by Peter Watson
According to his mathematical diary, Gauss was still quite young when he began to consider the possibility that the ancient Greeks—Euclid in particular—had got it wrong with some of their fundamental axioms in geometry. In particular, he had begun to have doubts about parallel lines. Euclid had set out the classical paradigm and identified the classical solution: if you draw a straight line and then a point off that line, there can be only one line that is parallel to the first line and runs through the point.14 When he was only sixteen, Gauss began to consider—daringly—whether there might be other geometries at variance with the Euclidean. He didn’t publish anything for years, fearing ridicule, because—if he were right—other things followed, such as the fact that the angles of a triangle would not always add up to 180 degrees. Gauss couldn’t get these subversive thoughts out of his mind: he even climbed to the summit of three hilltops to shine beams between them, to see if the angles added up to 180 degrees. This suggests that Gauss had some idea that light might bend in space, anticipating Einstein by nearly a century. It had occurred to Gauss that three-dimensional space might be curved in the way that the two-dimensional surface of the earth was. This thinking developed out of his observation that lines of longitude, along which the shortest path between two points on the surface of the earth is measured, all meet at the poles. They appeared parallel but were not. No one had considered that three-dimensional space might also bend.
Gauss was to be proved right, as Einstein was proved right, with Arthur Eddington’s confirmation of the bending of light in 1919, but once again Gauss never published his ideas and the friends this troubled man shared his thoughts with were pledged to secrecy.15
Noncommutative algebra is the mathematical description of noncommutative geometry, which emerged in the nineteenth century in relation to physics and chemistry. At its simplest it refers to the possibility that, in mathematics, xy, strange as it may seem, is not always equal to yx. We shall meet this phenomenon again in the case of isomers in chemistry and with the benzene ring, where “rightness” and “leftness” determine chemical properties. This, plus the second law of thermodynamics, considered in Chapter 17, which says that time is a fundamental aspect of space, shows that a purely mechanical (i.e., Newtonian) understanding of the universe has to be incomplete. Gauss’s noncommutative algebra was an early attempt to come to grips with this problem. Once more he was well ahead of his time.
Though the bulk of Gauss’s career was spent in the highly abstract world of numbers, it was bracketed by two very practical discoveries. The first—his calculation of the orbits of moving objects—has already been referred to. The second came when he was in his fifties and already had many abstract, imaginative discoveries to his name. Among the nonmathematical phenomena he was interested in (although of course he was interested in the mathematical aspects), was terrestrial magnetism, in particular the way it varied across the earth, and in the existence of magnetic storms.16 In 1831, stimulated by Michael Faraday’s discovery of induced current, Gauss collaborated (for once) with the brilliant experimental physicist Wilhelm Weber, one of the (liberal) “Göttingen Seven,” to investigate a number of electrical phenomena. They made several discoveries in static, thermal, and frictional electricity but kept their powder dry for a time since their main interest was terrestrial magnetism. This prompted the idea that a magnetometer might also serve as a galvanometer and that, in turn, it might be used to induce a current that could send a message. Weber managed to connect the astronomical observatory at Göttingen with the physics laboratory a mile away by means of a double wire “that broke ‘uncountable’ times as he strung it over houses and two towers.”17 The first words, and then whole sentences, were transmitted in 1833 and the results of this first “operating electric telegraph” was mentioned (briefly) by Gauss in a notice in the Göttingische gelehrte Anzeigen, for August 9, 1834. Gauss grasped the military and economic significance of the invention and tried unsuccessfully to persuade the government to take an interest. It was not until Carl August von Steinheil, professor of mathematics and physics in Munich, in 1837, and Samuel Morse in the United States, in 1838, developed more user-friendly techniques that the electric telegraph caught on. Being ahead of his time was an occupational hazard for Gauss.
Nonetheless, his contemporaries referred to him as princeps, and he is now generally elevated to the level of Archimedes and Newton, inspiring a later generation—August Ferdinand Möbius, Peter Gustav Lejeune Dirichlet, Bernhard Riemann, Richard Dedekind, Georg Cantor, and others. As Marcus du Sautoy has said, the collaboration between Gauss and Weber on the telegraph, and Gauss’s innovations with the clock calculator and its role in computer security, make them “the grandfathers of e-business and the Internet.” Their collaboration is immortalized in a statue of the two of them in the city of Göttingen.18
THE ADVENT OF HUMANE MEDICINE
In marked contrast, statues of Samuel Christian Friedrich Hahnemann have been erected in Washington, Paris, Leipzig, Dessau, and Köthen. In North America at the turn of the (twentieth) century, other memorials existed in the form of twenty-two homeopathic colleges, while homeopathic remedies were used by one in five doctors. By 1945, homeopathic universities existed in the United States, Hungary, and India. In the twenty-first century, there is a homeopathic college in Canada, as well as a National Homeopathic Center just outside Washington, D.C., a Homeopathic Society in India, an Oxford College of Classical Homeopathy, a professional journal Homeopathy, edited from Luton in England and published by Elsevier in the Netherlands, one of the world’s leading publishers of scientific journals. Dentists use homeopathy, it is employed in childbirth, on pets, and on farm animals.19
At the same time, there exist an organization and a Web site called “Homeowatch,” dedicated to exposing the “quackery” of homeopathy, to showing that the science on which it is based is wrong-headed, even fraudulent, and that the many products produced in its name are medically worthless. As Martin Gumpert puts it in his biography of Samuel Hahnemann, “is homeopathy merely an excrescence of science, or is its core genuine and useful, even if we may not like its covering?”20 Hahnemann’s name elicits hatred and scorn among most regular medical practitioners but at the same time he refuses to go away. At one stage, the British royal family appeared in thrall to homeopathic medicine.
Hahnemann (1755–1843) was the son of a painter in the Meissen porcelain factory, and like many contemporaries he was precocious: at an early age he could speak Latin and Greek, classify old coins, and catalog books. But medicine was his main love, and he graduated from the University of Erlangen in 1779. His first destination was Hettstedt, a mining town that lacked a doctor, and there he came across a mysterious copper sickness that often proved fatal. It was in studying this disease that he first began to have doubts about the traditional blood-letting technique that all doctors of the time used. The basic approach was to make patients excrete, so as to void their bodies of whatever poisonous substances they had accumulated. They were induced to sweat, prescribed laxatives, forced to gargle, vomit, or salivate. The most extreme was blood-letting.21
When Hahnemann moved on again, this time to Gommern, near Magdeburg, he had another fraught encounter when a patient of his, a cabinetmaker, broke down suddenly (as we would say now), and Hahnemann accompanied him to the mental hospital (again, as we would say). There, the cabinetmaker was strapped to a chair attached to a mechanism that enabled it to be rotated rapidly sixty times a minute. This “treatment” used the centrifugal force of the rotating chair to send blood rushing to the patient’s brain, producing dizziness, vomiting, evacuation from the bowels and kidneys, “while blood even oozed from the skin around the eyes.” Such brutality reduced even the wildest inmates to catatonics. These experiences eventually led to Hahnemann’s book Freund der Gesundheit (The Friend of Health; 1792), written after he had settled in Leipzig, where he put forward the idea of a public hygiene policy, becoming one of the first people to do so. And it was in Leipzig that
he translated Treatise on the Materia Medica, by William Cullen, a professor of medicine in Edinburgh.22 In the course of translating this work, Hahnemann had the idea for which the world now knows him.
Cullen had written the following sentence when he was discussing the properties of cinchona bark (the source of quinine), which he said was a “febrifuge,” a substance that drives away fever: “In this case the bark works by means of its fortifying effect on the stomach.” Hahnemann was brought up short. He knew that cinchona had never fortified his stomach. On the contrary, quinine made him very sick. Accordingly, he now decided on his own trial. “By way of experiment, I took four drams of good cinchona twice a day. My feet, my fingertips, at first became cold.” There was no sign at all of his stomach being “fortified.” “I grew languid and drowsy; then my heart began to palpitate, and my pulse became hard and small; intolerable anxiety, trembling (but without cold rigour), prostration through all my limbs. Then pulsation in my head, flushing of my cheeks, and, in short, all those symptoms which are ordinarily characteristic of intermittent fever, one after another made their appearance.” It was a little while later that he noted the observation that would change everything: “Substances which excite a kind of fever extinguish the types of intermittent fever.”
Fever cures fever. That was Hahnemann’s new doctrine. As Gumpert says, “We must remember that this was before the germ or cell theories of disease, and that Hahnemann’s new ideas were an alternative to the brutal current method of the evacuation of ‘pernicious juices.’”23 In 1796 Hahnemann offered a paper to the newly founded Journal of Practical Medicine titled, “Essay on a New Principle for Ascertaining the Curative Power of Drugs, with a Few Glances at Those Hitherto Employed.” His central idea was now clearly set out: “In order to cure diseases, we must search for medicines that can excite a similar disease in the human body.” “Similia similibus!”—this is the essence of homeopathy. 24
He formulated his views in full in Die Organon der rationellen Heilkunde (translated as The Organon of Homeopathic Medicine; 1810) and Theory of Chronic Diseases (1828–39), where he argued for the use of minute quantities of remedies that, in larger doses, produce effects similar to those of the disease being treated.25 His wilder views are revealed in his further belief that small doses of medication could be induced to have powerful effects by vigorous shaking (called succussion). He referred to this increase in potency as dynamization, which, for Hahnemann, released an “energy” that he regarded as “immaterial and spiritual.” Eventually, he thought patients need not swallow “dynamized” medicines at all; it was enough to sniff them.
Most doctors dismiss homeopathy now on the grounds that any active ingredients, such as they are, are diluted often by 10,000 times, reducing them to well below the levels at which any pharmaceutical capacity could exert an effect.
Hahnemann continued practicing homeopathy over a long life, dying in Paris (he had married a French patient in 1843, when he was eighty-eight). He was visited by patients from all over the world, and a Homeopathic Medical College opened in Philadelphia in 1848. By 1900 America had 111 homeopathic hospitals, those 22 homeopathic medical schools mentioned earlier, and 1,000 homeopathic pharmacies. Thereafter the fashion for homeopathic cures declined, only to resurface in the 1960s. It is now very popular in India, Latin America, and Europe, and Great Britain has five homeopathic hospitals, and homeopathic cures are covered—amid great controversy—on the National Health Service.
THE SCIENTIFIC DISCOVERY OF THE NEW WORLD
“[Alexander von] Humboldt has done more good for America than all her conquerors,” said Simón Bolívar, the Venezuelan-born general credited with leading the liberation of Venezuela, Colombia, Ecuador, Peru, Panama, and Bolivia. “[He] is the true discoverer of South America.” Ralph Waldo Emerson described Humboldt as “one of the wonders of the world, like Aristotle, like Julius Caesar…who appear from time to time as if to show us the possibilities of the human mind.” A recent biography of Humboldt says bluntly that it is “quite possible that no other European had so great an impact on the intellectual culture of nineteenth-century America.”26 In his day, Humboldt was as famous as Napoleon. He was friends with Goethe (who shared his interests in plants and mining), Schiller, and Gauss, and his brother Wilhelm founded the University of Berlin. The paleontologist Stephen Jay Gould described him as “the world’s most famous and influential intellectual.” He has also been described as “one of the greatest but least remembered figures in scientific history” and that is true too.27
Born in Berlin in 1769, he and his brother were tutored privately (their father was technically an aristocrat but had only recently been ennobled). All his life Alexander was a restless man. He was good at drawing and his self-portraits show a handsome face, though he wore his hair so as to conceal marks sustained from a childhood bout of smallpox. His brother found him “self-centered” and a “busybody,” which he feared others would construe as vanity.
Humboldt enrolled as a law student at Göttingen when he was twenty; the son-in-law of one of his professors was Georg Forster who, as a teenager, had accompanied his father on James Cook’s second voyage around the world.28 The younger Forster was already known for his highly acclaimed account of that adventure, and he and Alexander teamed up to make their own journeys across Europe, answering the latter’s restlessness but stimulating it too.
Humboldt’s most important teacher in his early years was someone he encountered after he left Göttingen and attended the Freiberg School of Mines: Abraham Werner. After studying with Werner, Humboldt joined the Prussian mining service where he had a distinguished career. Using his own fortune to good effect, he invented—among other things—a safety lamp and a rescue device for miners threatened with a reduced air supply belowground. (He tested these mechanisms on himself in potentially dangerous experiments.) He was an out-and-out empiricist—facts, numbers, measurement, these, not philosophical speculation, were for him the building blocks of science.
But it was Humboldt’s wanderlust that was to distinguish him from everyone else, and after a number of travels within Europe itself—looking at active volcanoes—he set out on the first of his two “great journeys.”
The first, and the more momentous, was to South America. On October 20, 1798, he left Paris with the French botanist Aimé Bonpland, who would be his traveling companion for the next six years. (Humboldt studied under Laplace and was fluent in French—most of his writings were in that language.)29 They went first to Marseille and then on to Madrid, where Humboldt was introduced to the Spanish king and managed to persuade him to allow a scientific expedition to South America. This was remarkable, first because there had only ever been six scientific expeditions to Spain’s New World colonies (she was almost exclusively interested in the gold and silver to be obtained there) and second, because Humboldt was a Protestant. But the royal passports Bonpland and he obtained in March 1799 gave them total freedom of movement in the colonies.30 They left from La Coruña and after breaking through the British blockade landed on July 16, 1799, in what would become Venezuela. Now began what has been called “the scientific discovery of the New World.”
The two men faced great hardships and considerable danger as they traveled—on foot, by packhorse, in native canoes, and oceangoing ships—across Venezuela, Cuba, Colombia (where Bonpland caught malaria and they were delayed for two months), Peru, Ecuador, and Mexico.31 In the process, they “recorded, sketched, described, measured, compared and gathered” some 60,000 plant specimens, 6,300 of them unknown in Europe. Humboldt, however, was not just interested in geography, geology, and botany: he studied ancient Indian monuments, population figures, social arrangements, economic conditions. He was appalled by the slavery he witnessed and thereafter campaigned against it. He navigated the Orinoco and Magdalena rivers (which run respectively west–east through Venezuela toward Trinidad, and south–north through Colombia to the Caribbean), and confirmed the bifurcation of the Casiquiare River, showing th
at it really did connect the Orinoco and the Amazon as had been rumored.32 Humboldt also set a new mountaineering record, climbing to 19,000 feet on Mount Chimborazo (Urcorazo, “snow mountain” in Quichua) in Ecuador in June 1802. He failed to reach the summit, but his record stood for nearly thirty years.
They traveled with forty-two instruments—thermometers, barometers, quadrants, microscopes, rain gauges, eudiometers (for measuring oxygen in the air)—each with its own velvet-lined box. They nearly lost all these more than once as they tried to negotiate the many fearsome rapids along the Orinoco. It was on the Orinoco and its countless tributaries that Humboldt discovered a form of rubber, and where he found that the “natives” could distinguish a river by the taste of its water.
In 1804 he returned to Europe via the United States, visiting Philadelphia and Washington, D.C., where he met President Thomas Jefferson in the White House and at Monticello and was elected a member of the American Philosophical Society.33 On his return to Europe, Humboldt brought with him quinine, curare (the nerve poison), and dapicho, a substance similar to rubber. He was the first to stress the glories of the Inca and Aztec civilizations. In Paris he met Simón Bolívar, with whom he was to correspond until Bolívar’s death in 1830. Both could see the need for scientists to help in the development of Bolivia, and Humboldt did all he could to help.
The journal he composed in the course of and after his travels was eventually published—in thirty-four volumes over twenty-five years. Not the least of its attractions were some 1,200 copperplates showing South American flora, fauna, and topography. He also wrote many specialist formal scientific treatises, in which he developed climatology as a science, established the specialities of plant geography and orography (the science of mountains), and initiated ideas that we still use today, such as mean temperature and the isotherm.
