Matter as Standard

If we wish to obtain standards of length, time, and mass which shall be absolutely permanent, we must seek them not in the dimensions, or the motion, or the mass of our planet, but in the wave-length, the period of vibration, and the absolute mass of these imperishable and unalterable and perfectly similar molecules.

The prisoner of Sèvres

For a hundred and thirty years the unit of mass of the known universe was a cylinder of metal the size of a plum. Thirty-nine millimetres in diameter, thirty-nine in height, an alloy of platinum and iridium, kept in the underground vault of the pavilion of Sèvres inside three concentric bell-jars, in a safe that opens only with the three keys of three different keepers and in the minuted presence of the authorities of the Convention. The metrologists call it familiarly le grand K: the International Prototype of the Kilogram, drawn by lot among its twins in 1889, and ever since — by definition, to the letter — the one thing in the world that weighed exactly one kilogram. Not by good manufacture: by logical decree. The kilogram was defined as the mass of that object; to ask whether the Grand K really weighed one kilogram was a question without meaning, like asking whether the walled standard of the first chapter was as long as itself [Quinn 2011].

Around the prisoner, the liturgy we now know: the six “witness” copies in the same vault, the eighty-odd national prototypes scattered across the world — each with its urn number and its certificate — and the great periodic verifications, the pilgrimages in which the kilograms of the nations return to Sèvres to compare themselves with the patriarch on balances able to feel the microgram: the first campaign at the turn of the twentieth century, the second around the war, the third between 1988 and 1992. The national prototypes, for that matter, aged each in its own way, according to the metrological character of its owner: those of the zealous countries, drawn often to calibrate the national pyramid, wore themselves out in service; those of the lazy ones slept untouched in their strongboxes — and at every verification the metrologists of Sèvres read in the drifts of the cylinders, as in a blood test, the administrative history of the nations. More than one director of the Bureau has confessed the recurrent professional nightmare of those years: the clumsy gesture, the drop of sweat, the cylinder rolling — the unit of mass of the universe dented on the floor of a cellar [Quinn 2011]. Before every comparison, the most delicate rite of all: the washing — chamois leather, ether and alcohol, distilled-water vapour, by a protocol codified in its least gestures — because a kilogram that lives in an atmosphere, however filtered, breathes: it adsorbs molecules, films itself with contaminations, grows fat on the world by fractions of a millionth of a gram. Already this should prick up the reader’s ears: a standard that must be washed by protocol is a standard whose identity depends on a procedure — the fixed point of the thermometers, again, but applied to the most sacred object of metrology.

The verifications themselves, then, were an exercise at the limit of experimental fakirism. To compare kilograms to the microgram — one part in a billion — means to work in rooms where everything is an enemy: the balances of the Bureau, jewels with equal arms suspended on agate knife-edges and then on metal flexures, had to be read from a distance, because the heat of a human body in the room is enough to create air currents that falsify the weighing; the exchanges between the pans were made with mechanical manipulators, the waits for stabilization were measured in hours, and the whole comparison campaign of a handful of cylinders lasted years. The reader of the sixth chapter will recognize the grammar: repeated weighings in crossed schemes, weighted averages, systematic errors hunted one by one — the theory of errors applied to its most precious object.

It was the third verification that minuted the unease that had been brewing for decades. Compared after a century of life, the official copies and the prototype no longer agreed as in 1889: the deviations had widened, with an average drift of the order of fifty micrograms — fifty billionths of the magnitude in play, the weight of a grain of fine dust, on a cylinder handled in a century fewer than a dozen times. And here the book asks the reader for all his attention, because the subtlety that follows is the heart of the chapter, and hasty popularization almost always smooths it away. The newspaper headlines wrote: “the kilogram is losing weight.” But that is exactly what could not be said. Balances measure only differences: the campaign had ascertained that the Grand K and its copies were moving apart — this, beyond any doubt — but to establish which was moving was in principle impossible, because the one yardstick of comparison was one of the contenders. If the Grand K lost a few atoms at each washing, the copies seemed to grow fat; if the copies adsorbed more than the patriarch, it was he who seemed to lose weight; and by definition the mass of the Grand K was one kilogram, today as in 1889 — so that, by strict juridical logic, it was the whole universe, copies included, that had moved around it. The reader recognizes the paradox of the toise of the Châtelet, the France that “lengthens” with respect to its own deformed iron: but this time it was no antiquarian’s quibble — it was the most precise measurement campaign ever conducted on a fundamental unit, and it returned a unit that fluctuated by an unknowable fifty micrograms a century, with the direction of the fluctuation theoretically inaccessible [Quinn 2011].

The embarrassment

Fifty micrograms on a kilogram: who would ever notice? Almost no one, and it is the first point to clear away: world commerce, the pharmacies, even ordinary laboratories could have lived forever with a wobbly kilogram of five parts in a hundred million. The embarrassment lay elsewhere, and it had three faces. The first was systemic: the kilogram does not work alone. In the International System the units are gears of a single mechanism — the newton is a kilogram times metre per second squared, the joule a newton times metre, the volt a joule per coulomb — and through these chains the uncertainty of the unit of mass filtered into half of physics: every electrical, energetic, mechanical magnitude of precision inherited, at the bottom of its own chain of calibration, the uncertain health of a cylinder in a French cellar. Even chemistry hung from the platinum: the mole, unit of amount of substance, was defined through twelve grams of carbon-12 — and the grams, in the last analysis, were fractions of the Grand K. The atoms of the whole world were counted, by way of definition, against an artefact of 1878. The second face was epistemological, and the reader knows it: a unit defined by an object is hostage to the biography of that object — a fire, a theft, a war, or simply the patient chemistry of the atmosphere, and the unit of mass of the universe changes with its fetish. The third face was almost aesthetic, but in science aesthetics is a serious symptom: at the end of the twentieth century the second issued from the atoms, the metre from the speed of light, the volt and the ohm from quantum mechanics — and amid this republic of natural laws there survived, anachronistic as a thaumaturge king at a congress of cardiologists, a single piece made by hand, cast in 1878 by a London firm [Crease 2011].

That the way out existed, in principle, had been known for over a century: it is the epigraph of this chapter. In 1870 Maxwell — the man we saw manufacture ohms and read light within a ratio of units — had indicated with prophetic exactness where to seek the eternal standards: not in the planet, which is a historical body with its ailments, but in the molecules, “imperishable, unalterable, and perfectly similar.” An atom of caesium does not wear out, does not adsorb, has no biography: every exemplar is identical to every other with the absolute perfection that quantum mechanics guarantees and that no foundry will ever reach — the standard at last democratic, possessed in infinite copies by anyone, anywhere, forever.

Between the programme and its execution, however, lay a century of instruments to be invented: to anchor the kilogram to the atoms one had to know how to count the atoms, or to weigh with the constants — and in 1900 no one knew how to do either. The history of twentieth-century metrology is the history of how both were learned.

Time leads the way

The first unit to make the leap was, as always in this book, time — and it is worth watching the manoeuvre closely, for it served as model to all the others. The second had for millennia been a fraction of the Earth’s rotation; but the Earth, measured with the quartzes of the thirties, proved a mediocre oscillator — it slows for the tides, jolts for the seasons, has the moods of its molten cores — and the astronomers fell back on a fraction of the year, more stable but as inconvenient as only a standard requiring decades of observation can be. Meanwhile, in the laboratories, Maxwell’s alternative was ripening: the atoms oscillate, and their transition frequencies are the spectral lines spectroscopy had been handling for a century. In 1955, at the National Physical Laboratory — the laboratory born of the metrological rivalry of the fourth chapter — Louis Essen and Jack Parry set going the first atomic clock worthy of the name, anchored to a transition of the caesium atom; a transatlantic collaboration with the astronomers allowed one to count how many oscillations of caesium stood in the astronomical second; and in 1967 the General Conference performed the formal act: the second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of caesium-133. The clocks that realize this definition have never ceased to improve — from the nineties the “fountains” of laser-cooled atoms, launched upward and interrogated in their slow fall, have brought the second to a handful of parts in ten million billion — but the definition, note, has not changed by a comma: the recipe withstands the progress of the cooks, which is exactly the merit for which it was chosen. Note the logical structure, for it is the template of all that will follow: first one measures the natural constant in the old unit, until the two are equivalent to the best of one’s ability; then one reverses the arrow — the measured number becomes exact by definition, and from that moment it is the old unit that is realized by means of the constant. The standard is not abolished: it is dematerialized into a recipe [BIPM 2019] .

Atomic time showed at once also how one lives, in practice, with a dematerialized unit. No clock is “the” standard: the official time of the planet is a weighted average — the reader of the sixth chapter will smile — of hundreds of atomic clocks scattered in dozens of laboratories, continuously compared by satellite, each with its weight proportional to its demonstrated stability; the result, International Atomic Time, is an accounting abstraction that no pendulum beats and that all pendulums pursue. And the old deposed mistress, the Earth’s rotation, underwent the fate this chapter reserves for sovereigns: measured against the atoms, it proved officially unreliable, and for decades civil time has pursued it by leap seconds — humanity stops its clocks for one second, now and then, to wait for the planet. The heavenly body that for all of history had defined time is today an object measured by time, watched like a shipboard chronometer of capricious rate: there is no sharper image of the reversal that twentieth-century metrology worked between nature and its judges.

The metre followed in 1983, and we have seen it: light promoted from measurand to measure, with the speed fixed once and for all. And the electrical units lived, from 1990, their anomalous transition that deserves a line, for it is one of the most instructive and least told episodes of recent metrology. The quantum effects discovered in the preceding decades — the Josephson effect junction and the quantized Hall effect of von Klitzing — offered voltages and resistances reproducible to the billionth, dictated solely by constants of nature; but the International System, anchored as it was to the kilogram, could not welcome them without inconsistencies. The laboratories of the world then did a thing formally scandalous and practically inevitable: they adopted for internal use conventional values of the two constants, and for nearly thirty years the volt and the ohm of every calibration certificate on the planet were units in fact outside the System — most exact, most reproducible, and legally stateless. The silent schism between the paper units and the laboratory units was the symptom that the dam was about to give way: quantum physics furnished standards better than those the system knew how to define, and the system, sooner or later, would have to surrender to the evidence. The obstacle was always the same, the last artefact: to give the quantum standards full citizenship one had first to dethrone the cylinder [Stock et al. 2019] .

To complete the picture of the pre-2019 oddities there was the ampere, whose old definition deserves a plaque in the museum of conventions: it was, from 1948, the current that, flowing in two parallel conductors rectilinear, of infinite length, and negligible cross-section, placed in vacuum one metre apart, produces between them a certain force. Infinite wires, in vacuum: the fundamental electrical unit of the planet was defined by an experiment impossible by construction, which no laboratory had ever performed nor could ever perform — it was realized by roundabout means, and the definition stood there like certain never-applied constitutional articles, venerable and useless. The new system has replaced it with the most concrete image there is: the ampere is a river of counted elementary charges — so many electrons per second — even if, an irony for connoisseurs, to count electrons one by one fast enough remains to this day so difficult that the ampere is realized in practice by way of the quantum volt and ohm. The definition has become thinkable again; the realization, as always, follows its own road.

Counting the atoms

For mass, the roads attempted were in fact more than two — and the dead roads deserve their epitaph, for they measure the difficulty of the enterprise. For years a German group pursued the most direct idea conceivable: to accumulate the mass atom by atom, depositing on a target a beam of gold ions counted electrically one by one — the kilogram built like a piggy bank, atomic coin upon atomic coin. The arithmetic was impeccable and the times inhuman: at attainable rates, to put together a few grams required months of uninterrupted beam and an accounting without a single error over quintillions of arrivals; the project was filed away with honour. There survived the two roads that could really reach the end, and their organized rivalry is one of the finest spectacles of recent science. The first is conceptually the most ancient: to realize Maxwell’s programme to the letter, to build a kilogram by counting the atoms that compose it. Put thus it is an absurdity — in a kilogram of matter the atoms are counted with twenty-five figures — but crystals offer the loophole: in a perfect single crystal the atoms are stacked with absolute regularity, so that it suffices to know the spacing of the lattice, the volume of the crystal, and a few atomic properties to know how many they are, as one counts the bricks of a wall by measuring the wall and a brick. The international Avogadro project turned the loophole into an epic of precision: kilograms of silicon enriched to extreme isotopic purity — the isotope 28 almost alone, bought from the Russian centrifuges — grown into single crystals without defects, and turned into spheres ten centimetres or so across that are, in all probability, the most perfectly spherical objects ever made: their dimensions deviate from the ideal sphere by a few tens of nanometres, and visitors are told that one of them, inflated to the dimensions of the Earth, would have between its highest peak and its deepest trench a difference in level of a few metres. The sphere is weighed against the prototypes; its volume is measured with optical interferometers that map its diameter from thousands of directions; the lattice spacing is measured with X-ray interferometry — and here the story speaks Italian too, for the determination of the lattice parameter of silicon was for years the specialty of the metrological institute of Turin, the heir of that tradition the fourth chapter saw arrive late and which here sits at the table of the decisive. And since at this level of fussiness nothing is trivial, even the skin of the sphere became a chapter of science: silicon exposed to air clothes itself in a few hours in a layer of oxide a handful of nanometres thick, and that layer — plus a few layers of adsorbed water and carbon — must be measured, modelled, and subtracted from the count, because the atoms of the rind are not stacked like those of the flesh. The counted kilogram thus ended by depending on the ellipsometry of an invisible film: the ritual washing of the Grand K, reincarnated in surface spectroscopy. By the mid-2010s the count was converging: the Avogadro number — the bridge between the world of atoms and the kilogram of the markets — was known to better than two parts in a hundred million [Stock et al. 2019] .

Weighing the light

The second road does not count: it weighs. In 1975 Bryan Kibble, a physicist of the National Physical Laboratory, conceived a balance of a new kind to realize the ampere, and realized — with the community — that the instrument could do much more: it could tie the kilogram to the Planck constant. The idea, stripped of details, is a balance between worlds: on one pan gravity — the weight of the kilogram to be measured — on the other an electromagnetic force, generated by a coil carrying current in a magnetic field. The stroke of genius lies in the double mode: the same coil, moved at a known speed in the same field, generates a voltage; and combining the two phases — weighing and motion — the geometry of the coil and the magnetic field, impossible to know well enough, cancel out of the equations, leaving a clean link between mass, speed, voltage, and current. The electrical magnitudes, in turn, are measured with the quantum standards of Josephson and von Klitzing, which speak the language of $h$: at the end of the chain, the balance weighs the mass directly against the Planck constant, with the speed of light and the time of caesium as the only other ingredients. Half a century of refinements — at the NPL, then at the American NIST, at the Canadian research council that bought the English machine and carried it to the finish, at the French laboratories and at the Bureau of Sèvres itself — brought these balances to weigh a kilogram with uncertainties of the order of one part in a hundred million. Every ingredient of the recipe had to rise to the same level, and one deserves mention because it reknits the book: gravity. The balance compares an electromagnetic force with the weight of the mass, and the weight is mass times local gravity — so $g$ must be known, at the exact point and at the exact hour of the weighing, to one part in a billion: it is measured by absolute gravimeters that time the fall of a mirror in vacuum, correcting for the terrestrial tides, for atmospheric pressure, even for the shifting of the water table beneath the building. Within every kilogram of the new regime there is therefore, hidden, a small geodesy: Richer and his slow pendulum at Cayenne, avenged to the billionth. When Kibble died, in 2016, the community renamed the instrument after him: the “watt balance” became the Kibble balance, the last eponym of our story and among the most deserved [Stock et al. 2019] .

Two roads, then: to count atoms of silicon, to weigh against Planck. Philosophically they are almost two metaphysics — mass as quantity of matter, in the manner of the ancients, against mass as coefficient in the laws, in the manner of the moderns — and that they must give the same number was not a courtesy but the brutal requirement fixed by the organs of the Convention: no redefinition until at least three independent experiments, of both families, had reached uncertainties of a few billionths and mutual agreement. For a decade the numbers wavered — there was a season, recounted with retrospective relief by the protagonists, in which the first leading measures of the two families differed among themselves by several times the declared uncertainties, the nightmare scenario of the sixth chapter: tight groupings of shots, sights in disagreement, and each group sifting its own systematics in the hope of finding the others’ first. It was a public process, conducted by workshops and cross-comparisons, and it resolved itself as serious things resolve themselves: by finding real errors, on both sides, and declaring them. Then, measure after measure, the convergence came; and in the summer of 2017 the international committee on data — the direct heir of Birge’s adjustments — performed the final special adjustment, fusing with least squares all the world’s measures of $h$, of $e$, of $k$, and of the Avogadro number into the four values destined to become eternal. The bell of Gauss, born to find a planetoid again, delivered the definitive numbers of the measurable universe [Stock et al. 2019] .

The final push, it must be said, did not come on its own: it was a campaign, conducted for fifteen years within the community by a generation of metrologists — Terry Quinn, former director of the Bureau and our reference historian, among the most obstinate — who had to overcome resistances anything but frivolous. There was the institutional prudence of those who remembered that a unit is changed once a century and one cannot get it wrong; there was the objection of the chemists, jealous of a mole that was changing nature; and there was the most popular and most serious objection of all, the pedagogical one: the old kilogram anyone could understand — it is that cylinder there — while the new one requires digesting the Planck constant, and a unit the citizen cannot understand seems to betray the democratic pact of the Revolution. The reformers’ answer is in this whole book: even the cylinder, on reflection, no one had ever seen — its comprehensibility was an affectionate legend — and the pact does not ask that everyone understand the definition, it asks that everyone be able to trust the chain that brings it into his home. On this organized trust, not on the anecdote of cylinders, the promise between strangers has always rested [Quinn 2011].

Versailles

On 16 November 2018, in the congress centre of Versailles — a few hundred metres from the palace in which Louis XVI had signed the passports of Delambre and Méchain — the twenty-sixth General Conference on Weights and Measures voted Resolution 1. The session was unusually crowded and unusually emotional for an assembly of metrologists: the proceedings broadcast live like a sporting event, the international press accredited, and in the hall, among the delegates, the veterans of the long march — there was also Klaus von Klitzing, the Nobel laureate whose quantum effect is kneaded into the new system, come to see his own discovery become universal law. The delegations of some sixty states approved unanimously, called one by one, and the hall — witness the films, which are worth seeking out — burst into a long applause, with embraces, a few tears, and the telephones raised as at a concert. There was reason: an arc opened a hundred and forty-three years earlier in the same city with the Metre Convention was closing, and for the first time since the night of the cubits humanity was taking leave of all its material standards at a single stroke [CGPM 2018] .

The text voted has the dry beauty of constitutions. The International System, it reads in substance, is henceforth the system in which seven constants of nature have numerical values fixed exactly, without uncertainty and forever. The frequency of the caesium transition, which defines the second: it is the clock. The speed of light, which with the second defines the metre: it is the ruler. The Planck constant — from that day exactly 6.626 070 15 times ten to the minus thirty-four joule seconds — which with metre and second defines the kilogram: it is the balance. The elementary charge, which defines the ampere: electricity reduced to a count of electrons. The Boltzmann constant, which defines the kelvin: temperature unmasked for what the fifth chapter let one glimpse, energy of agitation in disguise — the degree, after three centuries of fluids and fixed points, has become a fraction of a joule. The Avogadro number, which defines the mole: the accounting bridge between atoms and grams, at last loosed from every reference to the platinum kilogram. And the conventional luminous efficacy, which defines the candela: the ghost of the whale candle, pensioned off too inside a number. Seven numbers in place of seven stories of objects, rites, and keepers: the reader who has crossed the book can read in each the genealogy — the caesium of Essen, the light of Michelson and of 1983, the Planck of Charlottenburg, the electron of the telegraphers, the Boltzmann of Thomson and Regnault, the Avogadro of the Turin spheres, the candle of the photometrists — and knows that no table would do justice to how much civilization is compressed in those seven lines [BIPM 2019] .

The entry into force was fixed, with deliberate theatricality, for 20 May 2019: World Metrology Day, the anniversary of the signing of 1875. That morning the Grand K woke up pensioned off — and here the story gives its most exquisite reversal, which no novelist would have dared. The cylinder is still in the vault, with its bell-jars and its keys; but from that day it is, for the first time since 1889, an object like the others: a kilogram approximately. Its mass is now measured against the Planck constant, and therefore — for the first time in its life — it has an uncertainty: a few tens of billionths, the exact shadow of that drift which for a century had been logically unsayable. The king who could not err by definition has become a subject who errs like everyone; and the question that for a hundred years had had no meaning — “how much does the Grand K really weigh?” — is today a normal item of an experimental programme. I know of no image, in all the history of measure, that shows more cleanly what a definition is: not a property of things, but an office that things receive and lose by our decision.

The daily life of mass, in the new regime, resembles for that matter that of time: no object is the standard any longer, and the kilogram of the world emerges from the continuous comparison among the independent realizations — the Kibble balances of three continents, the silicon spheres — harmonized, until their small residual divergences close entirely, into a consensus value administered by the community. Mass too, the last of the magnitudes to surrender, has become what time had been already for half a century: not a thing kept, but an agreement maintained. And the kelvin has cashed in its dividend of democracy: as long as the degree hung from a privileged fixed point — the triple point of water, with its glass cells and its water of regulated isotopic recipe — every thermometry had to climb back to that particular temperature; now that the kelvin is a fraction of energy by way of the Boltzmann constant, one can realize it directly at any temperature, from the cryostat to the steel furnace, measuring the thermal agitation where it is needed. The scale has lost its sacred centre: this too, the reader of the first chapter knows, is a way in which the old regimes end.

What it means to define

For it is this, in the end, that Versailles forces one to think through: what did we do, exactly, when we “fixed” the constants? We did not measure them better: we stopped measuring them. Until November 2018 the Planck constant was a magnitude of nature with an uncertain value; from the day after it is a number exact by decree, and every future experimental refinement will no longer correct $h$ — it will correct the kilogram. The arrow of measure has been reversed by an act of assembly: before, the unit judged the constant, now the constant judges the unit. It is the gesture of 1872 — “the metre is the bar, in the state in which it is” — repeated at the deepest level conceivable: only that in place of the bar there is the very fabric of the physical laws, and in place of France there is the species. Convention, which the first chapter showed at the origin of every measure, has not been abolished by science: it has been promoted up to the constants of nature. Nature furnishes the regularities; which of them are exact — which numbers never err — we decide, by a show of hands, in a congress hall [Crease 2011].

An objection presents itself on its own, and it is well to look it in the face because the answer illuminates the whole edifice: if the Planck constant is exact by decree, has physics not tied its own hands? If one day $h$ “varied,” as certain speculative theories imagine, would we no longer notice? The answer is no, and the reason is a finesse the metrologists weighed long before Versailles. To fix $h$ means only to choose the dialect in which we describe the world: what nature has to tell us of itself lies entirely in the pure ratios — the numbers without units, like the fine-structure constant that governs the strength of electromagnetism — and those ratios no decree can touch: they are measured today as yesterday, and if one of them drifted in time, our realizations of the units would enter into disagreement among themselves in a revealing way, and we would notice indeed. Convention, however deep, stays in its place: it administers the language, it does not gag reality. It is the mature version of the lesson of the first chapter — convention is not arbitrariness — and the reason one can fix a constant without committing the sin of Procrustes.

There is a second reversal, subtler, which closes an account opened in the second chapter. The metre of 1799, it was said, was the child of an event: a historical expedition, irreproducible, dated and signed — measure anchored to history. The units of 2019 are the exact contrary: children of an eternal experiment, reproducible by anyone, anywhere, in any century, with the same result guaranteed by the uniformity of nature. No vault to inherit, no padded case on a journey, no London foundry: the mise en pratique of every unit is a text, and the text describes procedures that a competent laboratory of any planet could perform. For the first time, the promise between strangers no longer requires even that the strangers share an object: it is enough that they share the physics. If humanity were to lose tomorrow every standard, every laboratory, and every archive, but kept the books, the units would rise again identical: it is the definition of immortality that befits a convention, and no civilization had ever before attained it for any of its institutions — not for currencies, not for languages, not for laws.

And yet — the reader of the first chapter already suspects it — the most radical revolution in the history of measure was also the most invisible. On 20 May 2019 no balance in the world changed its reading: the values of the constants had been chosen on purpose so that the new kilogram should coincide with the old within the uncertainties, and the grocer, the pharmacist, even the industrial laboratory noticed nothing. Successful metrological revolutions are like this, and this book has by now a collection of them: the substance changes entirely, the façade does not tremble — because the purpose of the whole edifice, from the walled arm to the seven constants, has always been one alone: that the promise hold, and that no one should have to think about it. The difference is that now the promise no longer ages. There remains to ask what happens at the frontiers of this perfect edifice: how far down, how small, how finely one can still measure — and what happens when clocks become so precise that they feel the shape of spacetime, and measure meets the limits that nature itself imposes upon it. It is the last chapter, and it is the frontier of today.