Measuring the Invisible

When you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind.

The thermometer that could not trust itself

Everything the book has measured so far could, at least in principle, be touched: arms of cloth, arcs of meridian, hours of voyage. With this chapter the story changes nature, because the nineteenth century had to learn to measure magnitudes that no body embodies and no standard can wall in: heat and current, temperature and electricity. And the lesson begins with a paradox so clean it seems invented by a philosopher, while it is the practical problem number one of early thermometry. To build a thermometer one must calibrate it, that is, mark on its scale at least two points of known temperature; but to know that a phenomenon occurs “always at the same temperature” — that the ice melts today as yesterday, here as in Peking — one would need a thermometer already calibrated. The instrument presupposes the fixed point, and the fixed point presupposes the instrument: the circle is perfect, and thermometry was born inside it [Chang 2004].

The first to go round it were the experimenters of the seventeenth century. The thermoscopes of Galileo and his correspondents — a flask of air breathing over a little column of water — showed that something changed, not how much: they were instruments without a scale, as capricious as the atmospheric pressure that disturbed them. The Accademia del Cimento, in Florence, made the leap in quality in mid-century with its sealed spirit thermometers, blown by glassmakers so skilled that the surviving exemplars, compared today, still agree among themselves to a surprising degree: masterpieces of artisanal reproducibility, yet mute on the underlying question — fifty “Florentine degrees” of what? To anchor the scales, reliable natural phenomena were being sought, and the inventory of candidates is a small anthropology of trust: ice melting, water boiling, butter dissolving, the temperature “of the blood” of a healthy man, deep cellars — those of the Paris observatory, some thirty metres beneath the pavement, enjoyed for over a century the reputation of constant refrigerator of European physics, and generations of instruments descended there in pilgrimage of calibration. Every candidate had its devotees and its sceptics, and the sceptics were usually right: the human body, for one, was long defended as the most democratic of standards — everyone carries his own — until it was admitted what every doctor knew, that the blood of the healthy is not so punctual after all, and that fever, digestion, and seasons shift it enough to spoil a scale. Even Newton had tried his hand at the game: his scale of “degrees of heat,” published anonymously in 1701, chained a dozen fixed points in ascending order — from winter air to the human body, from melting tin to glowing coals — joining them with a hypothesis about the rate of cooling that still bears his name. The selection was, in short, a long, contentious, and instructive process, from which the surviving points — the ice, the vapour — emerged not because self-evident, but because they had passed a century of trials of intent [Chang 2004].

Instructive above all in the details, where as always the devil dwells. Take the most solid of the fixed points, the boiling of water: it boils at a hundred degrees, says the schoolroom. But it boils at a hundred only at the right pressure — the early thermometrists noticed this by taking their instruments up the mountains, where water boils sooner, and it was thus indeed that the thermometer became an altimeter too — and above all it boils at a hundred only if it is allowed to: very pure water, in very smooth vessels, heated very calmly, can exceed the threshold by several degrees without boiling, in a tense equilibrium that then breaks with a jolt. The eighteenth-century experimenters knew these caprices well, and the conclusion they drew from them is the first lesson of the chapter: a fixed point is not a fact one finds, it is a procedure one establishes. “The boiling point” became, by written convention, the temperature of the vapour over water boiling vigorously at standard pressure, in an apparatus made thus and so: nature furnishes the phenomenon, but reproducibility — the reader knows it from the first chapter — is furnished by the protocol. For the invisible too, to measure is first of all to agree on a rite [Chang 2004].

And where the rite did not reach, one invented from scratch. Above four hundred degrees the thermometers of the age simply died — the glass softens, the mercury boils — and it was a potter who first colonized that terra incognita: Josiah Wedgwood, the great English ceramist, who lived by high temperatures. His “pyrometric pieces” were little cylinders of calibrated clay that in the kiln contract in proportion to the firing undergone: by measuring their shrinkage, Wedgwood built a scale of furnace heats that industry adopted with gratitude — only to discover, when it was tried to join it to the scale of the thermometers, that Wedgwood’s degrees would not let themselves be converted: the clay told its own thermal history, not the temperature of any other instrument. The episode, which Chang has brought into full light, is the proof by absurdity of the whole chapter: a measure can be internally reproducible and remain an island — until it joins the network of the others, it is a dialect, not a language [Chang 2004].

The scales and the fluids

Within this workshop were born the scales we still carry about us. Daniel Gabriel Fahrenheit, a Prussian instrument-maker transplanted to Holland, was the virtuoso of the generation: his mercury thermometers — it was he who imposed the liquid metal, docile and brilliant — agreed with one another, a rare commodity at the time, and his scale spread with his instruments, as always happens: the standard travels inside the well-made object.

Anders Celsius, an astronomer of Uppsala, proposed in 1742 the centigrade scale between the two points of water — with a detail the schoolrooms ignore and which is a delight: his scale was inverted, zero at boiling and a hundred at the ice, and was overturned into the present form only after his death, by his colleagues. Réaumur had meanwhile divided the same interval into eighty parts, and Europe lived for over a century with three competing scales, converting degrees as Pegolotti’s merchant converted arms: temperature too had its age of local measures.

But the true divergence was deeper than the scales, and touched the heart of the matter. Two thermometers calibrated on the same fixed points — same zero, same hundred — agree by construction at the extremes; in between, they agree only if their fluids expand in the same way. And they do not: mercury marks its fifty where alcohol marks another figure, and the air thermometer another again. Which tells the truth? The Genevan Jean-André De Luc devised, to decide it, one of the conceptually finest experiments of the century: to mix equal weights of boiling water and freezing water — the mixture, he reasoned, must stand halfway — and to see which thermometer really marks the midpoint. Mercury won, and for a generation it seemed the last word; then it was understood that the experiment presupposed what it meant to prove — that mixing half and half gives the half, which is true only if the specific heat of water does not change with temperature, a thing that would need verifying… with a thermometer. The circle, driven out the door, came back through the window: the question “which fluid tells the truth” is a trap, because “the truth” would presuppose a notion of temperature independent of every thermometer, which is exactly what did not yet exist. The answer of the early nineteenth century was pragmatic and profound at once, and it was given above all by the Frenchman Victor Regnault, the most scrupulous measurer of the century: since we cannot ask the fluids for the truth, let us ask them for comparability — and gas, which of all fluids is the simplest and the most uniform in behaviour, was promoted to reference fluid not because “right,” but because gas thermometers, of different gases, agreed among themselves better than any other family. Hasok Chang, the philosopher-historian who has reconstructed this affair, has read in it the general model of metrological progress and has called it epistemic iteration: one starts from an imperfect instrument, uses it to establish provisional regularities, the regularities permit a better instrument, which corrects the regularities of departure — and so on, in a spiral. It is not a vicious circle: it is a circle that climbs. The knowledge of the invisible does not rest on a rock, it lifts itself by its own bootstraps — and it works [Chang 2004].

Temperature without a thermometer

That temperature had a floor, meanwhile, the most theoretical of thermometers had long been whispering. Already around 1700 Guillaume Amontons, observing how the pressure of sealed air fell regularly with the cold, had dared the extrapolation: if the falling continued at the same pace, at a certain point the pressure would vanish — and there, whatever it means, the cold could grow no further. The gas thermometer did not merely measure: it indicated a direction, it pointed like an arrow toward a zero that no laboratory knew how to reach. For a century and a half the arrow remained an arithmetical curiosity; the theory was lacking that would give it the right to mean something.

The conceptual stroke came in 1848 from a twenty-four-year-old of Glasgow, William Thomson, whom we shall find again shortly on a steamer in the middle of the Atlantic. Meditating on the work of Sadi Carnot — the abstract theory of ideal heat engines — Thomson overturned the problem of thermometry: instead of asking the fluids to define temperature, one could define it without naming any substance, through the work that heat can perform in falling from a hot body to a cold one. In the “absolute” scale that results, a degree is worth the same everywhere because it is defined by the efficiency of an ideal machine, not by the expansion of this or that liquid; and the scale has a zero not conventional but physical — the floor below which there is no more heat to give up — which the calculations placed at about two hundred and seventy-three degrees below the zero of the ice. Temperature, born prisoner of its instruments, received a definition that no thermometer could soil; the gas thermometers, from pretenders to the truth, retreated to the right rank: practical realizations, excellent earthly approximations of a celestial idea. In the following years Thomson settled the account with experience together with James Prescott Joule — the Manchester brewer who had weighed the equivalence between work and heat — measuring with him the minute deviations between real gases and the ideal: from that collaboration the absolute scale came out not only defined but calibrated, with tabulated corrections for translating the readings of earthly thermometers into the language of the Carnot machine. The reader will recognize the move, for it is the same this book has pursued from the first chapter: measure emancipating itself from the body — first from the king’s arm, then from the platinum bar, now even from the fluid in the capillary. There remained, of course, all the dirty work: to translate the ideal definition into instruments, fixed points, and practical tables, an enterprise that would occupy metrology for a century. Its mature form is the international temperature scale, agreed for the first time in 1927 and periodically renegotiated: a thermometric constitution that lists a scale of fixed points — the freezings and meltings of water, gallium, zinc, silver, gold, each with its ritual procedure — and prescribes with which instruments to interpolate between one and another. It is the old game of the fixed points of the seventeenth century, promoted to a planetary treaty: the difference lies not in the logic, which is identical, but in the network that holds it up. But the hierarchy was overturned forever: from then on theory defines, the instrument realizes — and when, in 2019, Thomson’s degree is reanchored to a constant of nature, the Boltzmann constant, it will be only the last bend in a road taken in that 1848 [Chang 2004].

The magnitude without a body

If temperature at least is felt on the skin, electricity grants not even this. It is not seen, not weighed, not unrolled on the counter: one sees its effects — the compass needle that deviates, the frog of Galvani that kicks, the spark, the shock — and every effect suggests an instrument and a possible definition. The eighteenth century had treated it as a drawing-room marvel, and had taken its leave of it with an Italian coup de théâtre: the dispute between Galvani, convinced he had found in frogs an animal electricity, and Volta, convinced that the electricity lay in the metals and the frog served only as an instrument of measure — the most sensitive electroscope of the age, in fact, was a frog’s leg. To win the dispute Volta built in 1800 the pile, the metal frog: and with direct current at his disposal, the early nineteenth century could begin to count. Coulomb had already weighed its forces with the torsion balance; Ørsted and Ampère chained it to magnetism; and in 1827 a then-obscure Bavarian professor, Georg Simon Ohm, enunciated the proportionality every electrician has recited ever since — voltage pushes, resistance holds back, current results, $V = RI$. But the magnitudes thus connected remained without shared units: each measured with his own galvanometer, in degrees of deflection of his own needle, and two laboratories could not compare a result without physically exchanging instruments. Electricity was a science with laws and without measures — until two Germans made the radical gesture. In the thirties, at Göttingen, Gauss and the physicist Wilhelm Weber showed that the magnetic and electrical magnitudes could be expressed in “absolute” units, that is, derived from length, mass, and time alone: the Earth’s magnetism measured in millimetres, milligrams, and seconds, with no dedicated electrical standard — no bar to wall in, no cylinder to keep, only definitions and mechanics. It was an idea of enormous power: for the magnitudes without a body, the standard would not be an object but a network of relations anchored to the fundamental units. And Gauss and Weber gave it at once also the social body it deserved: the Magnetic Union of Göttingen, a network of observatories scattered from Ireland to Siberia that, on days and hours prescribed by the common calendar, measured the Earth’s magnetism simultaneously with twin instruments and identical protocols — the first coordinated measurement campaign on a planetary scale in history, the “Göttingen hours” that made observers of ten nations rise at the same minute. The invisible, unable to have a walled standard, had first what all measures would have afterward: a synchronized network. There remained to convince the world to use the absolute units; and the world, as usual, was convinced only when imprecision began to cost [di) 1995].

The cost came with the telegraph, the first electrical industry in history. Within twenty years of the first lines, the wires covered the continents: the operators soon found that to locate a fault or specify a contract one had to say how much resistance, and each said it in his own way — in “miles of copper wire number such-and-such,” in the unit of his own company, his own country, his own supplier. The attempts to bring order retraced, in miniature, the whole history of standards: the physicist Moritz Jacobi sent his colleagues across Europe copies of a standard wire of his, so that the laboratories at least might speak the same resistance — the walled arm in postal format — and the great companies kept their own “house ohms” in padded boxes, calibrated against who knows what. It worked as far as it could: that is, as long as the wires stayed short and the contracts national. The first chapter of this book, transferred bodily into the industrial age: local units, embodied in the objects of work, perfectly functional as long as the network is local — and the network was about to cease being so in the most spectacular way possible.

The cable that died of imprecision

In August 1858, after two failed attempts and a summer of storms, two reconverted warships met in the middle of the Atlantic, spliced their respective halves of a cable three thousand kilometres long, and sailed in opposite directions, lowering it to the bottom. When the Agamemnon touched Ireland and the Niagara Newfoundland, the two worlds were electrically joined: Queen Victoria exchanged messages of jubilation with President Buchanan — the queen’s telegram, a hundred words or so, took sixteen hours to pass — and the American cities celebrated with fireworks the abolition of the ocean. Three weeks later, the cable was dead. It had transmitted, in all, a few hundred ever-fainter messages; then silence, and with the silence the collapse of the shares, the mockery of the newspapers, and the most stinging technological humiliation of the century [di) 1995].

The autopsy of that costly corpse is a watershed in the story we are telling. The chief electrician of the project, Wildman Whitehouse, a doctor by training and a self-taught experimenter of some worth, belonged to the last generation of the old school: electricity as a practical art, governed by experience and the practised eye, to be mastered by force — and faced with the signals the cable returned smeared and languid, his answer was to raise the voltage, with induction coils as tall as a man, up to thousands of volts that cooked the insulation of gutta-percha. The scientific consultant of the project, that William Thomson we left defining temperature, belonged to the new school: he had calculated in advance, with his theory of transmission in cables, that a submarine conductor of that length would necessarily smear every impulse — the delay grows with the square of the length — and that the way was not to shout louder, but to listen better: purer copper, calculated cross-sections, and at the receiver an instrument able to read minuscule currents. He built it himself: the mirror galvanometer, a fragment of magnet with a mirror glued to it as light as a fly’s wing, which translates the faintest current into a dance of a beam of light on a graduated scale. For twenty years the transoceanic telegraphers worked thus, in darkened cabins, reading letter by letter the trembling of a point of light — a craft halfway between the astronomer and the medium — until Thomson himself automated even that, with a siphon recorder that wrote the signals in ink on a strip of paper: the measure of the invisible, born as a perceptual virtuosity, ended as usual in a trace anyone can reread. The commission of inquiry that the government and the company instituted — the first great public technical inquiry in history, mother of all those on bridges, airships, and shuttles — distributed the blame with more equanimity than did the legend, which chose its roles at once: Whitehouse the charlatan, Thomson the prophet. The reality is more interesting than the caricature: Whitehouse was no fool but a man of the transition, formed in a world where electricity was not calculated — and his failure marked exactly the point at which that world became uninhabitable. After 1858, no serious electrical project would ever again entrust itself to the practised eye: electrical engineering was born quantitative or it was not born at all [di) 1995].

Among the reforms that prepared the revenge, one deserves a place of honour in the history of measure, for it inaugurated an institution today omnipresent: the testing of materials. Thomson had discovered, by measuring, that commercial “copper” was a lottery — batches of nominally identical wire conducted electricity differently by as much as half, according to the impurities of the lot — and imposed that every batch destined for the cable be verified at the galvanometer before purchase, with thresholds of conductivity written into the contract. Quantitative quality control was born: the supplier no longer sold “copper,” he sold copper with a number, and the number commanded a premium on the price. Within a few years the refiners, pursuing the telegraphers’ specifications, learned to produce a copper purer than chemistry could certify — the market, once again, had become a laboratory. The revenge came in 1866: the Great Eastern, the one hull in the world able to carry the whole cable, laid the new line — designed this time on measured specifications, with the copper tested batch by batch and the joints verified at the galvanometer — and even fished up, in an exploit out of a novel, the cable lost the year before, completing it. This time everything worked, and forever. Thomson gained the knighthood, a fortune in patents, and later the title by which physics knows him.

The factory of ohms

The institutional lesson of the disaster was drawn with notable speed. In 1861, at the urging of the telegraph engineers and with Thomson among the prime movers, the British Association for the Advancement of Science — the BAAS, the great itinerant synod of Victorian science — instituted a committee for electrical standards, entrusting its work to a team that today makes the wrists tremble: Thomson, the engineer Fleeming Jenkin as secretary and organizational soul, and a young Scottish theorist named James Clerk Maxwell. The mandate seemed technical — to choose and realize a unit of resistance — and was in fact constitutional: the committee decided that the electrical units of the empire, and ideally of the world, would be absolute in the sense of Gauss and Weber, derived by theoretical route from length, mass, and time on a metric basis, and at the same time materialized in practical standards to be distributed. Not the one or the other, but the theory plus the artefact: the ideal definition to guarantee universality, the tangible copies to serve the workshop. The reader recognizes the architecture, for it is the same as the metre’s — the ideal (the meridian, the Carnot machine, the absolute unit) above, the standard (the bar, the gas thermometer, the coil) below — and it is the architecture metrology has never since abandoned [Schaffer 1992].

The realization was a famous experiment: in 1863, in a basement of King’s College in London, Maxwell and Jenkin spun for months, at a timed speed, a coil in the Earth’s magnetic field, deducing the absolute resistance from the deflection of a little magnet suspended at the centre — the ohm extracted, to the letter, from the rotation of a coil and the magnetism of the planet. From the value thus fixed the committee had the standards built: coils of a stable alloy, sealed in paraffin, marked and numbered, sent to the laboratories and companies of half the world for a fee — and here the historian Simon Schaffer has shown the decisive other side of the coin: what rhetoric called the “absolute unit” issued in fact from a manufacture, with its workshop secrets, its production rejects, its more or less successful copies, and Britain, selling certified ohms to the planet, sold with them its hegemony over the new industrial physics. Precision, once again, was not only knowledge: it was a commodity, and the geopolitics of the commodity. Germany answered indeed with a rival standard — the unit of Werner Siemens, a column of mercury one metre long of calibrated cross-section, reproducible in any equipped laboratory without bowing to the English coils — and for twenty years electrical Europe lived its little schism of the ohms, with converters and polemics, until diplomacy did its work [Schaffer 1992].

Before leaving the committee, one must recount the gift its accounting made to physics, for it is perhaps the greatest metrology has ever made to pure knowledge. In the absolute system the electrical magnitudes can be founded in two ways — starting from the forces between charges at rest, or from the forces between currents — and the two roads give the same magnitude two different units. The ratio between the two units is not a pure number: it has the dimensions of a velocity, and nature was not bound to give it any particular value. When Weber and his colleague Kohlrausch measured it, in the mid-fifties, there came out a velocity enormous and strangely familiar: three hundred thousand kilometres a second, or nearly. Maxwell, who handled those units every day for the committee, took the coincidence seriously as only a metrologist could: if from the pure accounting of charges and currents — balances, coils, galvanometers, not a ray of light in the whole experiment — comes out the speed of light, then light is an electromagnetic phenomenon. The theory that followed, with its waves travelling at exactly that speed, is the monument of nineteenth-century physics; and it was born, to the letter, from a comparison between units of measure. Let the reader keep the episode in mind for the end of the book: the speed of light, entered into physics from a ratio between units, will end — in 1983, as we have seen — by defining the unit of length. The circle that closes upon itself over two centuries is the recognizable signature of this story [di) 1995].

To garrison the frontier between theory and workshop, meanwhile, a new institution had been born within the university itself. When Maxwell took over in 1871 the direction of the laboratory just founded at Cambridge with the money of the Duke of Devonshire — the Cavendish — he surprised those who expected a temple of theory: he put metrology at the centre of the programme, and for a decade the flower of England’s young physicists was trained redetermining the ohm, refining the value of the ratio between units, taming galvanometers — to the point that the odd colleague wrinkled his nose at so much “shop physics.” Schaffer has shown how strategic that choice was: the university laboratory was becoming a workshop of precision precisely while the workshop, as we have seen, was becoming a laboratory, and from the Cavendish of those years — not from its speculations, from its measures — would issue the instruments, the men, and the methods of the experimental physics of the next century. To measure the invisible had become the apprenticeship of the highest science [Schaffer 1992].

Diplomacy did its work in the international electrical congresses, the first and most important of which was held at Paris in 1881, while the International Exposition of Electricity dazzled the city with the absolute novelty of artificial light — the visitors climbed in crowds to see the chandeliers of Edison and Swan, and the delegates, a few steps away, discussed in what units those lamps were to be billed: rarely have theory and market kept an appointment with such punctuality. From it came the baptism of the system we still use: the unit of resistance was called the ohm, that of voltage the volt, and then ampere, coulomb, farad — one name per nation, with diplomatic nicety: the German Ohm, the Italian Volta, the French Ampère, the English Faraday. The Italian reader will note with some pride that every battery and charger on the planet bears engraved the name of a professor of Como; and the reader of this book will note the political finesse — the invisible units could not have a place, so they had a pantheon, distributed with the same equity as the urn of 1889. The following decades saw the customary pendulum: for practicality there was adopted in 1893, and refined in 1908, an “international ohm” defined precisely in Siemens’s manner, as the resistance of a column of mercury one hundred and six point three centimetres long — the material standard taking its revenge on the ideal — until, by mid-twentieth century, the absolute units took back the throne for good within the International System.

At the margins of the same congresses was fought, meanwhile, the third war of the invisible, the most ancient and the most domestic: that of light. The gas-lighting industry, and then the light bulb, sold brightness literally — and brightness was billed in “candles,” units that for decades were candles truly: the English candle of whale spermaceti, of regulation weight and consumption, to be burned by protocol and compared by eye on a photometric bench. The Germans answered with a precision amyl lamp, the Hefner flame, reproducible but sensitive to humidity and even to the carbon dioxide of the room; the French staked on the melting platinum proposed by Violle, a laboratory crucible in place of a drawing-room flame; and for half a century international photometry was a negotiation among national candles that repeated, in miniature and in light, the schism of the ohms. Peace came by degrees — an “international candle” agreed among the great laboratories in 1909, then in 1948 the new candle anchored to the black body at melting platinum, that same ideal radiator the previous chapter saw generate the quanta — until in 1979 even the last flame was put out: today’s candela is defined by purely physical means, through the power of the radiation and the conventional sensitivity of the average human eye. In the International System there survives thus, under the most domestic of all names, the ghost of a whale candle: no unit better tells the journey of this chapter, from the shop lamp to the constant of nature. Theory and artefact, ideal and workshop: electrical metrology oscillated between the two poles for ninety years, and it is exactly this oscillation that the next chapter on the kilogram will show concluded in the most radical way.

What exists because it is measured

There remains to gather the conceptual thread, for this chapter has stretched one of the deepest in the book. What are, in the end, temperature and resistance — magnitudes no one has ever seen, defined by ideal machines and spinning coils? The American physicist Percy Bridgman, meditating as an experimenter on these affairs, gave in the twenties the most drastic answer: a physical concept is nothing other than the set of operations by which it is measured — temperature is what the thermometer does. Operationism, taken literally, proves too much: if every instrument defined its own magnitude, the mercury thermometer and the gas thermometer would measure two different things, and science would crumble into so many instrumental solipsisms. But the history we have crossed suggests the right version of the intuition, and it is the one Chang has distilled: the invisible magnitudes exist for us as points of convergence — temperature is not what a thermometer measures, it is what ever more diverse thermometers, built on ever more independent principles, obstinately agree on. The convergence is never guaranteed in advance; when it arrives, it is the strongest sign we have of having hooked something real. The prime example is absolute zero: it was indicated by the extrapolation of Amontons from the behaviour of gases, deduced by Thomson from the thermodynamics of ideal machines, and is confirmed today by roads the two could not have imagined — electrical noise in conductors, the spectrum of radiation, Brownian motion — and all the arrows, shot from different physics, plant themselves at the same point of the scale. No conspiracy of instruments can explain such an agreement: at a certain degree of coherence, the most economical hypothesis is that nature truly has a floor, and that we have measured it. The realism of the metrologist is not the naïve faith in a world already measured: it is the wager, renewed instrument after instrument, that the coherence will go on growing — and for four centuries the wager has paid [Chang 2004].

There is a postscript, which prepares what follows. The electrical units begotten by the committee of the BAAS and baptized at Paris live today a life their fathers would not have dared dream: from 1990 the volt and the ohm of every laboratory in the world are realized no longer with coils nor columns of mercury, but with two quantum effects — the Josephson junctions and the Hall effect of von Klitzing — in which voltage and resistance issue in packets dictated by constants of nature, identical in every cryogenic refrigerator on the planet. The magnitude without a body has found the perfect standard: not an object, but a behaviour of matter, guaranteed by quantum mechanics in person. How this came about, and how it ended by sweeping away even the last of the artefacts — the platinum cylinder shut beneath three bell-jars at Sèvres — is the story toward which this whole book converges. But first a debt opened three chapters ago must be settled: all the measures we have spoken of, from the meridian to the coils, produced columns of numbers that never quite coincided — and we saw a man, at Barcelona, nearly die of it. How science ceased to be ashamed of its own deviations and made of them a mathematics, indeed the most useful mathematics of the modern age, is the next chapter.