This is the first in a series of essays. Subsequent installments will address waves and particles, relativity, nuclear physics, and the Standard Model.

In the seventeenth century, the English word Physics, Physicks or even Physiques had come to signify the study of the phenomena of nature, as used by Thomas Hobbes in 1656.1 Physics and natural philosophy were synonymous, both indicating, according to John Harris’s 1704 lexicon, the “Speculative Knowledge of all Natural Bodies and of their Proper Natures, Constitutions, Powers and Occupations.”2 Whether living things were to be included in “natural bodies” was left undecided. It is not far from my own favorite description of particle physics, which is a search for the basic building blocks of nature and for the rules by which they combine.

The full title of Isaac Newton’s 1687 masterpiece Philosophiae Naturalis Principia Mathematica recalls Galileo Galilei’s warning that physics is “written in the language of mathematics.”3 Just as Galileo and Johannes Kepler depended upon the mathematical techniques described by Euclid, so did Newton rely on his own development of the basic principles of calculus. Albert Einstein, too, needed tensor analysis and differential geometry. Today’s string theorists count on mathematics not yet invented.

I offer here an anecdotal journey along a few choice but interwoven strands of the magnificent tapestry that is physics, from its ancient beginnings to the present day. I will start with the early contributions of the mathematical sciences.

Quantitative science depends on measurement; measurements are expressed in numbers. Our number system is based on ten but some earlier systems were based on twenty—perhaps because our barefoot ancestors counted on their toes as well as on their fingers.4 Linguistic remnants, like eighty as four-score in English, firs in Danish and quatre-vingt in French, reveal their vigesimal ancestry, as did the once widely used Réaumur temperature scale, where the boiling point of water was set at 80 degrees. Many living languages, such as Mayan, Nahuatl, Georgian and Yoruba, retain heavily vigesimal counting schemes.

Computers, using simple yes-no binary numbers, are both digital and digitless.

Four millennia ago, our Sumerian and Babylonian antecedents favored 12 for the number of hours of daylight, signs of the zodiac, and months of the year. They also introduced the 360 degrees of the circle, and, for a while, used a year of that many days. Their number system was decimal from 1 to 59, but otherwise sexagesimal, using positive or negative powers of 60.5 It was a sophisticated choice.

In the measurement of time, early civilizations recognized three natural but incommensurate periods, each of which now has a precise meaning: 1) the solar year, the interval between successive vernal equinoxes, 2) the lunar month, that between successive new moons, and 3) the solar day, that between successive sunsets. The mean solar year is a few days more than 12 lunar months and a few hours more than 365 days. To deal with this issue the Sumerians came up with a remarkably accurate lunisolar calendar based on the near equality between 19 solar years and 235 lunar months. Their year consisted of 12 lunar months, one for each sign of the zodiac.

An additional intercalary month was added to the calendar seven times during each 19-year period, in what became known as the Metonic cycle, honoring the Greek astronomer Meton of Athens who had revived and codified the Sumerian system in 432 BCE. It became the basis of the ancient Attic calendar and persists today in both the Christian ecclesiastical and Hebrew calendars. The Sumerian correspondence between months and zodiacal signs is preserved intact in the Talmud.

Most people now use the same civil calendar with 12 months satisfying the refrain, “Thirty days hath September / April, June, and November…” and a year of 365 days. An extra day is added to February every four years, except for those divisible by 100 but not 400. The last stipulation was added by Pope Gregory XIII in 1582, thereby replacing the Julian calendar, which Julius Caesar had introduced in 46 BCE, which has a leap year every four years.

Let us compare lengths of the mean years of various calendars with the mean solar year in order of increasing precision.

The Islamic year is indifferent to seasons. It consists of 12 lunar months and is about 11 days short of the mean solar year. The 365-day year is six hours short, while the mean Julian year, still used by some churches, skimps by only 11 minutes. The mean Metonic or Hebrew year is closer but overshoots by seven minutes. Our brilliantly conceived Gregorian calendar boasts a mean year of 365.2425 days, a mere 26 seconds shy of its target. Over three millennia must pass before our calendar will err by a single day.

Consider the curious use of the word “second” to denote our smallest everyday time interval. Once again the sexagesimal Sumerians or their Babylonian brothers seem to be responsible. An often inconveniently long hour led them to use a smaller part as a unit of time: the sixtieth of an hour or the minute part of the hour or, even more briefly, the minute. Of course, much can happen in the course of a minute, so its sixtieth part—the second minute of an hour—became known to them, and eventually to us, as the second.6 Angular degrees were similarly subject to sexagesimal subdivision yielding arc-minutes and arc-seconds as small and smaller fractions of a degree.

The seven-day week is a convention that probably originated in Mesopotamia. Its days corresponded to deities responsible for the seven “wandering stars,” those heavenly bodies whose positions in the sky move relative to the “fixed” stars. The convention found its way to Rome where Sunday was linked to Sol, Monday to Luna, Tuesday to Mars, Wednesday to Mercury, Thursday to Jupiter, Friday to Venus, and Saturday to Saturn. Our English weekdays, however, recall Scandinavian usurpers of the Latin gods: Tir for Mars, Wodin for Mercury, Thor for Jupiter, and the goddess Frigga for Venus. English Mondays drop Luna for ancient Greek mena, which also gave us moon, month, and menses.

The days of the week also inherited links to the seven ancient metals and to the seven temperaments associated with the planets: gold to sunny Sunday, silver to lunatic Monday (Lundi in French), iron to martial Tuesday (Mardi), mercury to mercurial Wednesday (Mercredi), tin to jovial Thursday (Jeudi), copper to venial Friday (Vendredi), and lastly, lead for saturnine Saturday.

In China, the days of the week are simply numbered 1, 2, 3, etc., with a singular lack of romantic imagination.

The numbers scientists study are rarely pure; they bear dimensions, like those of power, pressure, force, density, or volume. A number per se means nothing to a physicist unless it is appended to an appropriate dimensional unit.

For our marvelous metric system of weights and measures, the primary units are seconds for time, meters for length, and kilograms for mass. Many compound units have names honoring famous scientists, like newtons for force, joules for energy, pascals for pressure, and watts for power.

The first primary unit to acquire an atomic definition is the second, which, for much of the nineteenth century and earlier, was simply 1/86,400 of the mean solar day. But tidal forces exerted by the sun and moon gradually retard the earth’s spin and lengthen the day by almost two milliseconds per century. So, in 1956, the second was redefined to be a certain fraction of the specific solar year 1900.

The change was short lived. Four years later the connection between time and the motion of heavenly bodies was forever sundered when the second was defined to be “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 the caesium 133 atom.”7 The day then became 86,400 seconds long, with one or two leap seconds added each year, to maintain a synchrony between the sun and the clock.

The meter was next.8 Before its introduction in 1793, dozens of different length units were in common use. The meter began as one ten-millionth the distance from the earth’s equator to its north pole. There were several attempts at readily-consultable precious metal artifacts on which this length standard could be precisely inscribed. In 1889, the first General Conference on Weights and Measures (CGPM-1) simply defined the meter as the distance between two lines etched upon the platinum-iridium International Prototype Meter, which was to be maintained indefinitely in Sèvres, France. The artifact had become the standard!

At CGPM-11 in 1960, the meter achieved an artifact-free atomic meaning as 1,650,763.73 wavelengths of light from a specific transition of the krypton-86 atom. This would also be short lived. At CGPM-17 in 1983, the speed of light in a vacuum was defined to be exactly 299,792,458 meters per second, and thus the meter became the distance traveled by light in 1/299,792,458 second.

For centuries, scientists strove to measure the speed of light with ever-increasing precision. No longer.

The kilogram was introduced by the French in 1795 as the mass of one liter of water at its freezing point of 0°C. Four years later it too was redefined. An International Prototype Kilogram (IPK) was fabricated with a mass as close as possible to that of one liter of water at 4°C, the temperature at which water achieves greatest density. This IPK, stored at Saint-Cloud, France, was ratified as the standard kilogram at CGPM-1 in 1889 and so it remains today.

Several plans exist to replace this artifact with an atomic definition, such as the mass of a specified large number of carbon-12 atoms or via a defined value for Planck’s constant. No such scheme has yet achieved the necessary precision, but the kilogram is scheduled for redefinition at CGPM-26 in November 2018.

Another Sumerian triumph was the creation some five millennia ago of what may have been the first written language, cuneiform. Hieroglyphics followed soon thereafter, but Egyptians wrote on fragile papyrus scrolls rather than carving on stone tablets; far less of their work survives. On a particular Sumerian clay tablet, called Plimpton 322 and made around 1800 BCE, are 15 Pythagorean triples: sets of three integers a, b, and c satisfying the equation a2 + b2 = c2 and thus forming the sides of a right triangle. The Plimpton triples range from two digits to five, a typical one being (4,601, 4,800, 6,649). We do not know their purpose, nor how the Sumerians managed to find them.

In the sixth century BCE, Pythagoras gave the first known proof of the theorem named after him, while the method for finding Pythagorean triples was devised by Euclid in the fourth century BCE.9 Yet some forgotten Sumerian scribe somehow created Plimpton 322 more than a millennium earlier!

Elisha Scott Loomis lists 370 distinct proofs of Pythagoras’s theorem, including ingenious examples by Euclid, by Einstein, and by, before his election as president of the US, James Garfield.10

There is a much more general question. While an infinite number of triples of positive integers a, b, c satisfy the equation ap + bp = cp for p = 1 or p = 2, Pierre de Fermat’s “last theorem” of 1637 stated that no such triples exist for any value of p beyond two. Fermat claimed to have proven his theorem, but there is no evidence that he had. He did establish its validity for p = 4, as Leonhard Euler would do for p = 3 in 1770. By 1839, it had been proven for p = 3, 4, 5, 6, 7, 10, and 14.

A general proof for all integers p greater than two was not found until 1995 when, after a heroic, decade-long struggle, Andrew Wiles triumphed.

Some mathematical problems take even longer. Euclid showed that there is no largest prime number and conjectured that there should be an infinite number of “twin primes” (primes differing by two, like 5 and 7, or 11 and 13). Even today Euclid’s conjecture has not quite been proven, though a rigorous proof may be in sight.

Of this sort of mathematics the famous number theorist G. H. Hardy wrote in 1940: “[t]here is one science … whose very remoteness from ordinary human activities should keep it gentle and clean.”11

The Elements of Matter

Matter appears in wondrous variety: gases, liquids, and solids, all imaginable shapes, forms and textures, colors, tastes, and smells. Empedocles of Sicily, in the fifth century BCE concluded that, contrary to appearance, there were just four “roots of all things”: fire, water, earth, and air, corresponding to four gods, Hera, Hades, Zeus, and Nestis. These elements, though he did not call them that, were subject to a force of attraction (love) and a force of competition (strife).12 Later Greek philosophers related them to the four humors of early medicine, the four seasons of the year, and to the four temperaments of man:

Element Season Humor Temperament
Air Spring Blood Sanguine
Fire Summer Yellow Bile Choleric
Earth Autumn Black Bile Melancholic
Water Winter Phlegm Phlegmatic

Other early societies had their own notions of primal elements. About twenty centuries ago, the Mesopotamian creation epic Enuma Elish proposed sea, earth, sky, and wind as cosmic elements, but in no sense were they to be the constituents of matter.13 Ancient Chinese philosophers chose five fundamental elements—fire, earth, water, metal, and wood—but these too were more akin to forces of nature, more becoming than being. There were Egyptian, Buddhist, and Hindu precedents as well. For example, the Ayurvedic elements of the first millennium BCE were linked to the five senses—earth to smell, water to taste, fire to sight, air to touch, and space to hearing—but these were more pertinent to medicine than natural philosophy. Only Greek ideas of atoms and elements would endure and provide the philosophical scaffold for Western science.

Alchemy was the forerunner of chemistry, metallurgy, and medicine.

We know little of the life of Mary the Jewess, one of its earliest practitioners, and what we do know comes primarily from the writings of the fourth-century Greek alchemist Zosimus. Like most later alchemists, she appears to have studied distillation, tried to transmute metals, and sought the elixir of youth. She is remembered by historians of science and chefs de cuisine, as at some point her name became associated with the double boiler, the bain-marie.14 Further steps toward the development of modern chemistry were taken by medieval alchemists such as the ninth century savant Jabir ibn Hayyan known to us as Geber. He proposed all metals and minerals to be made up of the four classical elements together with the metallic principles sulfur and mercury. In the 1500s, Paracelsus gave zinc its curious name and proposed the tria prima of sulfur, mercury, and salt as the basic elements of medical science.15

In the early seventeenth century, the Flemish Jan van Helmont spent five years patiently nurturing a tree, periodically weighing both it and the soil in which it grew. Seeing a considerable increase in the tree’s mass but negligible loss of soil mass, he concluded that the tree had transmuted the added water into wood. He would accept only two Paracelsian elements as fundamental, air and water. Van Helmont was among the first scientists to realize that not all air is the same. He recognized carbon dioxide, which he called “gas sylvestre,” as distinct, but did not know about photosynthesis and could not imagine a tree’s mass being made from thin air. He produced and described many different gaseous compounds, among them carbon monoxide, methane, and sulfur dioxide. Perceiving the utility of a collective noun, he coined the immensely useful word “gas” from the Greek word chaos.16

Hennig Brand of Hamburg may have been one of the last alchemists but he was also the first person known to have discovered a chemical element. In 1669, while searching for the secret to eternal youth, he was amazed to extract from human urine a waxy white material that would spontaneously catch fire. He had discovered phosphorus.17 The seventeenth-century British polymath Robert Boyle duplicated Brand’s synthesis and initiated its commercial use. He also mixed iron filings with acid, producing a gas he called “the inflammable solution of Mars [iron].”18 About a century later, Boyle’s mysterious effusion was identified by Henry Cavendish as the chemical element we call hydrogen. The principal components of air, nitrogen and oxygen, were identified soon thereafter. Cavendish showed that water is a compound of hydrogen and oxygen and Antoine Lavoisier—the French father of chemistry—proved that combustion is oxidation, thus eliminating the need for that mysterious and ineffable principle called phlogiston.19 Lavoisier listed thirty-three chemical elements in his 1789 textbook La Traité Elémentaire de Chimie. None of the ancient Greek four were included, and only two of his proposed elements, calorique (heat) and lumière (light), would fail ultimately to make the cut.

The pace of discovery quickened. By 1850 the number of known chemical elements had more than doubled, to seventy-three.

Many nineteenth-century scientists sought and found tantalizing patterns in the list of chemical elements. Among them were Johann Döbereiner’s law of triads in 1817, Alexandre-Émile Béguyer de Chancourtois’s spiral model in 1862, and John Newlands’s law of octaves in 1866. Three years later, Dmitri Mendeleev in Russia and Lothar Meyer in Germany independently presented the first periodic tables to include all the then-known elements. Mendeleev’s table, unlike Meyer’s, wisely left empty spaces for elements yet to be found, three of which were soon filled by patriotic Swedish, French, and German chemists who named their new elements scandium, gallium, and germanium. Each new element displayed precisely the chemical and physical properties predicted by Mendeleev, making him a world celebrity. Nonetheless, there were some flaws:

  • Atomic number, not atomic weight, is the correct parameter for the organization of elements in the periodic table; the latter does not always increase in sync with the element’s position in the periodic table. For example, nickel follows cobalt in the table but has the lesser atomic weight. Mendeleev can be forgiven this lapse because the atomic nucleus, without knowledge of which the notion of atomic number cannot be formulated, would not be found until 1911.
  • The chemically inert gas argon was discovered serendipitously in 1894; it comprises about 1% of Earth’s atmosphere. Soon afterward its inert cousins, helium, neon, krypton, xenon, and radon would show themselves. Yet Mendeleev resisted, arguing that argon was not an element but N3, a nonexistent nitrogenous analogue to ozone, until 1902, when he added a column to his table to account for elements with zero valence.
  • Far worse, Mendeleev never acknowledged that the systematic pattern he had discovered among the elements might suggest a more fundamental underlying atomic structure. Indeed, he viciously opposed such speculations:

    [F]rom the failures of so many attempts at finding in experiment and speculation a proof of the compound character of the elements and of the existence of primordial matter, it is evident … that this theory must be classed amongst mere utopias … [I]t really would be instructive to have them all collected together, if only to serve as a warning against the repetition of old failures.20

Unaccountably, Mendeleev insisted on the primacy of each and every distinct chemical element in his table.

Often in science the appearance of order leads to the discovery of structure. Regular patterns among quite different elementary particles led Murray Gell-Mann and George Zweig to quarks.21 Dan Shechtman’s observation of atomic patterns earned him a Nobel Prize for the discovery of quasicrystals. The integer laws of chemistry by weight and volume would demand the existence of atoms.

Of Mendeleev’s “old failures,” surely one was William Prout’s conjecture of 1816 that all elements are made up of hydrogen atoms, a hypothesis with more prescience than strict truth. When all isotopic masses were measured to be integers to within 1%, Francis Aston revived Prout’s notion. The citation to his 1922 Nobel Prize in Chemistry read in part, “for his enunciation of the whole-number rule.” The presentation speech at the award ceremony acknowledged that Prout’s “theory has now been restored to life … in a form different from that which its originator imagined.”22 We now know that the Prout–Aston rule of whole numbers follows trivially from the near equality of neutron and proton masses.

In the early 1930s, scientists had 92 different chemical elements, from hydrogen Z=1 to uranium Z=92, where Z denotes the element’s place in the periodic table as well as the number of its electrons and its protons. Only four had not yet been found in nature or created in the laboratory. The first synthetic element was created in 1937: technetium (Z=43). It now has numerous medical, chemical, and industrial applications. Two years later francium (Z=87) was found in the uranium decay chain by Marguerite Perey, a student of Marie Curie. None of its isotopes lives long enough to be useful. Astatine (Z=85), the least abundant of all elements found on Earth, was discovered in 1940, and five years later promethium (Z=61), another fission product of uranium, was first identified.

Enrico Fermi, one of the most brilliant stars of twentieth-century physics, used a beam of neutrons as an instrument of discovery, just as his predecessors had used alpha particles.23 His neutrons, upon striking atoms of a given element, would transmute them into atoms one or two steps higher in the periodic table. In 1934, he claimed to have created elements 93 and 94 by bombarding uranium with neutrons. He named his new elements ausonium and hesperium. Immediately the German chemist Ida Noddack warned that Fermi’s reasoning was not airtight. Imagining the possibility of nuclear fission, she wrote:

When heavy nuclei are bombarded by neutrons, it is conceivable that the nucleus breaks up into several large fragments, which would of course be isotopes of known elements but would not be neighbors of the irradiated element.24

Her paper was totally ignored. It might have been because of her earlier, false claim to have discovered masurium (Z=43) or because the concept of nuclear fission was too far out of the box. Perhaps it was because she was a woman.

On December 10 1938, Fermi received the Nobel Prize in Physics “for his demonstrations of the existence of new radioactive elements produced by neutron irradiation.” Just one week later the discovery of nuclear fission showed that Fermi had been wrong. Nonetheless his many seminal contributions to physics more than justify his misdirected Nobel Prize.25 Both neptunium (Z=93) and plutonium (Z=94) were first produced and discovered at a cyclotron in California in 1940. Since that time every element up to Z=118 has been found or made, named, and studied.26

The Atoms of the Elements

The history of the atomic hypothesis is intimately related to, interwoven with, but largely distinct from that of the chemical elements. The notion of matter formed of tiny, indivisible, indestructible, and eternal particles arose in sixth-century BCE India27 and, perhaps independently, a century later in Greece where it is usually attributed to Leucippus, thence to Democritus, who called his particles atoms, meaning uncuttable.28 He introduced one species for each of Empedocles’s four elements, fire, water, earth and air. Plato, like Pythagoras before him and Max Tegmark29 today, was convinced of the primacy of mathematics, that “all things are number.”30 Plato was much taken with the idea that there are precisely five regular polyhedrons and associated each element with one of these geometrical figures. Fire atoms had the sharp corners of tetrahedrons. Water atoms were regular icosahedrons that could easily slip around one another. Earth atoms were stackable cubes. Air atoms were octahedrons. Both Plato and Aristotle associated the dodecahedron with a fifth element—the quintessence or æther—of which the heavenly bodies were made and through which starlight could pass.31 It might even be the geometrical form of the universe itself. Some cosmologists have recently proposed a finite universe shaped like dodecahedron with its opposite sides identified. Sadly, subsequent studies of the cosmic background radiation offer scant support for their delightful hypothesis.32

The notion of atoms languished until the first century BCE when it was revived briefly by Lucretius in De Rerum Natura: “No rest is allowed the atoms moving through the depths of space, driven along in an incessant but variable movement … Entangled by their own close-coupled shapes, they make strong-rooted rock or the bulk of iron.”33

Galileo was puzzled by the failure of suction pumps or siphons to raise water by much more than ten meters. His own tentative explanation was an analogy with a long iron chain breaking due to its own weight. Just after his death, his disciple Evangelista Torricelli solved the problem. Inverting a mercury-filled glass tube sealed at one end over a mercury reservoir, he measured the pressure of air due to the weight of the atmosphere, writing to his colleague Michelangelo Ricci: “Noi viviamo sommersi nel fondo d’un pelago d’aria!” (We live submerged at the bottom of an ocean of air!)34

Torricelli never published his work, but he did demonstrate his mercury column to the visiting French philosopher and cleric Marin Mersenne, who had earlier measured the speed of sound in air, invented what are now called Mersenne primes, and was known as the father of acoustics. Learning of Torricelli’s column from Mersenne, Blaise Pascal arranged to have one brought to a mountaintop, observing the anticipated decline of air pressure with altitude, and in the process inventing the altimeter. He also showed air pressure to underlie such diverse phenomena as the flow of milk to nursing infants, the inhaling of air when breathing, suction, siphons, pumps and syringes, and the adhesion of polished metal blocks.35

Otto von Guericke, mayor of Magdeburg and a fine scientist, placed two hollow brass hemispheres, each half a meter in diameter, together, and pumped out the air inside with an air pump designed by himself. At the first of several public demonstrations, in 1654, at the court of the Holy Roman emperor at Regensburg, two teams of fifteen horses struggled but were unable to pull the hemispheres apart. 

Upon learning of these demonstrations, Boyle and Robert Hooke set out to explore this newly discovered phenomenon; they built a better air pump and confirmed that air is essential both to living animals and for the propagation of sound. They also found, in 1662, one of the first quantitative laws of physical science: the volume of gas at constant temperature varies inversely with its pressure, or, in simple terms, an air-filled cylinder with a piston behaves like a spring. “There is a Spring, or Elastical power in the Air we live in,” Boyle wrote, attributing this remarkable behavior to tiny springlike elements among the particles of air.36 Hooke would later formulate his own eponymous law.

Another important property of gases was revealed by the French physicist Guillaume Amontons in 1699. He showed that the pressure of a gas kept at constant volume increases with its temperature. Amontons was unable to formulate a quantitative law because temperatures could not be meaningfully measured until Daniel Fahrenheit devised his scale in 1834.

The history of Boyle’s law is fraught.37 Still, when Newton learned of it from reading Hooke, he proposed that a hypothetical new force acts among atoms:

If a fluid be composed of particles … [Newton hedges his belief in atoms] and the density be as the compression [a concise statement of Boyle’s law], the centrifugal forces of the particles will be reciprocally proportional to the distance of their centres. [His repulsive force between atoms] … But whether elastic fluids do really consist of particles so repelling each other, is a physical question. We have here demonstrated mathematically the property of fluids consisting of particles of this kind, that hence philosophers may take occasion to discuss that question.38

“That question” has been discussed and Newton’s explanation was ultimately found wanting. There is no such force in nature. The correct explanation is simple and elegant. In 1738, Daniel Bernoulli carefully drafted an image of a weighted piston upon a gas-filled cylinder. “Let the cavity contain very minute corpuscles which are driven hither and thither with very rapid motion,” he wrote, “so that these corpuscles, when they strike against the piston and sustain it by their repeated impacts, form an elastic fluid which will expand of itself if the weight be removed or diminished.”39 Bernoulli’s hypothesis enabled him to deduce that the product of the pressure and volume of a gas is a certain fraction of total kinetic energy carried by its molecules. Bernoulli’s simple model explained the laws of both Boyle and Amontons.

For many reasons, Bernoulli’s paper would be ignored for about a century, in favor of Newton’s explanation.40 Atoms, were they to exist at all, were expected to be stationary and not jiggling about as Bernoulli and Lucretius imagined. Heat was regarded as an impalpable substance called calorique until 1798, when Benjamin Thompson, Count Rumford argued persuasively that heat is energy in the form of molecular motion. The ideal gas law relating the pressure, volume, and temperature of a gas was first conjectured in 1834, but would not be deduced from microscopic principles until 1856.

Yet another gas law plays a critical role in our story. Charles’s law describes how a gas kept at constant pressure expands when it is heated.41 The effect was first observed in 1787 by the French physicist Jacques Charles, who never published it. In 1802, the law was confirmed by Charles’s countryman Joseph Louis Gay-Lussac in 1802 and found to apply in exactly the same manner to all gases by John Dalton.42 Gay-Lussac’s law was called upon to exonerate Tom Brady in the Deflategate football scandal.43

Now, to chemistry. In 1794, the French chemist Joseph-Louis Proust put forward his law of definite proportions, holding that the proportions by mass of elements in a chemical compound are always the same. Common salt from the Himalayas or from the Dead Sea contains the same percentages of sodium and chlorine,44 a simple and obvious truth from today’s perspective but heatedly debated at the time by Proust and the distinguished French chemist Claude-Louis Berthollet, who likened chemical compounds to mayonnaise, whose proportions of oil and egg yolk are determined by the chef. The dispute continued until 1811 when Proust’s law was accepted as a result of careful experimentation and the growing confidence in Dalton’s atomic theory.

Dalton was the first person to associate specific atoms to each chemical element. He found the carbon dioxide molecule, for example, to consist of one carbon atom and two oxygen atoms, CO2, and the salt molecule to consist of one atom each of sodium and chlorine, NaCl. Proust’s law is built into the theory. Dalton’s law of multiple proportions says that when two elements can combine to make several different compounds, their masses must be in a ratio of small whole numbers. For example, 12 grams of carbon can combine with 16 grams of oxygen to make 28 grams of carbon monoxide, or with 32 grams of oxygen to make 44 grams of carbon dioxide. Dalton’s law, like Proust’s, is an immediate consequence of his atomic theory, but the converse is not true; the truth of these laws does not imply the existence of atoms, but it is awfully difficult to imagine any other explanation.

Here is where gases again pose a vexing problem. Gay-Lussac was also an accomplished chemist and his careful studies of reactions among gases led him to another integer law, that of combining volumes: when gases react to form other gases, and when volumes are measured at the same temperature and pressure, the ratios between the volumes of the reactant gases and the gaseous products can be expressed in small whole numbers. For example, two liters of carbon monoxide react with one liter of oxygen gas to form two liters of carbon dioxide, simple enough, but difficult to reconcile with Dalton’s integer law of multiple proportions.

The problem was solved by the Italian chemist Amedeo Avogadro who concluded that molecules of gases such as hydrogen, oxygen, nitrogen, and chlorine are diatomic, each containing two atoms. Many chemists opposed this notion at first because they doubted identical atoms could have an affinity for one another. Even more important was Avogadro’s famous hypothesis that equal volumes of different gases under the same conditions of pressure and temperature contain equal numbers of molecules. He could offer no theoretical explanation. Conflicting measurements of atomic weights could now be resolved and the law of combining volumes became a triviality; just replace liter in the preceding example by molecule.

A simple but very useful construct, the mole (originally the gram-molecule) as a measure of the quantity of a compound, is needed to subsume the empirical laws of Boyle, Amontons and Charles into the ideal gas law. One mole of a compound is an amount whose mass equals its molecular weight in grams. Thus a mole of any compound contains the same number of molecules: Avogadro’s number, roughly 6 × 1023. A chemical equation, like 2H2 + O2 → 2H2 may be parsed as “two molecules of hydrogen plus one molecule of oxygen” or equivalently as “two moles of hydrogen plus one mole of oxygen.” If only my students could consistently distinguish moles from molecules! To ensure that this essay contain at least one equation, here is the long-sought ideal gas law:

PV = nRT,

where n is the number of moles of a gas, T is its Kelvin temperature and R is the universal gas constant.

Late in the nineteenth century, the science of thermodynamics arose to complement kinetic theory. It would explain the operation and limitations of steam engines, automobiles, air conditioners, and power plants. Scientists like Rudolf Clausius, James Clerk Maxwell, and Ludwig Boltzmann would not only deduce the ideal gas law from microscopic principles but would learn how, when, and why real gases depart from it. Maxwell’s synthesis of the laws of electricity and magnetism led to the electrification of society as well as the electrocution of some of its members. Light was found to occupy a tiny portion of a vast electromagnetic spectrum from radio waves to X-rays, both of which had then just recently been observed. In view of the triumphs of what we now call classical physics, many of its practitioners felt as if they were approaching the very end of their discipline:

The more important fundamental laws and facts of physical science have all been discovered, and these are now so firmly established that the possibility of their ever being supplanted in consequence of new discoveries is exceedingly remote,45

said Albert Michelson in 1899, whose own unexpected failure to detect evidence of the æther would soon be explained by Einstein’s theory of relativity.

Many unsolved problems remained, among them: what underlies the success of the periodic table? Why do heated objects turn from red to yellow to a blazing blue-white as their temperature increases? What fuels the sun and the stars? Can wildly conflicting estimates of the age of the earth by physicists and geologists be resolved? What role can electrons play in the structure of atoms? And just what is it that Henri Becquerel stumbled upon and M. and Mme. Curie call la radioactivité? I will return to these and many other fascinating questions in a sequel.

  1. Thomas Hobbes, Elements of Philosophy IV (1656), 290. 
  2. John Harris, Lexicon Technicum I (1704), not paginated. 
  3. See Richard Popkin, ed., The Philosophy of the Sixteenth and Seventeenth Centuries (New York: Simon & Schuster, 1966), 65. 
  4. See Giles Godefroy, The Adventure of Numbers, trans. Leslie Kay (Providence, RI: American Mathematical Society, 2004), 3. 
  5. See “Sumerian Babylonian Mathematics,” The Story of Mathematics
  6. The third minute of the hour, one sixtieth of a second, is today called a third in both Polish (“tercja”) and Turkish (“salise”). See Wikipedia, “Minute.”  
  7. The International System of Units (SI), 8th ed. (Sèvres: Bureau International des Poids et Mesures, 2006), 113. 
  8. The United States, virtually alone among nations, remains proudly and persistently non-metric with its inches, feet, quarts, bushels, gallons, acres, and miles both statute and nautical. Pity Americans who must remember that 231 cubic inches make a gallon, 5,280 feet a mile, and 43,560 square feet an acre! For most of the rest of the world such conversions are trivial: 1,000 cubic centimeters to a liter, 10,000 square meters to a hectare, and 1,000 meters to a kilometer. 
  9. Euclid, Elements X, xxix.  
  10. Elisha Scott Loomis, The Pythagorean Proposition, 2nd ed. (Ann Arbor, MI: Edwards Brothers, 1940). 
  11. G. H. Hardy, A Mathematician’s Apology (Cambridge: Cambridge University Press, 1940), 33.  
  12. See Internet Encyclopedia of Philosophy, “Empedocles (c. 492–432 BCE).” 
  13. See Internet Sacred Text Archive, “Enuma Elish: The Epic of Creation.” 
  14. Raphael Patai, The Jewish Alchemists: A History and Source Book (Princeton: Princeton University Press, 1994), 61. 
  15. See William Newman, “Alchemical and Chymical Principles: Four Different Traditions,” in The Idea of Principles in Early Modern Thought: Interdisciplinary Perspectives, ed. Peter Anstey (New York: Routledge, 2017), 77–97. 
  16. See, for example, Walter Pagel, Joan Baptista Van Helmont: Reformer of Science and Medicine (Cambridge: Cambridge University Press, 1982). 
  17. Ironically, his home town would be destroyed by phosphorus-based incendiary bombs 275 years afterward. See John Emsley, The Shocking History of Phosphorus (London: Pan, 2000). 
  18. See, for example, Lawrence Principe, The Aspiring Adept: Robert Boyle and His Alchemical Quest (Princeton: Princeton University Press, 2000). 
  19. See, for example, Alan Musgrave, “Why Did Oxygen Supplant Phlogiston? Research Programmes in the Chemical Revolution,” in Method and Appraisal in the Physical Sciences: The Critical Background to Modern Science, 1800–1905, ed. Colin Howson (Cambridge: Cambridge University Press), 181–210. 
  20. Dmitri Mendeleev, “The Periodic Law of the Chemical Elements,” Journal of the Chemical Society (London) 55 (1889) 634–56 (also delivered as the Faraday Lecture, June 4, 1889). A decade after Mendeleev’s pronouncement, electrons were discovered and recognized as atomic constituents. 
  21. See, for example, Andrew Watson, The Quantum Quark (Cambridge: Cambridge University Press, 2004). 
  22. Henrik Söderbaum, “Chemistry 1922: Presentation Speech by Professor H. G. Söderbaum,” in Nobel Lectures in Chemistry 1922–1941 (Singapore: World Scientific, 1999), 5. See also, “Francis W. Aston – Biographical.” 
  23. Alpha particles, each made of two protons and two neutrons, are the nuclei of helium atoms. They are emitted by the process of alpha radioactivity and were used by Ernest Rutherford to engineer artificial nuclear reactions and to discover protons and atomic nuclei. Chadwick used them to discover neutrons in 1932. 
  24. Ida Noddack, “Über das Element 93,” Angewandte Chemie 47, no. 37 (1934): 653.  
  25. See Emilio Segrè, Enrico Fermi, Physicist (Chicago: University of Chicago Press, 1970). 
  26. The lyrics to Tom Lehrer’s famous 1959 song include the names of all 102 then-known chemical elements. Of the 118 now known, 32 are named for places on earth, 25 appear necessary for human nutrition, 16 honor scientists, and nine each are named for colors, heavenly bodies, and mythological figures. A mere seven elements are mentioned in the Bible. Most of the 26 transuranic elements are too unstable to have any practical use whatsoever. Nevertheless the search for heavier elements proceeds, especially in Germany, Russia, Japan, and the United States, for a long-predicted but perhaps illusory island of relatively stable, long-lived, and maybe even useful new elements. I wish them all good hunting! 
  27. See, for example, Dick Teresi, Lost Discoveries: The Ancient Roots of Modern Science (New York: Simon & Schuster, 2003). 
  28. See Stanford Encyclopedia of Philosophy, “Ancient Atomism.” 
  29. Max Tegmark, “The Mathematical Universe,” Foundations of Physics 38, no. 2 (2007): 101–50, and Max Tegmark, Our Mathematical Universe: My Quest for the Ultimate Nature of Reality (New York: Knopf, 2014).  
  30. See, for example, Phillip Horky, Plato and Pythagoreanism (Oxford: Oxford University Press, 2016), 24. 
  31. See Plato, Phaedo 110b; Timaeus 53c–55c; Aristotle, De caelo III.8. 
  32. Jean-Pierre Luminet et al., “Dodecahedral Space Topology as an Explanation for Weak Wide-Angle Temperature Correlations in the Cosmic Microwave Background,” Nature 425 (2003): 593–95; Planck Collaboration et al., “Planck 2015 Results. XVII. Constraints on Primordial Non-Gaussianity,” ArXiv, doi:10.1051/0004-6361/201525836. 
  33. Lucretius, De rerum natura. For medieval atomic theories, see, for example, Tzvi Langermann, “Islamic Atomism and the Galenic Tradition,” History of Science 47 (2009): 277–95: Ruth Glasner, “Ibn Rushd’s Theory of Minima Naturalia,” Arabic Sciences and Philosophy 11 (2001): 9–26. 
  34. See Philip Robinson, “Evangelista Torricelli,” The Mathematical Gazette 78, no. 481 (1994): 37–47. 
  35. The Physical Treatises of Blaise Pascal, trans. Isidore Spiers and Alexander Spiers (New York: Columbia University Press, 1937). 
  36. Robert Boyle, New Experiments Physico-Mechanical Touching the Spring of Air (1662), 12. 
  37. Arguing “the folly of eponymy in science,” science historian I. Bernard Cohen, concluded: “This is a law discovered by Power and Towneley, accurately verified by Hooke, accurately verified again by Boyle (aided in some degree by Hooke), first published by Boyle but chiefly publicized by Mariotte.” I. Bernard Cohen, “Newton, Hooke, and ‘Boyle’s Law’ (Discovered by Power and Towneley),” Nature 204, no. 618 (1964). 
  38. Isaac Newton, Principia, Book II, Proposition XXIII. 301, 302–303. 
  39. Daniel Bernoulli, Hydrodynamica, 1738 [in Latin].  
  40. Stephen Brush, The Kind of Motion We Call Heat, Book 1 (Amsterdam: North Holland, 1976). 
  41. Incidentally but interestingly, the long-held dream of human flight was first realized on November 21, 1783 by a hot-air balloon designed and built by the Montgolfier brothers. It carried two human passengers a distance of nine km and reached an altitude of 910 meters. The deed was done years before Charles had discovered the law that permitted the ascent. Nonetheless, just a few days after this history-making event Professor Charles himself made a solo ascent in a hydrogen-filled balloon to an altitude of three kilometers. Not to be outdone, Gay-Lussac would reach over seven km in his own hydrogen balloon ascent of 1812. Comment sont-ils sportifs ces savants francais!  
  42. As Wikipedia has it: The law “became known wrongly as Charles’ law, then even more wrongly as Gay-Lussac’s law, but never correctly as Dalton’s law,” thereby providing another instance of the folly of eponymy. Wikipedia, “Charles’s Law.” 
  43. John Branch, “The Deflategate Scientists Unlock Their Lab,” The New York Times, September 21, 2016. 
  44. Salt samples from different sites do taste or appear sensibly different due to the varying presence of trace contaminants. So also for tap water, especially in Detroit. 
  45. Albert Michelson, Light Waves and Their Uses (Chicago: University of Chicago Press, 1903), 23–24.