The Top 100 Diagrams That Changed the World

Around the year 630, Isidore, bishop of Seville, was finishing the most ambitious reference book of the early Middle Ages. His Etymologiae set out to explain everything, from grammar and medicine to ships and stones, by tracing the origins of the words for them. In the part of the work dealing with the earth, copyists placed a small drawing that would become one of the most widely copied pictures of the world for the next thousand years: a circle containing a T.

The top of the map is east, not north. Paradise was believed to lie in the farthest east, and the rising sun was a symbol of Christ, so the place of honour went to the orient. This is where our verb "to orient" comes from: to orient a map once meant to turn it so that east was uppermost. Later versions put Jerusalem at the exact centre, at the point where the T meets, following the line in Ezekiel that God had set Jerusalem in the midst of the nations. Isidore's own diagram does not insist on this, but the idea fitted the shape so neatly that it became the norm.

That is precisely its importance. The T-O is the purest example of a diagram as argument. It discards distance, coastlines and scale and keeps only relationships: three parts, one ocean, one holy centre, sacred direction at the top. Its message, repeated in hundreds of manuscripts, was that geography and theology were the same subject. When Günther Zainer printed an edition of the Etymologiae at Augsburg in 1472, the little woodcut T-O became, by most accounts, the first map printed in Europe.

Claudius Ptolemy worked in Alexandria in the second century AD, in the city that still housed the greatest library of the ancient world. He is best remembered for the Almagest , his great handbook of astronomy, but around 150 he wrote a second book that may have mattered even more: the Geography . It is less a book of maps than a book of instructions for making them, and its central idea is so familiar that it is hard to see how radical it was. Every place should be fixed by two numbers.

He also explained how to flatten a sphere onto a page without making a mess of it. His first method draws the meridians as straight lines fanning out from a point above the map, and the parallels as arcs of circles around that point, a cone unrolled onto vellum. The parallel through Rhodes, at thirty-six degrees, keeps its true proportions, so the distortion is smallest across the Mediterranean, where most of his data came from. A second, more elaborate method curves the meridians as well, for a still more convincing look of a globe.

Whether Ptolemy drew maps himself is uncertain. The earliest surviving maps that follow his text were made by Byzantine scholars around 1300, after the monk Maximus Planudes went looking for the forgotten book. A Latin translation made in Florence around 1406 set off a sensation in Renaissance Europe, and printed editions with engraved maps followed from 1477. Columbus knew Ptolemy's world, and its shrunken ocean between Spain and Asia made his westward voyage look shorter than it was.

In the cathedral at Hereford, behind glass in a dim room, hangs the largest medieval world map to survive intact. It is drawn on a single calfskin about one and a half metres tall, and the neck of the animal still shapes the pointed top. A note in one corner asks everyone who reads the map to pray for Richard of Haldingham and Lafford, who made it. Scholars date it to around 1300 and argue about whether it was made in Lincoln, where a Richard of that name held office, or in Hereford itself.

Reading it is a pilgrimage in itself. At the top, outside the world altogether, Christ sits in judgement, the saved rising on one side and the damned dragged away on the other. Just inside the circle, at the far east, Paradise is a round, walled island, where Adam and Eve take the fruit. The Red Sea is coloured red. Noah's Ark rests on its mountain, the Tower of Babel rises in Mesopotamia, and Crete is marked by a neat circular labyrinth. Along the southern rim, crowded into Africa, are the monstrous races borrowed from ancient writers: people with one great foot, people with faces in their chests.

It would be easy to call this bad cartography, but that misses what it was for. The Hereford map is an index of everything its makers believed worth knowing, arranged in space so that a viewer could walk through history, geography and salvation at once. Time and place are folded together: Eden, the Exodus and the cities of the thirteenth century share the same sheet. A traveller would have used an itinerary, a written list of stages; this was for contemplation and teaching, perhaps placed beside an altar.

In the small town of Saint-Dié, in the Vosges mountains of Lorraine, a group of scholars under the patronage of Duke René II set out to update the world. Their leading mapmaker was Martin Waldseemüller; his collaborator was the poet and teacher Matthias Ringmann. In 1507 they published two things together: a short Latin book, the Cosmographiae Introductio , and a gigantic world map printed from twelve woodblocks. Pasted together, the sheets made a wall map well over two metres wide.

Across the southern part of that land, roughly where Brazil is today, the map prints a single word: America. The accompanying book explains the choice. Europe and Asia, it notes, were named after women, so there was no reason not to name this fourth part of the world after Amerigo Vespucci, the Florentine navigator who, the authors believed, had discovered it. They had been reading printed accounts of Vespucci's voyages, which argued that the lands across the Atlantic were not part of Asia at all but a new world. Some historians doubt that Vespucci wrote every letter printed under his name, or made every voyage claimed for him, but the printers in Saint-Dié took the accounts at their word. Columbus, who never accepted that he had reached anything but Asia, does not get the honour. At the top of the map the two authorities are given portraits: Ptolemy beside the Old World, Vespucci…

Waldseemüller himself seems to have had second thoughts. His later maps dropped the name. But a woodcut is a publishing machine, and the cosmographers were reportedly proud of having printed around a thousand copies. The word spread through other mapmakers' work, and in 1538 Gerardus Mercator applied it to both the northern and southern continents. After that it was settled.

Gerardus Mercator was a Flemish engraver, globe-maker and survivor. Born in 1512, he was arrested in 1544 on suspicion of Lutheran heresy and spent months in prison before being released, and in 1552 he moved to the quieter, more tolerant town of Duisburg on the Rhine. There, in 1569, he published a world map on eighteen sheets with a long Latin title that announced exactly what it was for: a new and more complete description of the earth, properly adapted for the use of navigators.

His solution was to make the meridians straight, parallel and evenly spaced, as if the globe had been unrolled from a cylinder. That stretches everything east and west, more and more towards the poles, since on the real earth the meridians converge. To keep angles true, Mercator stretched the map north and south by the same amount, spacing the parallels further and further apart as they climb. At sixty degrees north the stretching is twofold; near the poles it runs towards infinity, which is why the poles never appear at all.

The price of straight courses is the honesty of sizes. Because the scale grows with latitude, high northern lands swell. Greenland looks about as large as Africa, although Africa is roughly fourteen times bigger. Scandinavia looks larger than India. Alaska looks nearly as big as Brazil, which is about five times larger. For navigation none of this mattered; Mercator never designed his map for comparing countries. But wall maps in schools and offices adopted it anyway, and generations grew up with a picture of the world in which Europe and North America loom over Africa and South America.

In 1669 the astronomer Giovanni Domenico Cassini arrived in Paris from Bologna to run Louis XIV's new Observatory. He came for the stars and stayed for the ground. He, his son, his grandson and his great-grandson would spend more than a century turning France into the first country mapped by systematic measurement, and their name became the name of the map.

The early results were humbling. When Picard and Philippe de La Hire used astronomical positions to correct the outline of France in the early 1680s, the Atlantic coast moved east and the kingdom shrank. Louis XIV is said to have complained that his astronomers had cost him more territory than his enemies ever had. The remark is probably apocryphal, but the shrinkage was real.

That skeleton made the great map possible. Louis XV, reportedly impressed by Cassini's maps of the Flanders campaigns, ordered a survey of the whole kingdom at a scale of one line to a hundred toises, or 1 to 86,400. When the royal money dried up in 1756, Cassini formed a company of private subscribers to keep going. Surveyors filled in every village, road, mill and gallows. After Cassini de Thury died in 1784 his son Jean-Dominique carried on, until the Revolution took the map for the state. The roughly 180 sheets were finally completed in the early nineteenth century.

William Smith was born in 1769 in the Oxfordshire village of Churchill, the son of a village blacksmith who died when the boy was still small. He had little schooling, taught himself surveying, and in the 1790s found work on the Somerset Coal Canal near Bath. Digging a canal is a good way to look inside a hillside, and Smith looked very hard. In the cuttings and in the coal mines below, the rocks came in layers, and the layers always came in the same order.

In 1815, after some fifteen years of work, he published the result: A Delineation of the Strata of England and Wales, with part of Scotland . It was engraved by the London mapmaker John Cary on fifteen sheets at five miles to the inch, and assembled it stood well over two metres tall. Every copy was coloured by hand. Bands of colour sweep across the country from south-west to north-east: old slates in Wales and the north, coal measures, the red marls, the blue lias, the golden oolite of the Cotswolds and Bath, the chalk downs, and the clays of the London basin.

Recognition came slowly. The gentlemen of the Geological Society of London were wary of a self-taught surveyor, and a rival map produced under the Society's auspices borrowed heavily from his work. Smith's finances collapsed, and in 1819 he spent several weeks in the King's Bench debtors' prison. Only in 1831 did the Society award him its first Wollaston Medal, and Adam Sedgwick called him the father of English geology. Simon Winchester's book about him took its title from a fair description: The Map That Changed the World .

By the early 1930s the London Underground had a mapping problem. Its lines had been built by rival private companies, and the official maps showed them as they really ran: tangled together in the centre, then straggling out across the suburbs. Drawn to scale, the central stations were crammed into a smudge while the outer ends of the lines needed huge empty sheets. Previous designers had tried tricks, such as dropping the background streets, but the maps were still geographical at heart.

The only real-world feature he kept was the Thames, itself straightened into the same angles, because Londoners think of their city as north or south of the river. Stations became neat little ticks on the lines, and interchanges became small open symbols, diamonds in the earliest versions, later circles. It is often said that Beck borrowed the look of an electrical circuit diagram. Whether or not that was the inspiration, the resemblance is real: the map shows connections, not positions.

The idea that a map can be useful because it is inaccurate was the real breakthrough. Beck's diagram is a topological map: it preserves which things connect to which, and in what order, while freely distorting distance and direction. It answers the traveller's actual question, how do I get from here to there, and leaves out the rest. The side effects are famous. Some stations look far apart that are a short walk from each other, and tourists still take the Tube between stations that are quicker on foot.

In the early 1950s the floor of the ocean was the largest unmapped landscape on earth. Sailors had dropped weighted lines for centuries, but each sounding gave a single depth, and the abyss was imagined as a vast, featureless plain of mud. Then echo sounders, which time a ping of sound to the sea bed and back, began recording continuous profiles of depth along a ship's track. At Columbia University's Lamont Geological Observatory, on the Hudson north of New York, the job of turning those wandering lines into a picture fell to Marie Tharp.

To Tharp it looked like a rift, the kind of valley that forms where the earth's crust is being pulled apart. That would support continental drift, the idea that the continents move, which most American geologists then dismissed. Heezen at first dismissed it too; Tharp later recalled that he called it girl talk. What changed his mind was another dataset. Earthquake locations in the Atlantic, plotted at the same scale, fell neatly inside her valley. A line of shaking ground running down the middle of an ocean was hard to explain unless something was happening there.

The maps arrived just as other evidence piled up. Harry Hess proposed sea-floor spreading in the early 1960s, and magnetic stripes on the ocean floor soon showed new crust forming at the ridges and moving outward. By the end of the decade plate tectonics had become the foundation of the earth sciences. Tharp's rift was the place where the plates part.

In 1973 Arno Peters, a German historian with no background in cartography, held a press conference in Bonn to unveil what he presented as a new map of the world. It looked startling. Africa and South America hung down the page like long drips of paint, twice as tall as they were wide. Europe was squat and small, tucked up near the top. The familiar Mercator world, with its looming northern continents, was turned upside down in spirit if not in orientation.

The geometry was not new. Peters had produced a cylindrical equal-area projection, a family first described by Johann Heinrich Lambert in 1772. The particular version he used, with true shapes only along the parallels of forty-five degrees north and south, had been presented by a Scottish clergyman, James Gall, in 1855. The projection is now usually called Gall-Peters to give Gall his due. Everywhere except those two lines, shapes are distorted. Near the equator countries are stretched vertically, which is why Africa looks so long; near the poles they are squashed flat.

None of this slowed the Peters map in public. Churches, charities, aid agencies and schools adopted it; The Peters Atlas of the World followed in 1989, and the map became a fixture in classrooms and development offices. It even appeared in an episode of The West Wing , in which a fictional group of cartographers for social equality turn a White House meeting upside down. In 2017 the public schools of Boston adopted it for their classrooms, a decision reported around the world as a lesson in how maps shape what children think.

Around the middle of the second century, in Roman Alexandria, a scholar named Claudius Ptolemy set out to do something that sounds modest and was not: to predict where every wandering light in the sky would be, on any night, years in advance. The result was a thirteen-book treatise in Greek that Arabic translators later called al-Majisti , "the greatest", and which Europe came to know as the Almagest . Its diagrams are not pretty. They are lines and letters, the working geometry of a man doing sums. But one arrangement inside them, a small circle riding on a large one, held the universe together for well over a thousand years.

Ptolemy's contribution was to make the scheme fit the numbers. He moved Earth slightly away from the centre of the deferent. Then he added his boldest device, the equant: a point on the far side of the centre, at the same distance as Earth, from which the epicycle's centre appears to sweep round at a steady rate. Strictly this broke the rule of uniform circular motion, because the epicycle now speeds up and slows down along its track. Ptolemy accepted the cheat because it worked. Later astronomers, Copernicus among them, regarded the equant as an offence against good taste.

The popular story says the system finally collapsed under ever more epicycles piled on epicycles. That is largely a myth. Ptolemy's planetary models used one main epicycle each, and medieval astronomers did not keep stacking them. What eventually broke the system was not clutter but a better picture: first Copernicus's, then Kepler's ellipses, which needed no equant because planets really do change speed as they go round. Kepler, in effect, showed that Ptolemy's cheat had been pointing towards the truth all along.

The most consequential diagram of the Renaissance is barely a diagram at all. Early in the first book of De revolutionibus orbium coelestium , printed in Nuremberg in 1543, sits a woodcut of concentric circles with one word in the middle: Sol. Around it run numbered rings with brief Latin labels. Mercury returns in eighty days, Venus in nine months, Earth with its Moon in a year, Mars in two years, Jupiter in twelve, Saturn in thirty, and outermost of all is the motionless sphere of the fixed stars. It looks like a target. In a sense it was one.

What the diagram argued was elegant. With the Sun in the middle, the backward loops of the planets became a trick of perspective: Earth, on a faster inner track, overtakes Mars, and Mars seems to slip back, just as a slower car appears to drift backwards when you pass it on a motorway. The arrangement also fixed the order of the planets by their periods, and, using the Earth to Sun distance as a ruler, it gave their relative distances for the first time. Ptolemy's scheme could not do that. Copernicus valued this tidy harmony between size of orbit and length of year more than anything.

The book did not cause an instant uproar. Astronomers read it closely and filled its margins with notes, as the historian Owen Gingerich showed by tracking down hundreds of surviving copies. Tycho Brahe proposed a halfway house with the planets circling the Sun and the Sun circling Earth. Then Kepler and Galileo took the woodcut literally, and the stakes rose. In 1616 the Catholic Church placed the book on its Index of forbidden works until it was corrected, and in 1633 Galileo was tried for defending it.

Before Isaac Newton, almost everyone agreed that colour was something done to light. White sunlight was pure; a prism, a raindrop or a stained-glass window somehow stained or weakened it. Rainbows from prisms were a fairground novelty. Newton, a young fellow of Trinity College, Cambridge, bought a prism in the mid-1660s, the story goes at a local fair, and began asking a different question. What if the prism was not adding colour at all, but sorting it?

Then came the experiment he called the experimentum crucis , the crucial test, borrowing a phrase from Francis Bacon and Robert Hooke. He placed a board with a small hole just after the first prism, so that only one colour got through, and a second board with a hole further on to fix its direction. Behind that he put a second prism. If prisms created colour, the second one should have changed the colour again. It did not. Red light stayed red, green stayed green, and each was bent once more by exactly its own characteristic amount. Newton's conclusion was blunt: white light is a mixture of rays, each with its own fixed colour and its own degree of "refrangibility".

Not everyone was convinced. Hooke, the Society's formidable curator of experiments, criticised the paper sharply, and continental critics struggled to reproduce the result with poorer prisms. Newton, who hated public dispute, retreated, and did not publish his full account until Opticks in 1704, the year after Hooke's death. In it he named seven colours, deliberately including orange and indigo to match the seven notes of the musical scale. That is why schoolchildren still chant a sequence in which indigo is a colour few of them could pick out.

Some of the most powerful diagrams are thought experiments you can draw. Isaac Newton's cannonball is the best of them. It appears in A Treatise of the System of the World , published in English in 1728, the year after Newton's death, though he had written the text in Latin around 1685 as a more popular version of what became the third book of his Principia . He replaced it with something more forbiddingly mathematical, reportedly to keep out amateurs. The plain version survived, and with it a picture that has taught gravity to every generation since.

That is the whole idea of an orbit, and it was startling. For two thousand years, following Aristotle, the heavens and the Earth had obeyed different rules. Things on Earth fell and stopped; things in the sky circled forever, made of a different, perfect substance. Newton's mountain joins the two worlds with a single smooth sequence. There is no point where earthly falling turns into heavenly circling. The Moon, he argued, is just a very distant cannonball: always falling towards us, always missing.

Newton was careful to say the cannon was imaginary. The speed needed to circle just above the ground is about 7.9 kilometres a second, far beyond any gun of his day, and air resistance would ruin the shot at sea level anyway. Go faster still and the path stretches into an ellipse, and beyond about 11.2 kilometres a second the projectile escapes for good. All of this follows from the same law he used for the drawing.

The drawing endures because it does what equations cannot do alone. It lets a reader feel the answer before calculating it, and it removes a boundary, between the sky and the ground, that had shaped human thought since antiquity.

In the winter of 1869 Dmitri Mendeleev, professor of chemistry at St Petersburg University, had a practical problem. He was writing a textbook, Principles of Chemistry , and needed a sensible order in which to describe sixty-three known elements. Some families were obvious: lithium, sodium and potassium were soft, violent metals; chlorine, bromine and iodine were pungent and reactive. But no one had found a single scheme that held them all. Others had tried. John Newlands in London proposed a "law of octaves" in 1865 and was mocked for it, and Lothar Meyer in Germany was working on a similar table at the same time.

His arrangement did two things at once. Read along a row, the elements climb steadily in atomic weight. Read down a column, they fall into families with similar chemistry: the alkali metals stack up, the halogens stack up, and the valences repeat with a rhythm. The table is a grid in which both directions carry meaning, which is what makes it a diagram rather than a list. In his improved version of 1871 he labelled eight groups across the top, the layout redrawn here.

Nature obliged. In 1875 the French chemist Paul-Émile Lecoq de Boisbaudran found gallium, which matched eka-aluminium. When its measured density came out lower than predicted, Mendeleev wrote to say the sample must be impure; it was, and the corrected figure agreed with his. Scandium followed in 1879, discovered by Lars Fredrik Nilson in Sweden, and germanium in 1886, found by Clemens Winkler in Germany, its properties strikingly close to eka-silicon's. A theory that predicts what no one has yet seen is hard to argue with, and chemists stopped arguing.

Today it hangs in every chemistry classroom in the world, probably the most widely displayed scientific diagram ever made. Its quiet lesson is that an honest diagram should show what you do not know. Mendeleev's gaps were not failures of the table. They were its most powerful feature.

In 1905 Albert Einstein published special relativity as a paper full of trains, clocks and algebra. It showed that measurements of time and length depend on how fast the observer is moving, and that the speed of light is the same for everyone. It was revolutionary and, for many physicists, deeply strange. Three years later his former mathematics teacher turned it into a picture.

The union is spacetime, and the way to see it is to draw it. Put space along the horizontal axis, squeezing three dimensions into one, and time up the vertical. Choose units so that light travels one unit of distance in one unit of time, and every ray of light becomes a line at 45 degrees. Every event, a flash, a handshake, an explosion, is a single point. Every object becomes a line, its world-line, tracing where it is at each moment. A person sitting still is a vertical line; a person running leans over, but never by more than 45 degrees, because nothing outruns light.

This was more than a pretty illustration. The cone is the same for every observer, however fast they move, so it captures what survives when relativity has shuffled everyone's clocks and rulers. Simultaneity becomes relative, but the order of cause and effect does not, as long as nothing crosses the cone's walls. Einstein was reportedly unimpressed at first and regarded the mathematics as superfluous learnedness. Within a few years he changed his mind, and his general theory of relativity of 1915 is built on exactly this four-dimensional geometry, now curved by mass.

His diagram became the standard tool of relativity, taught to every physics student. Physicists tilt light cones near black holes to show why nothing escapes an event horizon, and Roger Penrose later compressed the whole of spacetime into finite diagrams built on the same 45-degree rule. Beyond physics, the light cone has become a quiet philosophical image: a picture of the limits of influence. At every moment there is a region of the universe you can still affect, a region that shaped you, and a vast elsewhere that, for now, is simply out of reach.

At the start of the twentieth century astronomers had a sky full of data and no story to tell with it. At Harvard College Observatory a team of women, led in classification by Annie Jump Cannon, was sorting hundreds of thousands of stellar spectra into classes now remembered as O, B, A, F, G, K and M, running from hot blue stars to cool red ones. Elsewhere, painstaking measurements of parallax were giving the true distances of a few hundred nearby stars, and with distance came true brightness. The obvious next step was to put the two together. It took two people, working independently, to do it.

The second was Henry Norris Russell at Princeton. In 1913 he presented a diagram plotting the spectral class of stars with measured distances against their absolute magnitude, a measure of true brightness in which smaller numbers mean brighter stars. Published in 1914, it showed the same striking pattern. Most stars lie on a diagonal band from bright blue at the top left to faint red at the bottom right. A separate cluster of bright red giants sits at the top right. And in Russell's diagram there was one awkward dot near the bottom left: a faint star, 40 Eridani B, whose spectrum was class A, white and hot. It should have been bright. It was so dim that it had to be tiny. It was the first recognised white dwarf.

The diagram turned from a catalogue into a clock. In a cluster, all the stars formed at about the same time, so the point where the main sequence peels off towards the giants reveals the cluster's age. That technique gave the first reliable ages for the oldest stars in the galaxy. The Sun, a G2 star with absolute magnitude of about +4.8, sits unremarkably in the middle of the band, roughly halfway through its time there.

By 1911 Ernest Rutherford, at Manchester, had shown that an atom is mostly empty space, with a tiny, heavy, positive nucleus and light electrons somewhere around it. It was a triumph with a fatal flaw. According to the physics of the day, an electron circling a nucleus should radiate energy continuously, spiral inward and crash in a fraction of a second. Atoms, on this reckoning, should not exist. Yet here we all are.

The key came from an old puzzle. In 1885 a Swiss schoolteacher, Johann Balmer, had found a simple formula that reproduced the wavelengths of the visible lines in hydrogen's spectrum: a red line, a blue-green one, and two in the violet. Nobody knew why it worked. Bohr later recalled that once he saw Balmer's formula, the whole thing became clear. In his model the energy of the nth orbit is proportional to minus one over n squared, and the radius grows as n squared. Balmer's lines were simply electrons falling into the second orbit from the third, fourth, fifth and sixth. From the charge and mass of the electron and Planck's constant, Bohr calculated the constant in Balmer's formula, and it matched the measured value closely.

The model was also plainly incomplete. It worked beautifully for hydrogen, with its single electron, but failed for helium and anything heavier, and it said nothing about why some lines are brighter than others. In 1925 and 1926 Werner Heisenberg, Erwin Schrödinger and others replaced it with full quantum mechanics, in which electrons do not travel on orbits at all but occupy clouds of probability. Bohr's energy levels survived intact. His neat circular tracks did not.

Bohr's diagram is a lesson in how science often moves forward, by a model that is clearly not the final truth but captures exactly enough of it to point the way. Its rungs became the foundation of the quantum world.

In 1920 astronomers could still argue in good faith about the size of the universe. In a famous debate that April, Harlow Shapley and Heber Curtis disagreed over whether the faint spiral nebulae were clouds inside our own Milky Way or entire galaxies far beyond it. Edwin Hubble settled the question. Working with the 100-inch Hooker telescope on Mount Wilson, then the largest in the world, he found Cepheid variable stars in the Andromeda nebula in 1923, and their regular pulsing gave him a distance. It was far beyond the edge of the Milky Way. The spirals were galaxies, and the universe was suddenly enormous.

In 1929 Hubble put the two measurements on one graph. His short paper in the Proceedings of the National Academy of Sciences plotted the velocities of 24 nebulae, most of them from Slipher's work, against Hubble's own estimates of their distance. The points were scattered but the trend was unmistakable. The farther away a galaxy was, the faster it was receding, in rough proportion. He drew a straight line through the dots, another through averages of grouped nebulae, and estimated a slope of about 500 kilometres a second for every million parsecs. Milton Humason, a former mule driver who had become one of Mount Wilson's most skilled observers, soon pushed the measurements to much greater distances, and the line held.

The credit is more tangled than the textbooks once admitted. In 1927 the Belgian priest and physicist Georges Lemaître had derived the expansion from Einstein's equations and estimated the rate from published data, in a little-read Belgian journal. When the paper was translated into English in 1931, that estimate was left out, by Lemaître himself, it later emerged. In 2018 the International Astronomical Union recommended renaming the relation the Hubble-Lemaître law.

Run Hubble's line backwards and every galaxy converges on a single moment. That is the root of the Big Bang. A scatter of two dozen dots, some of them misplaced, turned the universe from a fixed stage into something with a history.

In the late 1940s the quantum theory of light and electrons, quantum electrodynamics, was in a state of embarrassment. It worked to a first approximation, but every attempt to refine its predictions produced infinite answers. Then, in 1947, Willis Lamb measured a tiny shift in the energy levels of hydrogen that the simple theory missed, and the race was on to tame the infinities and calculate it. Three people succeeded more or less independently: Julian Schwinger at Harvard, Sin-Itiro Tomonaga in Tokyo, and a brash young professor at Cornell called Richard Feynman.

Feynman's own paper, "Space-Time Approach to Quantum Electrodynamics", appeared in the Physical Review in 1949, and its first figure is the diagram redrawn here. Time runs up the page and space runs across. Two electrons, drawn as straight lines with arrows, come in from the bottom. At a point called a vertex, one of them emits a photon, drawn as a wavy line; at another vertex the second electron absorbs it. The electrons recoil and leave at the top, pushed apart. That is electric repulsion, pictured as the exchange of a messenger particle.

The diagrams spread not through textbooks at first but through people. The historian David Kaiser has traced how postdoctoral researchers who learned them from Dyson at Princeton carried them to universities across America, Britain and Japan. Within a decade they were everywhere, used well beyond electrodynamics. In 1965 Feynman, Schwinger and Tomonaga shared the Nobel Prize in Physics. Feynman later had the diagrams painted all over his camper van.

Feynman diagrams are perhaps the purest example of a diagram as an argument. They are not illustrations of the mathematics; they are the mathematics, made visible, so that intuition and calculation finally run along the same lines.

In the summer of 1543 a Basel printer named Johannes Oporinus issued one of the largest and handsomest books of the century. Its author was a 28-year-old professor of surgery at Padua, Andreas Vesalius, and its title, De humani corporis fabrica , means "on the fabric of the human body". That same year Copernicus moved the Earth away from the centre of the universe. Vesalius carried out a quieter demolition: he showed that the most trusted map of the human interior was, in places, a map of a monkey.

Vesalius came down from the chair and held the knife himself, often on the bodies of executed criminals. What he found did not always match the book. The human lower jaw is a single bone, not two. The rete mirabile, a delicate net of blood vessels that Galen placed at the base of the brain, turns up in oxen and sheep but not in people. Each discrepancy was small; together they made a large point. Authority could be checked, and the way to check it was to look.

As a diagram, each plate works on two levels. The pose keeps the figure alive, so muscles read as working machinery rather than meat on a slab. And every structure carries a small letter, keyed to an explanation in the text, so the eye travels from picture to word and back again. Read in order, the series is a dissection unfolded in time: skin first, then the shallow muscles, then the deep ones, until only the skeleton remains, propped thoughtfully on a spade or leaning on a tomb to contemplate a skull.

What survived was the method. Padua became the place where anatomy was seen rather than recited, and its line of professors ran through Fallopius and Fabricius to a young Englishman named William Harvey. The Fabrica plates made a promise that medicine has spent five centuries trying to keep: that the picture in the book should be a picture of the body, and that when they disagree, the book gives way.

For fourteen hundred years, educated Europeans believed blood behaved rather like a sauce. In Galen's scheme the liver cooked it fresh from digested food, the veins carried it outwards to nourish the body, and the body used it up, much as a lamp burns oil. Some blood seeped through invisible pores in the wall of the heart to mix with air from the lungs. Nothing went round. There was no reason to think it should.

He spent years on the question, cutting open living animals, watching slow hearts in eels, snakes and frogs, and he lectured on the circulation to the College of Physicians from around 1616. In 1628 he published the argument in a slim Latin book of 72 pages, Exercitatio anatomica de motu cordis et sanguinis in animalibus , printed in Frankfurt. It contains just one plate, with four small figures of a human forearm. The images were adapted from Fabricius's own work on the valves, which makes them one of history's politer acts of intellectual theft: the teacher's picture used to prove what the teacher had missed.

The picture was half of the case. The other half was arithmetic. Harvey estimated how much blood the heart pushes out with each beat and multiplied by the number of beats; even with deliberately cautious numbers, the heart moved more blood in an hour or so than the whole body contains, and far more than any meal could replace. The blood had to be the same blood, going round. It was one of the first times a physiological argument was settled partly by doing a sum.

Its consequences arrived quickly. Within decades physicians at the Royal Society were experimenting with injecting fluids into veins and transfusing blood between animals. Medicine gained the idea of the body as a pump and pipes, measurable and testable, and the pulse became a reading of a system rather than a mysterious ebb. All of it rests on four small drawings of an arm, a band and a finger.

By the early eighteenth century naturalists were drowning. Explorers, traders and missionaries were shipping home seeds, skins and pickled specimens faster than anyone could describe them, and each new plant arrived wearing a different name in every country. A single species of rose might be known in Latin by a descriptive phrase a dozen words long, and two botanists could spend an afternoon arguing before discovering they meant the same flower.

The sheets did something startlingly simple. They divided everything in nature into three kingdoms, stones, plants and animals, and then split each kingdom into classes, each class into orders, each order into genera, and each genus into species. Every thing had a place and every place had an address. The animal table ran across the page in six columns: Quadrupedia, Aves, Amphibia, Pisces, Insecta, Vermes, the four-footed beasts, birds, amphibians and reptiles, fish, insects and worms. Whales, at this stage, were filed among the fishes.

The two-word name that the system is now remembered for came later. In Species Plantarum in 1753, and for animals in the tenth edition of Systema Naturae in 1758, Linnaeus attached a short "trivial name" to every species, so that a genus plus one word identified it anywhere: Rosa canina , Canis lupus , Homo sapiens . That tenth edition also coined the class Mammalia and moved the whales into it. Botanists and zoologists still treat those two books as the official starting points of their names.

Linnaeus was not modest about his achievement; a motto attributed to him claims that God created and Linnaeus arranged. He sent his students, his "apostles", around the world to collect, and one of them, Daniel Solander, sailed on Cook's Endeavour . The table itself is the lasting argument: that the bewildering variety of life can be filed, and that a good filing system is the beginning of science rather than the end of it.

In the summer of 1837, Charles Darwin was 28, living in rented rooms on Great Marlborough Street in London and drowning pleasantly in specimens. The Beagle had come home the previous October after almost five years at sea, and its cargo of fossils, birds and beetles was being parcelled out to experts. One of them, the ornithologist John Gould, had recently told Darwin that the mockingbirds he had collected on different Galápagos islands were not varieties of one species but distinct species. Darwin opened a new notebook, labelled it B, and began to think the unthinkable about where species come from.

Before Darwin, the dominant image of life was the ladder, the great chain of being, with simple creatures at the bottom, humans near the top and angels above. Species were fixed rungs. Even those who accepted some kind of transformation, like Jean-Baptiste Lamarck, tended to picture separate lineages climbing parallel ladders towards complexity. Darwin's sketch has no top. It has a root and many tips, and none of the tips is higher than any other.

What the sketch does not contain is a mechanism. Darwin did not read Thomas Malthus on population, the spark for natural selection, until the autumn of 1838, more than a year later. The tree came first, and the explanation for why it should grow that way came afterwards. Then he waited. He gathered evidence, raised pigeons, dissected barnacles for eight years, and published only when Alfred Russel Wallace's letter of 1858 forced his hand.

Notebook B had its own adventure. It and a companion notebook went missing from Cambridge University Library around 2000 and were returned anonymously in 2022, in a pink gift bag with a note wishing the librarian a happy Easter. The page is now among the most reproduced drawings in biology, printed on posters, mugs and the covers of textbooks. Its two hesitant words are the best advertisement ever written for thinking on paper.

In 1855 Henry Gray, a 28-year-old lecturer in anatomy at St George's Hospital in London, had an idea for a textbook. Medical students of the day bought expensive atlases or squinted at small, crowded plates in which the structures were marked with tiny letters and explained in a key somewhere else on the page. Gray wanted a book that was affordable, organised for the student preparing for an examination or an operation, and illustrated on almost every page. For the pictures he turned to a former student, Henry Vandyke Carter, four years his junior, a skilled draughtsman and the son of a painter.

Anatomy, Descriptive and Surgical appeared in 1858, published by John W. Parker and Son, with 363 figures. Its decisive innovation is visible at a glance. Instead of littering a drawing with letters and making the reader hunt through a key, Carter wrote the names straight onto the structures themselves, in clear capitals, following the shape of each bone or muscle. On the side view of the skull the word FRONTAL runs across the forehead, PARIETAL across the dome, TEMPORAL above the ear. The sutures where the plates of bone meet are named along their own wandering lines. Even the Victorian names, MALAR for the cheekbone and INFERIOR MAXILLARY for the jaw, sit exactly where a scalpel would find them.

Credit was not shared equally. The title page named "Henry Gray, F.R.S." prominently; Carter's name appeared below in smaller type, as the man who made the drawings. Carter was paid for his work rather than sharing in the book's long success. By the time the first copies were sold he had sailed for India, where he became professor of anatomy and physiology at Grant Medical College in Bombay and did respected research on leprosy and the fungal disease of the foot called mycetoma. Gray had less time still. In 1861, aged 34, he caught smallpox while nursing his nephew and died within days, as he was preparing the second edition.

There is a small irony in the fame. The book carries Gray's name, and even an American television hospital drama borrowed it as a pun. The lasting idea, though, belongs to the illustrator who decided that a label should live on the thing it names.

Ernst Haeckel read On the Origin of Species in 1860 and, by his own account, it changed his life. A young zoologist at the University of Jena, already famous among specialists for exquisite drawings of microscopic sea creatures called radiolarians, he became Darwin's loudest champion in Germany. Darwin's book contained a single abstract diagram of branching lines with letters for names. Haeckel set out to draw the real thing: the actual family tree of every living organism, with names on the branches.

That middle limb was a real advance. Microscopists had long struggled to decide whether a creature that swims like an animal and photosynthesises like a plant was one or the other. Haeckel's answer was that the question was wrong; there was a whole kingdom of life before the split. Modern biology has redrawn the base of the tree many times, with bacteria, archaea and fungi each claiming their own branches, but the habit of putting microbes on the tree at all owes much to him. The same book coined the words "phylum" and, more famously, "ecology", for the study of how organisms relate to their surroundings.

He was not always careful with the evidence either. His drawings of vertebrate embryos, intended to show that fish, chickens and humans look nearly identical early in development, were accused of exaggeration in his own lifetime, and his "biogenetic law", that development replays evolutionary history, did not survive. Worse, his trees of human races ranked peoples in a hierarchy, and that kind of ranking fed some of the most poisonous racial thinking of the following century.

Haeckel's trees are the clearest lesson in this book about how a layout makes an argument. Put the root at the bottom and one species at the top, and the picture says something the science never did.

Gregor Mendel did his arithmetic with peas. Over eight years in the monastery garden at Brno he raised tens of thousands of plants, sorting the offspring by colour, shape and height, and found that when two hybrids were crossed, the hidden "recessive" form came back in roughly one plant in four. He published in 1866 in the proceedings of the local natural history society, and almost nobody noticed. In 1900 three botanists independently rediscovered the work, and Mendel's laws became the most exciting idea in biology.

Punnett's solution, devised around 1905, was a checkerboard. Write the possible egg cells down one side and the possible pollen grains across the top. Each parent that carries one dominant form, call it A, and one recessive form, a, makes two kinds of gamete in equal numbers. Fill each box with the pair you get where a row meets a column. For a cross of Aa with Aa there are four boxes: AA, Aa, aA and aa. Three of the four contain at least one A, so three in four offspring show the dominant trait; only one is aa and shows the recessive. Mendel's 3 to 1 ratio drops out of the grid with no algebra at all, as does the less visible 1 to 2 to 1 ratio of underlying types.

Punnett's square also nudged one of the founding results of population genetics. Around 1908 he was puzzled by a question from a critic: if a trait is dominant, why does it not gradually spread through the whole population? He put the problem to his friend and cricket companion, the mathematician G. H. Hardy, who quickly showed that, without other pressures, the proportions of dominant and recessive forms stay steady from generation to generation. Hardy published the result in a short letter to Science , apparently regarding it as rather obvious; the German doctor Wilhelm Weinberg reached it independently the same year, and it is now the Hardy-Weinberg principle.

The square has limits. Real traits often depend on many genes, and on environment, and do not fall into neat boxes. That is exactly why it is so useful as a first diagram: it makes the simple case so clear that everything more complicated can be seen as a departure from it.

In June 1937 Hans Krebs posted a short paper to Nature . A few days later it came back with a polite note: the journal had enough letters to fill its correspondence columns for weeks and suggested he try elsewhere. Krebs sent it to the Dutch journal Enzymologia , where it appeared that year with his student William Arthur Johnson as co-author. Sixteen years later the work won him a Nobel Prize, and he liked to show the rejection letter to young scientists as proof that editors are not infallible.

The question in the 1930s was how cells burn food. Everyone knew the overall result, sugar and oxygen in, carbon dioxide and water out, energy released, but not the route. Several pieces were known. Albert Szent-Györgyi in Budapest had found that small amounts of certain four-carbon acids, such as succinate and fumarate, made minced muscle consume far more oxygen than they could account for themselves; they seemed to act as catalysts. German chemists had worked out how citric acid could be converted into α-ketoglutarate. Krebs and Johnson, working with minced pigeon breast muscle, among the most energetic tissues available, added the missing link: citrate is formed when oxaloacetate combines with a fragment derived from food. The four-carbon acid is not used up. It comes back round.

Krebs did not know the exact identity of the fuel fragment. In 1937 he wrote of a "triose", a three-carbon product of sugar breakdown. Over the next fifteen years Fritz Lipmann discovered coenzyme A and Feodor Lynen showed that the entering fragment is acetyl-CoA, which can come from sugars, fats or proteins alike. That made the cycle the common furnace of metabolism, the hub where all the body's fuels meet. Krebs and Lipmann shared the Nobel Prize in Physiology or Medicine in 1953.

Generations of students have cursed it, learned it by mnemonic and forgotten it after the exam. But the picture of a wheel, rather than a production line, changed how biochemists thought. Metabolism is full of cycles, and a cycle is a diagram of something that sustains itself: what goes round, comes round.

The most famous molecular structure of the twentieth century was announced on a single page of Nature on 25 April 1953, and its only illustration was drawn by an artist. Odile Crick, Francis Crick's wife, took her husband's description and his rough sketches and produced a simple pen-and-ink figure: two ribbons spiralling around a vertical line, joined by horizontal rods like the rungs of a twisted ladder. Its caption begins with an unusual piece of modesty: "This figure is purely diagrammatic." It was, and it was also exactly right.

In Cambridge, at the Cavendish Laboratory, James Watson, a 24-year-old American, and Francis Crick, a 36-year-old British physicist still finishing his doctorate, were building models out of wire and cardboard. The decisive evidence came from King's College London, where Rosalind Franklin and her student Raymond Gosling were making X-ray diffraction photographs of DNA fibres. One of them, taken in May 1952 and later known as Photo 51, shows a bold X-shaped pattern of spots: a signature of a helix. In January 1953 Maurice Wilkins, Franklin's colleague and rival, showed the photograph to Watson without her knowledge. Crick also saw a report containing Franklin's measurements. Together they gave the dimensions the model had to obey: a repeat every 34 ångströms along the axis, a width of about 20, and the sugar-phosphate backbones on the outside.

The paper says so in what may be the most celebrated understatement in science: "It has not escaped our notice that the specific pairing we have postulated immediately suggests a possible copying mechanism for the genetic material." Unzip the ladder, let each half collect new partners, and you have two identical molecules. In 1958 Matthew Meselson and Franklin Stahl showed that DNA does indeed replicate that way. The same issue of Nature carried papers from Wilkins's group and from Franklin and Gosling, whose data supported the model; few readers realised how much the model had depended on them.

Odile's two ribbons became the emblem of modern biology: on textbook covers, corporate logos, sculptures and t-shirts. Behind the emblem lie the genetic code, DNA sequencing, the Human Genome Project, forensic fingerprinting and gene editing. All of them follow from a shape that, once drawn, explained itself.

In September 1957 Francis Crick gave a lecture at University College London to a symposium of the Society for Experimental Biology. Published the following year as "On Protein Synthesis", it is one of the most influential papers in biology, and almost entirely speculative. Four years after the double helix, nobody knew how the sequence of bases in DNA was turned into the sequence of amino acids in a protein. Crick set out, in the plainest terms he could, what the answer had to look like.

The choice of word caused trouble for decades. Crick later admitted that he had thought a dogma meant an idea for which there was no reasonable evidence, a bold guess rather than an article of faith. Many readers took it in the religious sense, as something to be accepted without question, and some critics complained that biology had acquired a creed. Crick had meant almost the opposite: here is a strong claim, go and test it.

It is the one redrawn here. DNA, RNA and protein sit at the corners of a triangle. Solid arrows mark the "general" transfers that occur in all cells: DNA copies itself, DNA is transcribed into RNA, RNA is translated into protein. Dashed arrows mark "special" transfers found only in particular circumstances: RNA copying itself in RNA viruses, RNA copied back into DNA by reverse transcriptase, and DNA read directly into protein, which had only been coaxed to happen in a test tube. And three arrows are missing altogether. Nothing leaves protein. Reverse transcription, Crick pointed out, had never been forbidden; the true dogma was intact.

The central dogma is a good example of a diagram as a set of bets. Its value lies not only in the arrows Crick drew but in the ones he dared others to find. Draw the direction of flow in your field, and the most interesting experiment is usually the one that would draw an arrow going back.

Somewhere in Alexandria, around 300 BC, a teacher we know only as Euclid gathered the geometry of the Greek world into thirteen books and called them the Elements . Almost nothing is known about the man himself. What survives is a chain of definitions, axioms and propositions so tightly linked that it served as the model of rigorous thought for more than two thousand years. The first book climbs steadily towards a single summit. Proposition 47 states that in a right-angled triangle, the square on the side opposite the right angle is equal to the squares on the two sides that contain it.

The figure is unmistakable once seen. A right triangle sits in the middle. A square hangs off each side, the largest one dropping below the hypotenuse, the two smaller ones tilting up and out like ears. A line falls from the right angle straight down through the big square, cutting it into two rectangles. Then four long, thin lines cross the drawing from corners to far corners, and the whole thing starts to look like a set of sails. Students nicknamed it the windmill, and in some traditions the bride's chair.

In 1847 an Irish surveyor and mathematician called Oliver Byrne published the first six books of Euclid with the letters stripped out and replaced by colour. In The First Six Books of the Elements of Euclid , printed in London by William Pickering, triangles became blue, squares red and yellow, and the reader followed the argument by matching shapes rather than labels. It was expensive, it sold poorly and Victorian schoolmasters ignored it. Today it looks like a forecast of Mondrian and the Bauhaus, and facsimiles sell steadily to people who never expected to enjoy a geometry book.

In the summer of 1654 Blaise Pascal, a thirty-one-year-old Parisian prodigy who had already built a calculating machine, began exchanging letters with Pierre de Fermat, a magistrate in Toulouse who did mathematics for pleasure. Their subject was a gambling puzzle known as the problem of points: if a game of chance is interrupted before anyone has won, how should the stakes be fairly divided? The two men solved it by different routes, and in solving it they founded the mathematics of probability. Pascal's route ran through a table of numbers.

Pascal did not discover it, and he never claimed to have done so. The pattern turns up in Indian writing on poetic metre, where counting the ways to combine long and short syllables leads straight to it; the commentator Halayudha described it clearly in the tenth century. In Persia, al-Karaji and later Omar Khayyam used it to expand powers of a sum. In China it appears in the work of Jia Xian in the eleventh century, was preserved by Yang Hui in 1261, and was drawn handsomely by Zhu Shijie in 1303, with rods for numerals, reaching down to the eighth row. Italians still call it Tartaglia's triangle. Pascal's name stuck because he did the most with it, proving its properties systematically and using an argument that is now recognised as one of the earliest clear uses of mathematical induction.

What the drawing changed was the idea that chance could be calculated. Pascal and Fermat's letters led directly to expected value, insurance mathematics and the statistics that sit under every opinion poll and drug trial. Isaac Newton, still in his early twenties, extended the binomial pattern to fractional and negative powers, a step that fed his calculus. Gregor Mendel's peas, splitting into ratios like 1, 2, 1, were the triangle's third row growing in a monastery garden.

There is a much-loved story that René Descartes, lying in bed late one morning, watched a fly crawl across the ceiling and realised he could pin down its position with two numbers: its distance from one wall and its distance from the other. It is a charming tale, and there is no good evidence for it. What is certain is that in 1637, in Leiden, Descartes published his Discours de la méthode with three scientific essays attached as demonstrations of his method. The third of them, La Géométrie , about a hundred pages long, joined two subjects that had been kept apart since the Greeks.

He tested the idea on a classical puzzle, the problem of Pappus, which asks for the path of points whose distances from several given lines stand in a fixed ratio. The ancients had solved only special cases. Descartes produced a general method, and in the process set notations we still use: letters from the start of the alphabet, a, b and c, for known quantities, and letters from the end, x, y and z, for unknowns.

The plane works because it gives every point an address and every rule a shape. Write down a point, say four along and three up, and the picture and the numbers are the same thing. Write down an equation and you can see whether it climbs, curves or closes into a loop, and where two curves cross is where two equations are solved at once. The four quadrants record the signs, so that being behind or below the origin is simply a matter of a minus.

Königsberg, the Prussian port on the River Pregel, had a pastime that cost nothing. The river split around a central island called the Kneiphof, ran on either side of a second stretch of land to the east, and divided the town into four pieces joined by seven bridges. Could a person stroll through the city crossing every bridge exactly once? Residents tried and failed, and the question travelled. By 1736 it had reached Leonhard Euler, a young Swiss mathematician at the Academy in St Petersburg who was already on his way to becoming the most prolific mathematician in history.

His own figure was still a map: a schematic river with the four land areas lettered A, B, C and D, and the bridges marked a to g. The bridge lengths, the street plan, the shape of the island, all were irrelevant. Euler recorded a walk as a string of capital letters, one for each land area visited, so a route crossing seven bridges would be written with eight letters. Then he counted. Island A touches five bridges, so any walk using each once must visit A three times. B, C and D touch three bridges each, so each must appear twice. Three plus two plus two plus two makes nine letters, one more than any seven-bridge walk can contain. The stroll was impossible.

The famous picture of four dots joined by seven curved lines is not Euler's. That stripped-down drawing, which we now call a graph, appeared only in the later nineteenth century, when the puzzle became a staple of recreational mathematics books. Yet it is exactly what Euler's argument had already made: a map with the geography deleted, leaving nodes, edges and the number of edges at each node, which mathematicians call its degree.

Abraham de Moivre was a French Protestant who had fled to London as a young man after Louis XIV revoked the Edict of Nantes. Despite friendship with Isaac Newton and a fellowship of the Royal Society, he never secured a university post, and he earned his living tutoring and advising gamblers and insurers, reputedly from a table at Slaughter's Coffee House in St Martin's Lane. His speciality was the doctrine of chances, and one problem in it was brutally tedious: if you toss a coin hundreds or thousands of times, how likely is it that the number of heads falls within a given range? Answering exactly meant adding up enormous binomial numbers, the rows of Pascal's triangle far beyond any patience.

The curve's second birth came from the sky. In 1801 the newly discovered dwarf planet Ceres slipped behind the Sun and astronomers lost it. Carl Friedrich Gauss, then twenty-four, predicted where it would reappear, and he was right. In 1809, in Theoria motus , his treatise on the motion of heavenly bodies, he explained his method of least squares and justified it by assuming that errors of observation follow exactly this curve. Pierre-Simon Laplace soon showed why such a law should be so common: add up many small independent effects and the total tends towards the bell, whatever the individual effects look like. That is the central limit theorem, and it is the reason the curve appears everywhere.

In the nineteenth century the Belgian astronomer Adolphe Quetelet found the curve in the chest measurements of Scottish soldiers and built a social science around the average man. Francis Galton built a board of pins, the quincunx, down which falling shot piled into a bell, and he was among those who popularised the name normal. Later in the twentieth century the curve was printed, with a portrait of Gauss, on the German ten-mark banknote.

For more than two centuries, mathematicians used the square root of minus one while apologising for it. Gerolamo Cardano met such quantities in 1545 while solving cubic equations and called the business as subtle as it was useless. Rafael Bombelli showed in 1572 that calculating with them could produce perfectly real answers. Descartes gave them the dismissive name imaginary. They worked, but nobody could say what they were. A number ought to be a quantity, a length, something you can point at on a line, and no point on the line squares to give minus one.

The payoff is in what the operations look like. Adding two such numbers means placing their arrows end to end. Multiplying them means multiplying their lengths and adding their angles. Multiplying by i adds a right angle: it turns any arrow a quarter turn anticlockwise without changing its length. Do it twice and the arrow points the opposite way, which is precisely what multiplying by minus one does. So i times i equals minus one, and the mystery dissolves. The square root of minus one is simply the number that, applied twice, turns you round.

Once the plane existed, complex numbers stopped being a trick and became a tool. William Rowan Hamilton recast them as pairs of real numbers and then hunted for a three-dimensional version, which led him in 1843 to quaternions, now used to rotate objects in computer graphics and spacecraft. At the end of the nineteenth century Charles Proteus Steinmetz showed engineers how to treat alternating currents as rotating arrows in exactly this plane, which made the design of power grids practical.

John Venn was a Cambridge man to the bone: a fellow of Gonville and Caius College, an ordained clergyman, a historian of his university and a logician of the careful, unflashy kind. In July 1880 he published a paper in the Philosophical Magazine with the unglamorous title On the Diagrammatic and Mechanical Representation of Propositions and Reasonings . It contained the drawing that now appears in more classrooms, business presentations and internet jokes than perhaps any other diagram ever made.

His solution was a change of philosophy. Instead of drawing the facts, draw every possibility first. Three overlapping circles, x, y and z, divide the plane into eight compartments: things that are x and y and z, things that are x and y but not z, and so on, down to the region outside all three circles for things that are none of them. The diagram makes no claims at all until you start shading. Each premise then tells you which compartments are empty, and you shade them out. Whatever remains unshaded is all the possibility left in the world of the argument.

Venn knew the method's limits. Three circles give eight compartments, but four circles cannot show all sixteen combinations, so he proposed ellipses for four classes and admitted that beyond that the drawings become ungainly. A century later a fellow of the same college, the statistician Anthony Edwards, devised a way of extending the diagram to any number of sets, and the college put a three-circle Venn diagram into a stained-glass window in its hall. The name Venn diagram itself was in print by the early twentieth century, though Venn himself was careful to credit Euler and other predecessors.

Georg Cantor spent almost his entire career at the University of Halle, a respectable provincial institution far from the mathematical centres of Berlin and Göttingen, and from there he did something that many of his contemporaries regarded as close to madness. He took the infinite, which mathematicians since Aristotle had treated as a direction rather than a thing, and began to count it. Two collections are the same size, he argued, if their members can be paired off one to one with nothing left over. By that test the even numbers are exactly as numerous as all the whole numbers, which is strange but harmless. The real question was whether every infinity was the same infinity.

Cantor did not use decimals, as most retellings do. He used infinite sequences made of just two symbols, m and w. Suppose, he said, someone claims to have listed all such sequences: E1, E2, E3 and so on forever, each one written out as a row. Now walk down the diagonal of that grid, taking the first symbol of the first row, the second symbol of the second row, the third symbol of the third, and swap each one, m for w and w for m. The result is a new sequence, which he called E0.

The reaction was fierce. Leopold Kronecker in Berlin, who believed mathematics should rest on the whole numbers alone, opposed Cantor's work and obstructed his career, and Cantor suffered repeated bouts of severe depression. Others were captivated. In 1926 David Hilbert declared that no one shall expel us from the paradise that Cantor has created.

In the winter of 1961 Edward Lorenz, a meteorologist at the Massachusetts Institute of Technology, wanted to look again at a run of his weather simulation. The model was a toy atmosphere of twelve variables, running on a Royal McBee LGP-30, a desk-sized computer that filled the office with noise and heat. Rather than start from the beginning, he typed in numbers from halfway through an earlier printout and went for a coffee. When he came back an hour later, the new run had matched the old one for a while, then drifted, then diverged completely, until the two weathers bore no resemblance to each other.

To understand why, Lorenz stripped the problem down further. Taking a model of convection, warm fluid rising and cool fluid sinking, that his colleague Barry Saltzman had been studying, he reduced it to just three equations in three variables, x, y and z, with three fixed numbers that he chose as 10, 28 and 8/3. Nothing in them is random. Given a starting point, the future is fixed forever. Yet when he followed the solution through time, it never settled down and never repeated. In March 1963 he published the result in the Journal of the Atmospheric Sciences under the deliberately plain title Deterministic Nonperiodic Flow .

The paper was read by meteorologists and then largely forgotten for a decade, until mathematicians discovered it. In 1971 David Ruelle and Floris Takens coined the term strange attractor, and in 1975 Tien-Yien Li and James Yorke gave the field its name, chaos. In 1972 Lorenz gave a talk in Washington whose title, supplied by the session's organiser, asked whether the flap of a butterfly's wings in Brazil could set off a tornado in Texas. The butterfly effect has been a household phrase ever since, helped along by the happy accident that the attractor itself looks like a butterfly. Whether the shape really existed, rather than being an artefact of rounding, was proved only in 2002, by the mathematician Warwick Tucker, using a computer-assisted proof.

Benoit Mandelbrot liked to describe himself as a nomad by choice. Born in Warsaw in 1924, raised in France, and for most of his career employed at IBM's Thomas J. Watson Research Center in Yorktown Heights, New York, he wandered across economics, linguistics, hydrology and physics looking for the same thing everywhere: roughness. In a 1967 paper he asked how long the coast of Britain is, and answered that it depends on the length of your ruler, growing without limit as the ruler shrinks. In 1975 he coined a word for shapes that look equally complicated at every scale. He called them fractals.

The rule is almost absurdly short: take z, square it, add a fixed number c. Start with z equal to zero and repeat. For some values of c the numbers stay small forever; for others they grow and rush off to infinity. Choose c equal to minus one and the sequence goes 0, minus one, 0, minus one, trapped in a loop. Choose c equal to one and it runs 0, 1, 2, 5, 26 and away. The Mandelbrot set is the collection of every point c in the complex plane, the Argand plane of chapter thirty-six, whose sequence stays bounded.

Its anatomy rewards patience. The large heart-shaped body is the main cardioid, the points whose sequences settle on a single value. The disc to its left holds the points that settle into a two-step cycle like minus one. Smaller bulbs sprout everywhere, each with its own cycle length, and along the spike to the left sits a miniature Mandelbrot set, one of infinitely many. The boundary is so intricate that its dimension was proved in 1991 to be two, as rough as a line can possibly be.

William Playfair was a Scot with a gift for ideas and a weakness for schemes, and in 1786 the first of these produced something that outlived all of the second. In The Commercial and Political Atlas , published in London that year, he set out to show the trade of England not as columns of figures but as pictures. Most of the plates were line charts, the ancestors of every stock price graph, tracing imports and exports across the eighteenth century. One plate was different, and it was different because Playfair had run out of data.

It sounds trivially obvious now, which is exactly the point. Before Playfair, a merchant comparing Scotland's dealings with Norway and with Holland had to read two numbers, hold them in mind and do the subtraction himself. After Playfair, the eye did the arithmetic. Length turns out to be the quantity human beings judge most accurately, far better than area, angle or shade, a fact that perception researchers would measure carefully two centuries later. Playfair simply guessed it, or stumbled on it while improvising.

He was not, it must be said, an entirely reliable character. His later career took in a disastrous scheme selling land in Ohio to French emigrants, a string of failed businesses and periods of real poverty. He died in 1823 with his graphical inventions largely ignored in Britain, where statisticians preferred their numbers neat and unillustrated. The idea survived anyway, because once a reader has seen a bar chart, a table never looks quite the same.

There is a pleasing irony in the origin. The most successful chart form ever devised was a workaround, a way of salvaging a page for which the proper data did not exist. Playfair thought of it as second best.

Fifteen years after inventing the bar chart, William Playfair invented the pie chart, and he did it in a book with the deliberately businesslike title of The Statistical Breviary , published in London in 1801. Europe was at war with Napoleon, maps were being redrawn by the season, and Playfair wanted to show at a glance how the great powers measured up against each other in territory, people and money.

Then came the slice. The Turkish Empire straddled three continents, and Playfair wanted readers to see how its territory divided between Europe, Asia and Africa. So he cut its circle into coloured sectors, each sized in proportion to its share. That single divided circle is generally accepted as the first pie chart, though nobody would call it that for another century. The French, more poetically, call it a camembert .

Yet the pie keeps surviving its critics, and Playfair's original shows why. He used it for one specific job: showing the parts of a single whole, with only a handful of parts, where the question was roughly how big each share was. A reader seeing the Turkish circle does not need to know that Asia was precisely this fraction; what matters is that the empire was mostly Asian, with a European shoulder and an African wedge. For that question, a pie answers instantly, and it carries a built-in reminder that the slices add up to one hundred per cent, which a bar chart does not.

Playfair died poor and largely ignored, but within half a century his circles had travelled. Florence Nightingale's famous rose is a pie whose slices grow outwards rather than across, and every progress ring on a phone screen is a single slice of his.

In the last days of August 1854, cholera exploded in Soho. Within roughly ten days more than five hundred people living within a few hundred yards of one another were dead, many of them within a day of falling ill. Shops were shuttered, families fled, and the authorities blamed what they had always blamed: miasma, the foul air rising from London's filth.

His suspicion fell on a public pump on Broad Street. On 7 September he put his case to the local Board of Guardians, and on 8 September the handle of the pump was removed. The outbreak was already waning by then, as Snow himself acknowledged, because so many had died or fled. The handle has since become a legend; the map was the real achievement.

The map's real power lies in its exceptions. A large workhouse on Poland Street, surrounded by stricken houses, had its own well and lost only a handful of its several hundred inmates. The brewery on Broad Street had its own well too, and its workmen drank beer; none of them died. Meanwhile a widow in Hampstead, miles away, died of cholera after having Broad Street water carried to her because she preferred its taste. Each exception is the kind of detail a table buries and a map displays.

Officialdom was slow to agree. The Board of Health rejected the water theory at first, and Snow died in 1858 with his idea still disputed. But within a generation the bacterium had been seen under the microscope, London had built its great sewers, and Snow's map had become the founding image of epidemiology: a discipline that begins by asking where.

Florence Nightingale is remembered as the Lady with the Lamp, gliding through the wards of Scutari at night. She deserves at least as much fame as the lady with the protractor. Her most important weapon in the long campaign to reform the British Army's medical care was not a lamp at all but a diagram, and she drew it to be understood by people who would never read a table.

The diagram she produced in 1858 consists of two circles, each divided like a clock face into twelve wedges, one for each month. The right-hand rose covers April 1854 to March 1855, the left-hand one the following year. Every wedge is really three wedges laid over one another, all measured from the centre: blue for preventable disease, red for wounds, and black for all other causes. Crucially, Nightingale made the area of each wedge, not its length, proportional to the death rate, so a month with four times the deaths looks four times as large rather than sixteen.

Nightingale knew exactly whom she was drawing for. The diagram appeared in her privately printed Notes on Matters Affecting the Health, Efficiency, and Hospital Administration of the British Army and was circulated to ministers, officials and Queen Victoria herself. She wanted, in effect, to show a busy man in one glance what pages of figures would never make him feel.

The diagram worked. It helped drive the Royal Commission's reforms, the founding of an Army Medical School and a new emphasis on sanitation in barracks and hospitals across the Empire. In 1858 Nightingale became the first woman elected a Fellow of the Statistical Society of London. Every public health infographic since owes something to her conviction that statistics are useless until someone in power can see them.

In 1869 a retired French civil engineer named Charles Joseph Minard, then in his late eighties, published a lithograph about a disaster that had happened more than half a century earlier. It showed Napoleon's invasion of Russia in 1812, and it compressed one of the great catastrophes of military history into a single beige ribbon and a single black one. A century later the statistician Edward Tufte would write that it may well be the best statistical graphic ever drawn, and few people who study it closely disagree.

The chart reads from left to right. At the Niemen river, on the western border of the Russian Empire, the beige band is enormous: 422,000 men, according to Minard's figures. As it heads east towards Moscow it narrows relentlessly, losing men to disease, desertion, garrison duty and battle. Detachments peel away. By the time the band reaches Moscow it is less than a quarter of its starting width, about 100,000 soldiers, arriving in a city that had been abandoned and was soon burning.

Below the map runs the element that turns a good chart into a great one. Minard added a line graph of the temperature during the retreat, recorded in degrees Réaumur, with each reading linked by a thin vertical line to the place and date on the black band above. As the line plunges through November and into December, to around thirty degrees below zero on the Réaumur scale, the reader can see the army freezing to death in step with it.

He died in 1870, as France slid into another war. His map was later championed by the physiologist Étienne-Jules Marey and, much later, by Tufte, whose books made it a poster on the walls of design schools everywhere. It remains the benchmark against which every ambitious data graphic is quietly measured.

Francis Galton was the kind of Victorian who measured everything. He counted fidgets in lecture audiences, devised a whistle to test the upper limit of hearing, drew the first weather map printed in a newspaper and helped turn fingerprints into a means of identification. Half-cousin to Charles Darwin, he became obsessed with a question Darwin's work had sharpened: how exactly are traits passed from parents to children? In 1886 he answered part of it with a diagram that gave the world a new word.

To find out he collected the heights of families, partly through a scheme that offered cash prizes for the best family records. He ended up with 928 adult children of 205 sets of parents. To compare men and women on one scale he multiplied every female height by 1.08, and he averaged each father and adjusted mother into a single "mid-parent". Then he laid the data out as a table, mid-parents down one side, children across the other, each cell holding the number of families that fell there.

The line through the centres of each row, Galton found, had a slope of about two-thirds. Exceptional parents tended to have children whose height sat around two-thirds as far from the population average. He called this "regression towards mediocrity" in his 1886 paper, mediocrity then simply meaning the middle. Two years later he generalised the idea into what he called co-relation, and his protégé Karl Pearson later turned it into the correlation coefficient taught in every statistics course.

Galton's legacy carries a darker strand, and it should be stated plainly. He coined the word eugenics in 1883 and championed breeding a "better" humanity, ideas that fed some of the cruellest policies of the twentieth century. The same man who taught science how to reason about variation also showed how badly it could be misused. The tilted ellipse remains, a tool far more honest than the uses he imagined for it.

In the 1880s London was the richest city on earth and, according to a growing chorus of reformers, home to a vast submerged population living in misery. Nobody actually knew how large. Charles Booth, a wealthy Liverpool shipowner settled in London, decided to find out, and the map he produced in 1889 turned an argument about impressions into an argument about streets.

Their best source was an army of School Board visitors, officials whose job was to knock on doors and chase up children missing from school. They knew who lived where, how many children each family had, what the father did and how regularly he worked. Booth's team interviewed them street by street and, in later rounds, walked the streets themselves, often accompanied by local policemen, filling notebooks with sharp and sometimes judgemental descriptions.

What the map revealed was not simply that the East End was poor. It showed the texture of poverty. Main roads were frequently red and pink, lined with shopkeepers and clerks, while the dark blue and black lay a few yards behind them, down alleys and courts invisible from the omnibus. Rich and poor lived far closer together than the respectable West End supposed. And the poverty was enormous: Booth estimated that around 30.7 per cent of Londoners lived in poverty, below a threshold he called the line of poverty, a phrase that has been in use ever since.

Booth's notebooks and maps survive at the London School of Economics, now searchable online, and historians still use them to trace individual streets. The categories were moralising and crude by modern standards. But the principle he established, that you cannot fix a social problem you have not measured, and cannot measure it from an armchair, became the foundation of social science.

At the Paris Exposition of 1900, among the electric lights, moving walkways and palaces of industry, a small room in the Palace of Social Economy held one of the most startling exhibits of the fair. Its walls were hung with photographs of Black Americans in their homes, schools and businesses, and with a series of hand-drawn charts in bold reds, golds, greens and blacks. They had been made under the direction of W. E. B. Du Bois, and they set out to show the world, in numbers, what Black Americans had achieved in the thirty-five years since slavery ended.

They came in two sets. One surveyed Black Americans across the whole country; the other, titled The Georgia Negro: A Social Study , focused on a single state, the one with the largest Black population. Labels were written in both English and French, for the Paris audience. The subjects were deliberately concrete: population, literacy, occupations, the acres of land owned and the value of household furniture, the growth of businesses and churches.

Every chart argued against a lie. White supremacist science of the period insisted that Black Americans were incapable of progress and doomed by freedom. Du Bois answered with figures gathered rigorously and displayed with a clarity no reader could dodge, showing property accumulated, illiteracy collapsing, and lives built in the teeth of segregation and violence. In The Souls of Black Folk three years later he would write that the problem of the twentieth century is the problem of the colour line; in Paris he had drawn that line.

Captain Matthew Henry Phineas Riall Sankey had a name almost as long as the energy losses he set out to draw. An Irish-born engineer who had served in the Royal Engineers, he spent much of his career on the practical question that obsessed the late Victorian age: how to get more work out of a ton of coal. In 1898, in a paper on the thermal efficiency of steam engines for the Institution of Civil Engineers, he included a figure that looked like a river breaking its banks.

Sankey drew the heat as a band whose width was proportional to its quantity. It enters wide on one side, the whole energy content of the fuel. As it travels, branches peel away from it, each one as wide as the loss it represents, curving off the main stream like distributaries from a delta. What remains at the far end, a surprisingly thin ribbon, is the useful work. Because widths are conserved, every unit of heat is accounted for: the streams going out add up exactly to the stream coming in.

Sankey was not the first to draw quantities as flowing bands. Charles Joseph Minard had done so decades earlier for goods, passengers and, most famously, Napoleon's army. But Sankey applied the idea to energy, where the law of conservation gives the bands a satisfying rigour, and his name stuck. By the early twentieth century German engineers were drawing what they called Sankey diagrams for factories and power stations, and the term spread with them.

There is a quiet moral in the picture. Most systems are far less efficient than their owners assume, and the waste is rarely where people are looking. A table lets a reader skip the uncomfortable row; a Sankey diagram makes the loss as wide as it really is. Sankey drew it to improve steam engines. His real legacy is a way of seeing where things go.

John Wilder Tukey was a mathematician who could not leave a word or a number alone. At Bell Labs and Princeton he co-invented the fast Fourier transform, an algorithm that now runs inside nearly every phone and radio. He is credited with coining the word "bit" for a binary digit and with one of the earliest printed uses of "software". But his most visible legacy is a humble drawing of a box with two whiskers, devised so that anyone with a pencil could understand a pile of numbers.

The box plot, which Tukey called a schematic plot, was its signature tool. It condenses a batch of numbers into five values: the median in the middle, two "hinges" marking off the middle half of the data, and the extremes. The box spans the hinges, so its height shows at a glance how spread out the typical values are. A line across it marks the median, which is far less easily dragged around by a few wild values than the familiar average. Lines called whiskers reach out towards the extremes.

The design was deliberately cheap. Before personal computers, Tukey wanted methods that could be done with a pencil, a sheet of paper and a little counting, which is why medians and hinges, found simply by ordering the data, are used instead of means and standard deviations. He was building on earlier work: the economist Mary Eleanor Spear had drawn a similar "range bar" in a 1952 charting manual. Tukey's version added the fences and the outliers, and gave the whole idea a rigorous rationale.

Statisticians have since added notches, violins and swarms of dots, and the plain box can conceal a distribution with two humps. Yet Tukey's five-number sketch remains the quickest honest summary of a batch of data ever devised, and the habit he championed, of looking before testing, has become the first rule of data science.

Around 1490, in Milan, a man in his late thirties took a sheet of paper about the size of a modern A4 page and tried to solve a puzzle that was fifteen centuries old. The puzzle came from Vitruvius, a Roman engineer who served under Augustus and wrote the only architectural treatise to survive from antiquity. In Book III of De architectura , Vitruvius claimed that a well-made human body fits two perfect figures. Lie a man on his back, spread his limbs, put the point of a compass at his navel, and his fingers and toes will touch the circle. Stand him upright with arms outstretched and his span equals his height, so he also fits a square.

Then he drew one man in two poses at once. Arms level and feet together, he fills the square. Arms raised and legs splayed, he touches the circle. Four arms and four legs on one torso ought to look monstrous, and somehow it looks calm, the face as steady as a portrait. Beneath the figure Leonardo ruled a scale bar divided into palms and fingers, and around it he wrote, in his habitual right-to-left mirror script, a précis of Vitruvius's measurements: four fingers make a palm, four palms a foot, six palms a cubit, four cubits a man's height, and so twenty-four palms in all. The head, from chin to crown, is an eighth of the whole.

The sheet, roughly 34 by 25 centimetres, now lives in the Gallerie dell'Accademia in Venice, where it is rarely shown because light damages it. Its absence from the walls has not slowed it down. It has become the logo of the Renaissance itself, printed on the Italian one-euro coin, stamped on medical textbooks, borrowed by space agencies and gyms, and parodied with cartoon characters in the same splayed pose.

A machine drawn whole is a riddle. You can see the crank and the drum, but the interesting things, the pinion hidden behind the wheel, the pawl that stops the load from running back, the shoulder on the axle that holds everything in place, are buried inside. Renaissance engineers knew this problem well. Mariano Taccola and Francesco di Giorgio filled notebooks with hoists, mills and siege engines, and their readers often had to guess how the pieces met. Leonardo da Vinci, working in Milan in the 1480s and 1490s and later in Florence and France, found a different way of looking.

The idea sounds obvious only because we have grown up with it. It required two leaps. First, the draughtsman has to imagine the object in three dimensions with complete confidence, since nothing on the page can be copied from life; nobody has ever seen a winch hanging in pieces. Second, he has to accept that a drawing need not show a moment that could exist. Leonardo's exploded view is a picture of a process, the assembly, frozen into a single image. Space on the page stands in for time on the workbench.

So the exploded view had to be reinvented, and it was, many times. Agricola's De re metallica of 1556 printed mining machinery with parts laid out and lettered. By the nineteenth century, engineering drawings, patent figures and gunsmiths' manuals used separated components with dotted lines showing where each one goes, and by the twentieth the exploded view was the standard grammar of the workshop manual, the spare-parts catalogue and the aircraft maintenance handbook.

By 1788 the campaign against the British slave trade had plenty of facts. Thomas Clarkson had spent months in Bristol and Liverpool collecting testimony from sailors, ship's surgeons and port records. The Society for Effecting the Abolition of the Slave Trade, founded in London in 1787, could quote mortality rates and tonnages to anyone who would listen. The trouble was that numbers slide past people. A Member of Parliament could hear that a ship carried several hundred Africans and nod, without ever picturing what several hundred bodies in a hold actually meant.

What makes the image so terrible is its calm. It is drawn exactly as a shipwright or a cargo clerk would draw a stowage plan, in plan and section, with rooms labelled for men, boys and women and measurements in the accompanying text. The men are allotted a space six feet long by one foot four inches wide. Half-decks or platforms run round the sides so that a second layer of people could be laid above the first. Nothing is exaggerated, and nothing needs to be. The slavers' own logic, that people were freight to be packed efficiently, is simply shown back to the public at scale.

The image travelled with astonishing speed. The committees printed it by the thousand as posters and pamphlets, hung it in coffee houses and pubs, and sent it to supporters across Britain and America. Clarkson took it to Paris in 1789, where the Comte de Mirabeau is said to have had a wooden model of the ship made. Alongside Josiah Wedgwood's cameo of a kneeling man asking Am I not a man and a brother? , it became one of the first mass-produced images of a political campaign. Britain abolished its slave trade in 1807 and slavery in most of its colonies in 1833; the plan did not do that alone, but it gave the argument a face that could not be explained away.

Gaspard Monge was the son of a knife-grinder and pedlar from Beaune, and in the 1760s he was a lowly draughtsman at the royal engineering school at Mézières, a place that trained officers for the French army's corps of engineers. Gentlemen studied there; Monge drew for them. One of the tedious jobs of the school was the problem of defilade, the design of a fortification so that its walls hid the interior from guns on surrounding high ground. It required long chains of arithmetic, and Monge, so the story goes, came back with an answer so quickly that his superiors at first refused to believe he had done the work.

From those two views alone, any solid can be reconstructed exactly, and better still, problems about it can be solved by construction. Where does a sloping roof meet a cone? How long is this edge in reality, rather than as it appears? At what angle must this stone be cut? Monge showed how to answer such questions with ruler, compass and further projections, without heavy algebra. Masons and carpenters had long used rough versions of plan and elevation; Monge turned the craft into a rigorous branch of geometry.

That is roughly what happened, though not only in France. Descriptive geometry became the backbone of engineering education in Europe and, through West Point, in the United States. The two-view drawing evolved into the standard orthographic sets of front, top and side views, with first-angle projection in much of Europe and third-angle in America, and a small symbol in the title block still tells the reader which one is in use. Without that shared grammar, the interchangeable parts, machine tools and railways of the nineteenth century would have had no reliable way to travel from one workshop to another.

By the 1790s the firm of Boulton and Watt had a commercial problem dressed up as a scientific one. Their steam engines were far more efficient than the old Newcomen engines, thanks to James Watt's separate condenser, and the firm charged customers a royalty based on the fuel those engines saved. That made the precise performance of each engine a matter of money. How hard was the steam actually pushing, at each moment of each stroke, inside a cylinder nobody could see into?

The picture is a closed loop. At the start of the stroke steam rushes in, and the pencil climbs and then runs along a high, flat line. At cut-off the steam valve closes and the steam inside expands, so the line curves gently down as the piston moves on. At the end of the stroke the cylinder opens to the condenser and the pressure drops below the atmospheric line, and on the return the pencil runs back along the low line before the cycle begins again. Leaky valves, sticky pistons and badly timed cut-offs all showed up as dents, humps and rounded corners that an experienced eye could read like a doctor reading a pulse.

Boulton and Watt treated the indicator as a trade secret, and for years only their own engineers used it. It leaked into wider practice in the early nineteenth century, and then into theory. In 1834 the French engineer Émile Clapeyron took Sadi Carnot's purely verbal account of an ideal heat engine and redrew it as a loop on a pressure-volume diagram, the indicator diagram's abstract twin. That picture became one of the founding images of thermodynamics, the science of energy and entropy that would later underpin everything from refrigerators to the study of stars.

In 1842 an Italian engineer, Luigi Menabrea, published a paper in French describing Charles Babbage's Analytical Engine, a steam-age design for a general-purpose calculating machine with a store for numbers, a mill for arithmetic and instructions fed in on punched cards borrowed from the Jacquard loom. Augusta Ada King, Countess of Lovelace, the daughter of Lord Byron and a friend of Babbage's for nearly a decade, translated it into English. Babbage suggested she add some notes of her own. When the translation appeared in 1843 in Richard Taylor's Scientific Memoirs , signed only with her initials, A.A.L., her seven notes, lettered A to G, were around three times as long as Menabrea's article.

The notation is ingenious. Each variable is written as a V with a subscript naming its column in the Store and a small index above recording how many times its value has changed. So the first three variables hold the data, the numbers 1, 2 and n, with n set to 4 in her example. The first operation multiplies V2 by V3 to give 2n and sends the answer to three working variables at once; the next two produce 2n minus 1 and 2n plus 1; a few rows later the table has assembled the first term of the formula. Then comes the crucial idea. Operations thirteen to twenty-three are to be repeated, with the variables updated, for each further term, so a short block of instructions does work of any length. That is a loop, drawn on paper a century before electronic computers existed. After twenty-five operations the Engine would deliver the Bernoulli number she called B7.

Above all, Lovelace saw that the Engine manipulated symbols, not merely numbers. In Note A she wrote that it “weaves algebraical patterns just as the Jacquard loom weaves flowers and leaves,” and she speculated that it might one day compose music if the relations of pitched sounds could be expressed in its notation. In Note G she added a caution that has echoed ever since: the Engine “has no pretensions whatever to originate anything.” Alan Turing answered it in 1950 under the heading of Lady Lovelace's Objection.

Before the blueprint, every copy of an engineering drawing was made by hand. In the drawing offices of shipyards and railway companies, rows of tracers sat over the originals, copying every line, dimension and note onto fresh sheets. It was slow, it was expensive, and every copy was a fresh chance for error: a slipped decimal, a hole in the wrong place, a revision that reached the foundry but not the fitting shop. Large projects needed dozens of identical drawings, and they could not have them.

Herschel's friend Anna Atkins used the process almost at once, laying seaweeds directly on the paper to make Photographs of British Algae , begun in 1843 and often called the first book illustrated with photographs. But for engineers the trick lay in the negative. Draw a design in black ink on translucent tracing cloth, lay it on the sensitised paper under a sheet of glass, and put the frame out in the sun. The ink lines block the light, so they print white; everything else turns deep blue. No drafting skill is needed to make the copy, and every copy is exactly the size and shape of the original.

The effect on how things were built was enormous and largely invisible. Steel ships, bridges, skyscrapers and factories involve hundreds of trades working at once, and the blueprint let every one of them hold the same drawing, to scale, with the same revision letter in the title block. It made the drawing, rather than the master craftsman's memory, the authority on site. It also gave English a metaphor: by the twentieth century a blueprint meant any detailed plan, from economic policy to the genetic code.

The most famous patent in aviation does not show an aeroplane with an engine. United States Patent 821,393, Flying-Machine , granted to Orville and Wilbur Wright on 22 May 1906, illustrates their unpowered glider of 1902. That was no accident. The brothers filed the application on 23 March 1903, nine months before their first powered flight at Kitty Hawk, and what they claimed was not the motor or the propeller but something subtler: a way of keeping a flying machine under control.

Sheet one of the patent is drawn in the formal style of the age: fine ink lines, a careful perspective, the inventors' names, the filing date and the patent number lettered in the margins. It shows the box-like biplane truss of struts and diagonal wires, the horizontal front rudder that we now call an elevator, and the vertical rudder behind. Further figures show the arrangement of cords and pulleys that warped the wings. In flight trials of 1902 the brothers had discovered that warping alone could make the glider slew into a dangerous spin, and that turning the rudder at the same time cured it. The patent claims that combination: roll control by changing the angle of the wing surfaces, coordinated with the rudder. In the language of later engineers, it is control about all three axes, roll, pitch and yaw, the basis of every aircraft since.

The patent war consumed much of Wilbur's last years before his death in 1912, and many historians believe it slowed American aviation, which fell behind France and Germany before the First World War. It ended only in 1917, when, with the United States entering the war, the government pressed the industry into a patent pool, the Manufacturers Aircraft Association. Warping itself soon vanished, replaced by the ailerons the Wrights had fought.

When President Kennedy told Congress in May 1961 that America should land a man on the Moon and return him safely before the decade was out, nobody at NASA yet knew how. The obvious plan, called direct ascent, was the one from science fiction: build an enormous rocket, fly the whole ship to the Moon, land it tail-first, and launch it home again. Every kilogram carried down to the surface had to be lifted back off it, which meant a rocket, known in planning as Nova, far larger than anything that could be designed, built and tested in time. A second plan, favoured by Wernher von Braun's team in Huntsville, Alabama, would assemble the spacecraft in Earth orbit from several launches.

Many senior engineers were horrified. A rendezvous a quarter of a million miles from home, with no possibility of rescue, sounded like a gamble with three lives. Houbolt kept presenting his diagrams, and in November 1961 he went over the heads of his superiors with a long letter to the agency's associate administrator, Robert Seamans, asking in effect whether NASA wanted to go to the Moon or not. Studies through the following spring kept pointing his way, von Braun's centre came round in June 1962, and NASA announced its choice of lunar orbit rendezvous that July.

The arithmetic was the argument. By leaving the mothership in orbit and building a lander that never needed to carry a heat shield or the return fuel, the mass that had to be launched from Earth shrank enough to fit on a single Saturn V. It also allowed two spacecraft to be designed separately, each optimised for its job, and it gave the crew of Apollo 13 a lifeboat when their service module failed in 1970.

The most widely read technical drawings on Earth are not engineering plans or patent figures. They are the folded leaflets that tumble out of IKEA boxes, and most of them contain almost no words at all. That was a deliberate answer to a very practical problem. A company that sells the same bookcase in dozens of countries, to customers who speak dozens of languages and have never held a cam lock, needs instructions that anyone can follow and nobody needs to translate.

IKEA's illustrators, working at the company's headquarters in Älmhult, built a visual language that borrows from almost every chapter in this part of the book. The first page is an inventory, each part drawn with its count beside it, 8x, 4x, 1x, like a miniature parts catalogue. Assembly steps use exploded views in the Leonardo tradition, with dashed lines showing exactly where each dowel slides. Circular magnifications pull out details too small to read at full scale. Large step numbers, ticks and crosses replace sentences. Drawings are clean line art, closer to Monge's orthographic views than to a photograph, because a photograph is full of shadows and texture that distract from the one thing that matters at each step.

The approach is not perfect, and the instructions are a staple of comedy for good reason. Wordless diagrams demand that readers notice small differences, a hole on the left rather than the right, a panel with its finished face turned outwards, and a misread step can mean dismantling half a wardrobe. Yet the sheets work well enough that IKEA's bestsellers have been assembled by many millions of people, and the Billy bookcase alone, launched in 1979, has sold in the tens of millions. Psychologists even have a name, the IKEA effect, for the finding that people value objects more highly when they have assembled them themselves.

In the spring of 1936 a 23-year-old fellow of King's College, Cambridge, sent a paper to the London Mathematical Society with a title that promised very little excitement: On Computable Numbers, with an Application to the Entscheidungsproblem . Alan Turing had been chewing on a question posed by David Hilbert, and put to him in Max Newman's lectures the year before. Was there a mechanical procedure that could decide, for any statement of mathematics, whether it could be proved? Answering it meant first saying what "mechanical" actually meant, and nobody had done that.

Here is the surprise for anyone who has seen the textbook picture: the paper contains no drawing of a tape and a head at all. Turing's diagrams are tables. His first example has four states, written in Gothic letters as b, c, e and f, and its table runs to four lines: print 0 and move right, move right, print 1 and move right, move right, then start again. Left running, it writes 0 1 0 1 on alternate squares for ever. The redraw above sets that tape beside that table, which is how generations of lecturers have chalked it since.

Then he turned the machine on itself. Using a diagonal argument borrowed from Georg Cantor, he showed that no machine can examine the description of an arbitrary machine and decide whether it will ever print a 0. Today a close cousin of that result is called the halting problem, a name added later by others. It meant Hilbert's dream was impossible: there is no general mechanical test for provability. Church had reached the same verdict a few weeks earlier by a different route, and Turing went to Princeton to work with him, but it was the tape-and-table picture that convinced people, because anyone could imagine operating it.

In 1947 there were only a handful of electronic computers in the world, and most of them were being programmed by people plugging cables into boards. At the Institute for Advanced Study in Princeton, Herman Goldstine and John von Neumann were designing a machine that would hold its instructions in memory, and they faced a problem nobody had faced before. How do you plan a long sequence of coded orders, full of jumps and repetitions, so that it does what you meant? Their answer filled a report with the unglamorous title Planning and Coding of Problems for an Electronic Computing Instrument , and at its heart was a new kind of picture: the flow diagram.

So they invented a small vocabulary of boxes. Operation boxes held the arithmetic. Substitution boxes, written with an arrow, changed the value of a counting variable, so that "i + 1 → i" meant add one to i. Alternative boxes tested the sign of a quantity and had two exits, one marked plus and one marked minus, so control could flow down either path. Assertion boxes, drawn off to the side, did not do anything at all: they stated what must be true at that point, a remarkably modern habit that later grew into the formal proof of programs. Boxes were numbered with roman numerals, joined by arrows that showed the flow of control.

The flow diagram spread almost as fast as the computers did. By the 1950s it was the standard way to design and document programs, and in 1970 the American National Standards Institute fixed the shapes, while IBM and others sold green templates so that anyone could draw them neatly. Generations of programmers drew the flowchart first and wrote the code second, and when managers wanted to see what a program did, the flowchart was what they were shown.

The most influential computer design of the twentieth century was written for a machine that did not yet exist, by a man who was not on its payroll, in a document that was never finished. In the spring of 1945 John von Neumann, the Princeton mathematician already busy at Los Alamos, was a consultant to the team at the University of Pennsylvania's Moore School that had built ENIAC. ENIAC was fast but maddening: to change its problem, operators rewired it by hand, which could take days. The team, led by J. Presper Eckert and John Mauchly, was already planning a successor called EDVAC that would keep its instructions in memory. Von Neumann wrote up its logic, reportedly in part on train journeys to Los Alamos, and on 30 June 1945 Herman Goldstine circulated the result: First Draft of a Report on the EDVAC .

The decisive idea is in the memory. Instructions and numbers are stored side by side, in the same kind of cells, encoded the same way. The control fetches an instruction from memory, the arithmetic unit carries it out, and the control fetches the next one, which may be the one after it or one found by a jump. Because programs are just numbers, a program can be loaded as quickly as data, and could even modify itself. Something like this had been anticipated by Turing's universal machine, and Eckert and Mauchly would argue that the stored program was theirs; the draft carried only von Neumann's name, which caused a lasting rift.

Circulating the draft so widely had a consequence its authors may not have intended: it counted as publication, so the stored-program idea could not be patented, and anyone could use it. People did. The Manchester "Baby" ran the first stored program in June 1948, EDSAC in Cambridge began regular work in 1949, and the design spread to Princeton, to IBM and eventually to every desktop and pocket.

In July 1948 the Bell System Technical Journal , a house magazine for telephone engineers, published the first half of a paper by a 32-year-old mathematician called Claude Shannon. A Mathematical Theory of Communication ran across two issues, July and October, and near its start sat a figure of disarming plainness: five boxes and a small square, joined by arrows. Shannon labelled it "Schematic diagram of a general communication system". It may be the most consequential line drawing ever to appear in a trade journal.

It could do this because Shannon deliberately threw meaning away. In a line that is quoted wherever information theory is taught, he wrote that "the fundamental problem of communication is that of reproducing at one point either exactly or approximately a message selected at another point." The semantic content, he added, was irrelevant to the engineering problem. All that mattered was that the message had been selected from a set of possible messages, and the size and statistics of that set could be measured. The unit of measurement he called the bit, crediting the word to his colleague John Tukey; the paper is the first place it appeared in print.

Shannon did not say how to reach that limit, and the hunt took half a century. Error-correcting codes protected pictures sent back by space probes; compression shrank music into MP3s and photographs into JPEGs; turbo codes in the 1990s finally came close to the capacity he had predicted. Every mobile phone, Wi-Fi router and broadband modem is an engineered attempt to live up to that formula.

In 1937 a 21-year-old graduate student at the Massachusetts Institute of Technology noticed something that, in retrospect, seems almost too neat. Claude Shannon had been tending Vannevar Bush's differential analyser, a room-sized calculating machine controlled by hundreds of relays, and he had also taken a philosophy course on the logic of George Boole. In his master's thesis, A Symbolic Analysis of Relay and Switching Circuits , published in 1938, he showed that the two were the same subject. Switches in series behave like Boole's AND; switches in parallel behave like OR. Any circuit could be written as an equation, simplified on paper, and built with fewer parts.

The United States military, which was buying a great deal of digital electronics for missiles and computers, codified the shapes in a standard called MIL-STD-806B in 1962. Industry followed, partly because military contracts demanded it and partly because the shapes were simply easy to read. A rival system of rectangles marked with "&" and "≥1" was later adopted by the International Electrotechnical Commission and the IEEE, but on most engineers' whiteboards the old distinctive shapes never went away.

This is not merely a curiosity. Chip makers love a single universal gate because one well-made transistor pattern can be repeated millions of times. The Apollo Guidance Computer that helped land men on the Moon was built from integrated circuits that each held simple NOR gates, around 5,600 of them doing all the work. Today's processors contain billions of gates, and their designers still sketch the critical paths in D-shapes and bubbles.

At the start of the 1960s the most frightening question in American defence planning was not whether a nuclear war could be started but whether it could be controlled. If a Soviet first strike destroyed the telephone exchanges, how would anyone order a response, or order a halt? At the RAND Corporation, the Air Force's think tank in Santa Monica, a soft-spoken engineer called Paul Baran set himself the task of designing a communications network that could take a beating and still carry messages.

The figure makes its argument before a word is read. Knock out the hub of the star and every node is isolated. Knock out a few hubs of the second network and whole regions fall silent. Knock out nodes in the fishnet, even a great many of them, and there is usually still a path from one side to the other. Baran backed the intuition with computer simulations, and found that a surprisingly modest amount of redundancy, each node linked to three or four others, gave a network that survived very heavy damage. It did not need expensive, hardened equipment, just plenty of cheap links.

The idea met resistance. AT&T, whose network was built for continuous circuits, was unconvinced that a digital, chopped-up system could work, and the Air Force project eventually stalled. But the memoranda were public, and in 1967 Lawrence Roberts, planning a computer network for the Pentagon's Advanced Research Projects Agency, heard of both Baran's and Davies's work. Packet switching became the foundation of the ARPANET, and so of the internet.

The internet's first map fits on the back of an envelope, and very nearly was drawn on one. In December 1969 someone sketched the entire network of the Advanced Research Projects Agency in a few strokes of a pen: four large circles, four small ones, and a handful of lines. The note at the top reads, in effect, "The ARPA Network, Dec 1969, 4 nodes". The authorship of the sketch is uncertain, but its content is not. It shows everything there was.

The small circles were the trick that made it possible. Rather than teach each big computer to handle the network, the designers gave every site an Interface Message Processor, a ruggedised Honeywell minicomputer whose only job was to pass packets of data along leased telephone lines running at 50 kilobits per second. The contract to build these IMPs went to Bolt Beranek and Newman, a consulting firm in Cambridge, Massachusetts, where a small team under Frank Heart delivered the first one to UCLA at the end of the summer of 1969, ahead of schedule.

What makes the drawing remarkable is what came after. The network's builders kept redrawing its map as it grew, at first monthly, adding nodes as university after university came online. By 1970 it had reached the east coast; by 1973 links reached across the Atlantic to Norway and London, and the logical maps were crowded with dozens of sites. By the 1980s nobody could draw the whole thing on a single sheet any more, and when the network became the internet the very idea of a complete map became absurd.

By the early 1970s the world's databases were in a muddle of competing philosophies. Some stored records in rigid hierarchies, like IBM's IMS, built for the Apollo programme. Others, following the CODASYL committee, stored them as networks of pointers. And in 1970 Edgar Codd at IBM had proposed something cleaner: the relational model, in which everything lived in simple tables. Each camp had its own vocabulary, and the managers who were paying for the systems understood none of them. What was missing was a way to describe the data itself, before anyone argued about how to store it.

The notation was disarmingly simple. Entity sets, the nouns, were rectangles: employees, departments, projects, suppliers, parts. Relationship sets, the verbs, were diamonds placed on the lines between them. The numbers written on the lines told you how many of one thing could be linked to the other. A department employs many employees, but each employee belongs to one department, so the line reads 1 at one end and N at the other. Employees and projects are linked many to many, M and N. A diamond could join three entities at once, as when a supplier supplies a part to a project. A double-bordered box marked a weak entity, such as an employee's dependants, which only make sense attached to someone else.

Popular memory credits Chen with the crow's foot, the three-pronged mark that shows the "many" end of a relationship. That is a myth of convenience: the crow's foot came from Gordon Everest's notation of the same year and was popularised in the 1980s by the Information Engineering methods of James Martin and Clive Finkelstein. Chen wrote numbers and letters on his lines. The crow's foot simply proved easier to sketch quickly, and today most diagramming tools offer both.

In the late 1970s computer networks were multiplying like dialects. IBM had its Systems Network Architecture, Digital Equipment had DECnet, the American research community had the ARPANET and its new TCP/IP protocols, and each worked beautifully with itself and not at all with anyone else. In 1977 the International Organization for Standardization, based in Geneva, set up a committee to fix this, under the chairmanship of Charles Bachman, an American database pioneer. Its task was to define Open Systems Interconnection: a way for any computer to talk to any other, whoever made it.

The diagram's genius was a rule of manners. Each layer talks only to the layer directly above and below it in the same machine, and, conceptually, to its opposite number in the other machine. The transport layer in one computer need know nothing about whether the bits travel by cable or satellite; it simply trusts the layers beneath. The routers in the middle of the journey need only the bottom three layers. Data heading out is wrapped in a new header at each layer on the way down, and unwrapped on the way up, like a letter inside an envelope inside a mail sack.

Yet the seven-layer picture won a stranger victory. TCP/IP does not quite fit it, having merged the top layers into one and never really used a session layer, but engineers adopted the model as a common vocabulary regardless. A switch is a "layer 2" device; a firewall that reads web traffic is "layer 7"; a fault in the cabling is "a layer 1 problem". Students still learn the stack with mnemonics such as "Please Do Not Throw Sausage Pizza Away", and help desks joke about "layer 8", the user.

In March 1989 a 33-year-old British software engineer at CERN, the European particle physics laboratory on the edge of Geneva, handed his manager a document called Information Management: A Proposal . Tim Berners-Lee was not proposing to discover a particle. He was complaining, politely, about paperwork. CERN employed thousands of physicists and engineers who came and went on short contracts, working on dozens of incompatible computers, and when someone left, what they knew tended to leave with them. Nobody could find out who had written which program, which experiment used which magnet, or where a crucial note was kept.

The argument was aimed at trees. Most information systems of the day filed things in hierarchies, like folders inside folders, and Berners-Lee pointed out that real organisations do not work that way. A person belongs to a group, works on a project, uses a machine and wrote a paper, and each of those things connects to many others. What was needed was a web of nodes and links in which anything could point to anything, and in which the links themselves were stored and followed by the computer. He had built a small private version years earlier, a program called ENQUIRE, named after the Victorian household guide Enquire Within Upon Everything .

By Christmas 1990 he had written the three inventions that still hold it together: a way of naming any document anywhere, now called the URL; a simple protocol for fetching documents, HTTP; and a markup language for writing them, HTML. The first web server, info.cern.ch, ran on that NeXT machine with a handwritten label warning people not to switch it off. In 1991 the software was released to the wider internet, and on 30 April 1993 CERN put it into the public domain, so that anyone could use it without paying a licence fee.

In the late 1880s a Spanish doctor with an artist's hands sat at a microscope in Barcelona and drew what he saw, and what he saw settled one of the great arguments in biology. Santiago Ramón y Cajal had wanted to be a painter as a boy. His father, a surgeon who taught anatomy, wanted something more respectable, and took the boy to graveyards to study bones. Cajal compromised in the way stubborn people do: he became an anatomist who drew.

Cajal saw the method in Madrid in 1887 and set about improving it, staining thicker sections twice and choosing young animals, especially birds, whose nerve fibres had not yet been wrapped in fatty insulation. In 1888 he published his findings on the cerebellum in a journal he founded and paid for himself. His conclusion was blunt. The fibres ended freely. They touched other cells, wrapped around them, crowded against them, but they did not fuse. Each nerve cell was an independent unit. Golgi had used his own stain to see a net; Cajal used it to see individuals.

Persuasion took time and travel. In 1889 Cajal carried his slides to a meeting of anatomists in Berlin and won over the eminent Albert von Kölliker, who learned Spanish to read him. In 1891 Wilhelm Waldeyer gathered the evidence into what became the neuron doctrine, and in 1897 Charles Sherrington named the meeting point between cells the synapse. In 1906 Cajal and Golgi shared the Nobel Prize in Physiology or Medicine, an awkward double act: Golgi used his lecture to defend the reticular theory. The gap itself could not be seen until electron microscopes revealed it in the 1950s, by which time Cajal had been proved right for decades.

For several years in the late 1870s and early 1880s, a young German philosopher named Hermann Ebbinghaus spent his days memorising gibberish. He built some 2,300 syllables of the form consonant, vowel, consonant, such as DAX, BOK and YAT, chosen precisely because they meant nothing. He shuffled them into lists, read each list aloud at a steady pace until he could recite it twice without error, and wrote down how long it had taken. Then he waited, and tested himself again. He was the experimenter, the subject and the entire laboratory.

His cleverest move was the measure itself. Asking whether he remembered a list was too crude, since after a day almost nothing could be recited. Instead he relearned the list and timed how much faster the second learning went. If the first round took 1,000 seconds and the relearning took 600, he had saved 40 per cent. This savings method detected traces of memory too faint to recall, rather as a sensitive balance weighs what the hand cannot feel.

The drawing does a great deal of work. As a table, the figures look like a list; as a curve, they become a law of nature, and the steepness of that early cliff is a visceral shock. The curve also carries a second message that Ebbinghaus found in his data. Spreading repetitions over several days was far more efficient than cramming them into one sitting, and every relearning left the slope gentler than before. Drawn as a sawtooth of reviews, the forgetting curve becomes a recipe.

His results have also survived the hardest test in science, repetition. In 2015 the Dutch researchers Jaap Murre and Joeri Dros spent some seventy hours relearning nonsense syllables in the same way and produced a curve strikingly similar to the original, complete with its early cliff. Ebbinghaus, a single subject armed with a stopwatch and an extraordinary tolerance for boredom, had founded the experimental psychology of memory, and his curve remains one of the few in psychology that almost anyone could redraw from memory.

Ask almost anyone to draw Freud's theory of the mind and you will get an iceberg. A small white peak glints above the waves, labelled conscious; a band just under the surface is the preconscious; and below, vast and dark, sits the unconscious, the part that sinks ships. It is one of the most recognised diagrams in psychology. Freud never drew it.

At the top of the egg is a cap marked for perception and consciousness, the surface that meets the world. Just below lies a dotted region for the preconscious. The ego occupies the upper part of the egg, and its lower part merges without a hard border into the id, which fills the bottom. On one side a pair of lines marks off the repressed, cut off from the ego by resistance but still part of the id. On the other side perches a little cap for hearing, since Freud thought words reach the ego largely through the ear. The super-ego, central to the same book, is missing from the drawing; Freud added it to a revised version in his New Introductory Lectures of 1933, and even then warned readers not to take the proportions literally.

The iceberg won because it makes an argument the egg only hints at. It says that what you are aware of is a small fraction of what drives you, and it says so through physics everybody knows: roughly nine tenths of floating ice lies beneath the surface. It turns a contested claim into common sense. It also quietly drops the parts of the theory that are hardest to picture, such as the conflict between agencies, in favour of a single fact about depth.

That makes Freud's Iceberg a peculiar entry in a history of diagrams. It is a picture that improved upon the theory it illustrated, at least as rhetoric, and then outlived it. The original drawing in Vienna was a careful, uncertain attempt to combine two models. The famous one belongs to everybody and to nobody, which is exactly why it is so hard to sink.

The figure is grotesque, which is why nobody forgets it. A little man lies draped over a slice of brain, his head hanging down the side, his feet tucked over the top. His legs and trunk are spindly. His hand is enormous, a vast palm and fingers like sausages. His lips swell into a great pout, and his tongue lolls out below. He is the homunculus, Latin for little man, and he is a map of how much brain is devoted to feeling each part of the body.

The answers were startlingly specific. Stimulating one point produced tingling in the thumb, another a twitch of the lips, another a sensation in the toes. Over hundreds of operations a pattern emerged. Along a strip of cortex just behind the central fissure, called the postcentral gyrus, the body was laid out in order, toes and legs near the top of the head, then trunk, arm and hand, then face, lips and tongue down the side. A matching strip just in front controlled movement. The body was mapped onto the brain, but the map was wildly distorted.

The drawing makes an argument that a list of stimulation sites could not. The size of each part is not its size on the body but its importance to the brain. Hands and lips are huge because they are our finest instruments of touch and manipulation, packed with receptors, while the broad back is a desert. Penfield himself regarded the drawing as a cartoon rather than an exact chart, and later studies showed that the map is messier, with overlapping territories.

The map is still being revised: in 2023 researchers using brain imaging reported regions linked to whole-body action interleaved among the classic motor territories. Yet the little man remains on posters, in textbooks and as lumpy sculptures in science museums, a portrait of ourselves as the brain sees us.