheets at Windsor, made in his last years in France, showing cities engulfed at the centre of enormous vortices, rocks flung by wind, water rebounding off water. They are usually filed as apocalyptic fantasies and I think that is a misreading. They are the point at which his fluid vocabulary becomes the drawing language itself. He is not illustrating a flood. He is using a notation he developed over thirty years to render a physical system at a scale nobody could observe—which is what a simulation is for.

Two Things He Reasoned Out Correctly From Almost Nothing

The Codex Leicester—eighteen double sheets, mostly about water, bought by Bill Gates in 1994 for a little over thirty million dollars—contains two arguments worth setting out, because in both cases his *reasoning* is better than his conclusion.

The first is fossils. Marine shells were being found high in the mountains, and the standard explanation was the biblical Flood. Leonardo argued against it, and his decisive evidence is the good bit. The shells carry trace fossils—borings, and tracks left by animals moving—which a rock spontaneously generating a shell-shape could not produce; these were living creatures. And worm traces appear between successive layers of rock, which requires successive episodes of deposition separated by time in which something was alive, not one catastrophic year.

That is stratigraphic reasoning, arrived at around 1508, and a 2019 paper in a stratigraphy journal argues that essentially all the core concepts of the discipline are present in his writings. Nicolas Steno gets the credit, a hundred and fifty years later. Steno published.

The second is the moon. When the moon is a thin crescent you can often see the whole disc faintly, the dark part glowing ashen. Leonardo proposed that this is sunlight reflected from the Earth onto the lunar surface—the first known attempt at an explanation, roughly a century before Kepler. He thought the reflecting surface was the oceans. It is mainly the clouds. He got the mechanism right and the mirror wrong, which given that he was reasoning about planetary albedo in about 1508 seems a reasonable score.

The Optics That Became the Painting

Now the part where the science and the art stop being adjacent and become the same thing.

Leonardo left the oldest known clear description of the camera obscura, around 1502, and he explicitly compared its working to the eye. He studied shadow obsessively, classifying its kinds. He understood that vision away from the centre of the visual field does not resolve edges—that peripheral vision is soft.

Sfumato is what happens when you take that seriously in paint. Not blurriness, and not a mood: a systematic refusal to draw an edge where the eye would not perceive one. And in 2010 a team using X-ray fluorescence measured how he actually did it—working directly on the paintings, in situ, with no sampling at all, across seven Louvre works and nine faces spanning about forty years of his career.

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The numbers are extraordinary and they are not the numbers usually quoted. Individual glaze films of one to two micrometres. A total stack of no more than thirty to forty micrometres—thinner than a human hair, for the entire transition from lit flesh to shadow. The often-repeated “up to thirty layers” is not in the paper. It is what you get by dividing the total by the film thickness, and it has hardened into a fact through repetition. Give the micrometres; they are more impressive anyway.

The chemistry is where you see the method. In the Mona Lisa’s shadows he used a manganese-bearing glaze over lead white. Manganese is unusual in oil painting precisely because it dries aggressively—which is a problem, unless what you want is to lay one gossamer coat over another without disturbing what is underneath, in which case it is exactly right. Later analysis found he used two distinct grades of lead white, chosen for how they scatter light, and that the Mona Lisa’s ground contains a lead compound implying he deliberately treated the pigment—a treatment absent from the Last Supper. His technique was not a style he had. It was a set of decisions he made per painting.

One correction while we are here. Aerial perspective was not his invention. The effect appears in Roman wall painting from the first century BC, and Alberti described it before him. What Leonardo did was name it, systematise it, and turn it into a repeatable procedure other painters could execute—which is a different and arguably more useful kind of contribution.

The Reason None of It Mattered

Here is the part of the story that ought to be taught alongside the achievements, because it is the most useful lesson in it.

He published nothing. Not one page, in his lifetime. The papers passed to his pupil Francesco Melzi, then to Melzi’s son, who did not understand what he had. In the 1590s Pompeo Leoni acquired over two and a half thousand sheets and dismembered the original notebooks to rearrange them by subject, which is why the Codex Atlanticus exists and why reconstructing his actual sequence of thought is now a scholarly speciality.

The anatomical drawings reached the English Royal Collection in the seventeenth century and sat there. The first facsimiles appeared in 1898 and 1901, covering sixty-one sheets. The remaining seven hundred or so drawings were published between 1911 and 1916, in an edition of two hundred and fifty copies. Four hundred years.

Meanwhile, in 1543, Andreas Vesalius published De humani corporis fabrica and became the foundation of modern anatomy. In several respects Leonardo’s drawings are better. It made no difference whatever, because anatomy is a discipline and a discipline is a conversation, and he was not in it. The treatise on painting that Melzi assembled after his death was compiled from eighteen of his books; two-thirds of those books are now lost entirely.

Vesalius published in 1543 and changed medicine. Leonardo, in several respects better, changed nothing—because a discipline is a conversation, and he was not in it.

The same pattern runs through the paintings. The Adoration of the Magi was abandoned in 1482 at the underdrawing stage; a technical study concluded that essentially none of the visible paint is his. The Battle of Anghiari was abandoned in 1505 when his experimental medium ran down the wall and heating it with braziers failed to fix it. The Sforza horse existed as a full-size clay model, exhibited in 1493 to general astonishment; the bronze was requisitioned for cannon, and in 1499 French soldiers used the clay for archery practice until the weather finished it.

Fifteen to twenty paintings, depending which attributions you accept. Several thousand surviving manuscript pages. The ratio is the man.

Four Things He Did Not Do

Since this territory is thick with invention, four brisk corrections.

He did not compose with the golden ratio. He drew sixty polyhedra for Luca Pacioli’s book about divine proportion, which is where the association comes from. There is no scholarly consensus that he used it compositionally in any painting, and the drawing most often cited as proof—the Vitruvian Man—has been given perfectly good constructions from an octagon or a vesica instead.

He did not discover the circulation of the blood. That is Harvey, in 1628. Leonardo understood the valves brilliantly and did not understand the circuit.

He did not predict plate tectonics. He argued that former seabeds had been raised. That is uplift, which is a real and correct inference, and it is not the same claim.

And the claim that he had a squint which helped him render depth—proposed in 2018 from measurements of eyes in six works—drew two published rebuttals in the same journal, including the observation that printed images from etchings are horizontally flipped, which reverses the apparent deviation and undoes the measurement. It is a nice story. It is not established.

The Drawing That Explains the Rest

Which brings me back to the Vitruvian Man, and to what it is actually for, because almost nobody knows and the real answer is better than the mystical one.

Vitruvius, writing about a thousand five hundred years earlier, had said that the human body fits both a circle and a square, and had been read as placing both figures on the navel. Illustrators before Leonardo followed him literally, and their drawings do not work—the proportions simply refuse.

Leonardo’s solution was to notice that the received text was wrong and to fix it. He used two different centres: the navel for the circle, the groin for the square. And he changed the pose between them—arms and legs spread for the circle, arms horizontal and feet together for the square—so that a single body could satisfy both constructions. That is why the figure has four arms and four legs. It is not a diagram of cosmic harmony. It is a correction, published as a picture.

And that, in the end, is the through-line. What separates him from the many other capable people of his century is not the range of his interests. Plenty of Renaissance men were curious about several things at once. It is that he treated drawing as an instrument rather than a record—something you do in order to find out, not something you do once you know. The glass aorta with seeds in the water, the skull opened like a building, the water drawn until the eddies made a vocabulary: these are all the same act.

The hand that worked out how a valve closes itself is the hand that had to decide what colour to make a shadow. He did not experience those as different problems, and the notebooks make clear he could not have explained why anyone would. Four hundred years later we built two separate cultures out of the distinction he never made, and we have been trying to talk across it ever since.