3
MOVING THE LINE

EVEN BY THE WEIGHTY standards of nineteenth-century scientific German, Max Rubner’s Die Gesetze des Energieverbrauchs bei der Ernährung was a notoriously indigestible slab of prose. But Max Kleiber, who was sent a copy by his professor, was one of the few people to read it all the way through. Kleiber had nothing better to do at the time—he was in prison.

To call Max Kleiber a free spirit hardly does him justice. Serving in the Swiss army during the First World War—Switzerland was neutral but mobilized its military to defend its borders—he was dismayed to learn that senior Swiss officers had been passing information to the Germans. Believing that the chain of command had been corrupted, he felt he could no longer pass on orders; Kleiber ignored his next call-up and was arrested and jailed.

The initial military summons had found him in Canada. After a year of university in Switzerland, during which he scandalized Zurich society by walking about town hatless, in sandals, and with an open collar, Kleiber decided the academic life was not for him and that he wanted to get as far as he could from people and their works. Along with two friends, he tried to make a life as a homesteader in Alberta,



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In the Beat of a Heart: Life, Energy, and the Unity of Nature 3 MOVING THE LINE EVEN BY THE WEIGHTY standards of nineteenth-century scientific German, Max Rubner’s Die Gesetze des Energieverbrauchs bei der Ernährung was a notoriously indigestible slab of prose. But Max Kleiber, who was sent a copy by his professor, was one of the few people to read it all the way through. Kleiber had nothing better to do at the time—he was in prison. To call Max Kleiber a free spirit hardly does him justice. Serving in the Swiss army during the First World War—Switzerland was neutral but mobilized its military to defend its borders—he was dismayed to learn that senior Swiss officers had been passing information to the Germans. Believing that the chain of command had been corrupted, he felt he could no longer pass on orders; Kleiber ignored his next call-up and was arrested and jailed. The initial military summons had found him in Canada. After a year of university in Switzerland, during which he scandalized Zurich society by walking about town hatless, in sandals, and with an open collar, Kleiber decided the academic life was not for him and that he wanted to get as far as he could from people and their works. Along with two friends, he tried to make a life as a homesteader in Alberta,

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In the Beat of a Heart: Life, Energy, and the Unity of Nature Max Kleiber (1893–1976). Credit: University of California, Davis. Canada, the most remote and least populated place they could afford to get to. When he arrived, however, he found that he had over-estimated his disdain for society. “This experiment made me aware of the degree to which Homo sapiens is a social animal,” he wrote, and tutoring his neighbor’s daughter in mathematics turned out to be more rewarding than eking out a living from the land. However, Kleiber was reluctant to give up the agrarian ideal and had another go at farming

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In the Beat of a Heart: Life, Energy, and the Unity of Nature back in Switzerland, once his four-month prison sentence was over. He founded a commune with a group of like-minded conscientious objectors. But arguments within the radical collective, and his own realization that he was a better scientist than farmer, sent him back to university. His prison term had led to his expulsion from Zurich’s College of Agriculture, but the same professor who had sent him Rubner’s book pulled strings to get Kleiber readmitted. In 1929, Kleiber emigrated with his wife and daughter to the United States, to work at the University of California’s agricultural college in Davis, a town about two hours’ drive inland from San Francisco. He got the Davis job only because a more eminent European animal physiologist wanted more money than the University of California was willing to pay, but he ended up spending the rest of his life there. Neither emigration nor age dimmed his radicalism. In 1949 he became the first University of California professor to refuse to sign an anticommunist loyalty oath, which the university heads had demanded from all their employees. He demonstrated against the atom bomb and took on Edward Teller, the father of the U.S. nuclear weapons program, in public debate. Late in life he protested against the Vietnam War. “I seem to have a greater than average allergy against mental schisms,” he wrote. “This means an enhanced tendency to express my ideas through actions.” He now has a building on the Davis campus named after him. Davis originally hired Kleiber to build and use equipment to measure respiration and metabolic rate in cattle. This is an important agricultural question because understanding how metabolism changes with size helps farmers calculate how much feed their livestock need and the rate of growth and the quantity of milk and meat that they should expect from an animal. Soon after he arrived at Davis, Kleiber used these measurements to provide biology with one of its few universals and one of its most enduring mysteries. Cracks in the Surface Law In the late 1920s, Rubner’s surface law of metabolism was dogma. Scientific studies often gave metabolic rate as a function of body surface

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In the Beat of a Heart: Life, Energy, and the Unity of Nature area alone, without reporting body size or height. Studies that didn’t conform to the law’s predictions were usually dismissed as inaccurate. But the anomalous measurements were beginning to stack up, and researchers were realizing that previous studies of both metabolism and surface area might not be as accurate as they had thought. For a start the techniques and procedures used for measuring metabolic rate had moved on since Rubner’s day. Rubner’s calorimeters were excellent, but his experimental set ups were not the best suited to getting consistent, reproducible results. Rubner’s sample sizes—one experiment involved 2 men, 5 dogs, 5 rabbits, 3 guinea pigs, and 12 mice—would today be seen as too small to allow for firm conclusions. During his time, beer was a routine part of experimental diets—now a strict no-no, as I discovered when I had my own metabolic rate measured. His favored period for taking measurements was 24 hours, a long time to expect a dog to sit still. And his favored experimental temperature was 16°C; his animals must have been shivering to keep warm. Eugene DuBois said this about the experiments underpinning the surface law: “Like much of Rubner’s work, the experiments and measurements contained many mistakes, but Rubner was a genius who drew correct conclusions from data inadequate for any other man.” DuBois was reluctant to let the surface law go, which was not surprising given all his work on surface area, but even his backhanded compliment to Rubner’s methods came to look increasingly optimistic. This isn’t to say that Rubner was a fraud or an incompetent. Like many a scientist before and since, he may have been reluctant to let the data get in the way of a good theory, but he and the other biologists taking these measurements in the nineteenth century were exploring uncharted territory. You can now buy off-the-shelf apparatus and software to measure metabolic rate and process the results, but Rubner’s generation had to find out for themselves what worked. There were no standard recipes to follow. For example, it became common to measure metabolic rate in fasting animals. But how long should this fasting period prior to measurement be? A rat will empty its stomach more quickly following a meal than a dog will. A ruminant such as a cow never really stops digesting, and its gut contents make up a good

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In the Beat of a Heart: Life, Energy, and the Unity of Nature percentage of its weight. The diversity of methods made it hard to compare results coming out of different laboratories. So it took a while to work out a standard measurement for metabolic rate. But these difficulties were trivial compared with measuring surface area. Each measuring device, technique, and approximation formula gave wildly different results. Skin has many folds and is immensely stretchy—how taut should it be when measured? If you are measuring the skin taken from an animal, should you peg it out for measurement or let it lie slack? Does surface area include internal surfaces that interact with the outside world, such as the lungs and intestines? If so, how do you measure or estimate their area? The exposed area of a living animal is changing all the time, as it curls up for warmth or stretches out. Some biologists took a back-to-front approach, using the surface area measurement that gave the best fit with the surface law. Rabbits’ metabolic rates, for example, gave a better fit if you ignored their ears, so they were dismissed as anomalous appendages. DuBois suggested that surface area measurements were reliable as long as the same method was used for each study. But even this turned out to be untrue: Samuel Brody found that, in the hands of three different investigators, the same technique for measuring the surface area of a rat gave answers varying by 60 percent. It was starting to look as if Rubner’s conclusions were no more accurate than his measurements. It was chaotic—Kleiber compared the situation to the measurement standards of the middle ages, when the length of a foot varied from town to town. Biologists began to despair of measuring surface area accurately enough to tell whether it matched an animal’s metabolic rate. But when Rubner’s law began to fall from favor, it was replaced not by chaos but by a different form of order that was much harder to explain. From 2/3 to 3/4 Never one to let a received opinion go unchallenged and never afraid to dissent, Kleiber was the first to spot this new form of order. He compiled measurements of metabolic rate in different-sized animals,

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In the Beat of a Heart: Life, Energy, and the Unity of Nature Max Kleiber’s 1932 analysis showed that metabolic rate is proportional to body mass raised to the power of 3/4. ranging from a rat to a steer, and published the results in the Davis house journal Hilgardia in 1932, the same year that Rubner died in Berlin. Kleiber’s graph from this paper, of metabolic rate plotted against body mass, remains well known among biologists to this day. Metabolic rate does not correspond simply with body mass, so the graph should be a curve. Kleiber made the curve a straight line by plotting the logarithm of mass against the logarithm of metabolic rate. The notches on a logarithmic axis mark proportions—instead of ticking along from 1 to 2 to 3, they go from 1 to 10 to 100 to 1,000. Because metabolic rate changes at a constant rate as body mass increases, plotting the axes in this fashion produces a straight line, making it a lot easier to compare animals at different points on the graph. The logarithms help us see past the absolute weight difference between two animals to see how many times bigger or smaller one animal’s mass and metabolic rate are than another’s. As we can see, Kleiber’s comparison revealed a regular relationship between the logarithms of mass and metabolic rate, getting rid of any

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In the Beat of a Heart: Life, Energy, and the Unity of Nature need for complicated and almost certainly unreliable measurements of body surface area. Although the points for each species on the graph of mass versus metabolism are averages—any one animal is unlikely to lie right on the line—the correlation is nevertheless tight. Correcting for body mass accounts for more than 90 percent of the variation in metabolic rate. But in cutting through the empirical muddle, Kleiber created a theoretical problem. The gradient of this line is 0.74—nearly three-quarters, not the two-thirds, or 0.67, the surface law predicts. In other words, bigger animals produce more heat per unit of surface area—mice are a bit cooler than cats after all—and bigger animals need more food than the surface law would predict. The surface law predicted that metabolic rate rises 100-fold for every 1,000-fold increase in body mass; Kleiber found a metabolic rise of 180 times for the same weight gain. “Modern American animals take the surface law less seriously than did the earlier European animals,” he concluded. “The gadgeteers who designed apparatus for surface measurements, the statisticians who derived formulas for calculating ‘true’ surface areas, and the theoretically inclined biologists who discussed the proper interpretations of the surface law seem not to have been interested in the reliability of the surface law itself.” To work out how many calories a warm-blooded animal burns each day, said Kleiber, calculate the 3/4 power of its mass (i.e., cube it and then take the fourth root of that number. For example: ) and then multiply by 70. Doing this sum for me, with my weight of 76 kilograms, gives a figure of 1,802 calories per day—very close to Dr. Costarelli’s measurement. Kleiber chose three-quarters because the relationship he found of 0.74 is statistically indistinguishable from 0.75, and raising a number to the power of 3/4 was much easier to calculate using slide rules, which at that time were biologists’ most powerful computational aid. To translate Kleiber’s law into the language of mathematics, metabolic rate = constant × mass3/4. Kleiber might have been the first to publish this relationship in a scientific journal, but as he was later delighted to learn, it was first aired in 1699. In Gulliver’s Travels the Lilliputian king allocates Gulliver a daily ration equivalent to the diet of 1,724 of his subjects. Assuming

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In the Beat of a Heart: Life, Energy, and the Unity of Nature that the average Lilliputian was about the same size as Gulliver’s index finger, Kleiber calculated that Gulliver was 26 times higher than a Lilliputian and so weighed 263, or 17,600 times more than him or her; 1,724 is 17,600 to the power of 0.76. Allometry The relationship between metabolic rate and body weight is an example of a biological pattern called allometry, which compares how the value of any biological trait, such as metabolic rate or leg length, changes with the total size of a plant or animal. It shows whether, as things get bigger, they become proportionately bigger or smaller—in other words, if their shape changes. For example, baby humans have big heads relative to their bodies. But as they grow up, their heads grow more slowly than their bodies, bringing about adult proportions. Other features get proportionately bigger as an animal’s size increases. Deer antlers show this trend: The biggest British species, the red deer, is about 4 feet tall at the shoulder, and its antlers span about 3 feet. The largest deer that ever lived, the extinct Irish Elk, was less than twice as tall, standing at about 7 feet. But its antlers were 12 feet across. In other words, big deer have really big antlers. But although the Irish Elk’s antlers might seem freakish, the allometric comparison across all deer species shows that they are as big as we should expect for a deer of that size. The power of allometry is that it allows us, literally, to put things in proportion. The examples of babies and deer show that allometric comparisons can be made within species, between species, at different times during an individual’s growth, or across a population of animals of the same age. Allometry can be applied to fossils, to study a species’ changes in shape through time. It can also show when something is anomalously small or large. Perhaps no other theory in biology can match its sheer usefulness: The data for allometry studies are usually easy to collect and interpret; the analysis is transparent—biologists are not generally good at mathematics, but anyone can understand the allometric equation—and its results can be presented in a visual fashion. Allometry is one of the most powerful techniques for revealing patterns in biology.

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In the Beat of a Heart: Life, Energy, and the Unity of Nature To pop back into math-speak: The general form of the allometry equation is y = axb. If b is greater than 1, the trait gets proportionately larger, like deer antlers; if b is less than 1, as is the case with metabolic rate or babies’ heads, it gets proportionately smaller. A good analogy is two bank accounts with different rates of compound interest. Over time the amount of money in each account will diverge, but each account’s growth rate will remain the same. If b is zero, the y variable remains constant. If b is less than zero, the trait declines with increasing size. This is what happens with relative, or cellular, metabolic rate. If metabolic rate is proportional to body mass raised to the power of 3/4, then relative metabolic rate will be proportional to this figure divided by simple mass (mass to the power of 1, in other words)—which equals body mass to the power of −1/4. Kleiber helped show that the idea of allometry could extend beyond solid anatomical features such as horns and into the body’s processes, such as metabolism. Biological growth, like financial growth, is a compound process—another reason logarithmic axes are more useful than arithmetic ones when comparing different-sized animals. Cells arise from cells, so the amount of new material added in any time period depends on how much there was before. Allometries are an example of something called a power law, so called because the y variable depends on the x variable being raised to some power. Power laws spread far beyond biology. The frequencies of earthquakes and landslides of different sizes follow them, as do the sizes of air bubbles in a breaking ocean wave or the length of the waking periods during a night’s sleep. They also apply to social patterns, such as the length of time patients in the United Kingdom wait for a hospital operation, or the size of changes in financial markets, the frequency with which words appear in a language, and the popularity of people’s names. Power laws also relate the size of an event to its frequency. Crudely put, in systems that follow negative power laws, the probability of events such as quakes or stock market crashes declines at a constant rate as those events get bigger. Small earth tremors are common; city-destroying catastrophes are rare. As for sleep patterns, most of the time you will wake up long enough only to turn over and go back to sleep, but occasionally you’ll lie awake for what seems like half the night.

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In the Beat of a Heart: Life, Energy, and the Unity of Nature Allometry is most closely associated with the British biologist Julian Huxley. Huxley was an intellectual aristocrat. He was the grandson of Thomas Henry Huxley, the Victorian biologist whose robust defense of evolutionary ideas earned him the nickname Darwin’s bull-dog; the brother of author and philosopher Aldous, whose books include Brave New World; and the half-brother of Andrew, who won a Nobel Prize in 1963 for his studies of how nerves work. Julian made the rest of the clan look like slackers. He did important work on animal behavior, genetics, evolution, and developmental biology. He was a busy media scientist, producing a stream of journalism, books (including a volume of poetry), and radio and television broadcasts, and codirecting a pioneering nature film about seabirds, The Private Life of the Gannets, which in 1937 made him the only biologist so far to win an Oscar. Huxley was active in social, political, and philosophical debates and campaigned against the bogus reasoning of Nazi race science and Soviet genetics. He argued for better diets and against air pollution. He lobbied for, and helped establish, the first government-supported nature reserves and national parks in Britain. He advocated a humanist view of religion that excluded the supernatural, which caused him to be branded “Europe’s leading atheist” in the United States. He ran the London Zoo before and during the Second World War—once chasing down an escaped zebra during an air raid—and became the first director general of UNESCO, the United Nations Educational, Scientific, and Cultural Organization. In 1924, Huxley published what became a seminal paper showing that male fiddler crabs develop one disproportionately large claw because this claw grows about six times more quickly than the rest of their bodies. He offered the allometry equation as the mathematical description of this pattern. Huxley went on to show that this equation applied to other features of animals’ anatomy, including deer antlers, and that different animal forms were the result of different growth rates. By the late 1920s and 1930s, measuring patterns of this sort was one of the hottest areas of biology. Huxley wrote a book on the subject, Problems of Relative Growth, and dedicated it to D’Arcy Thompson, the most passionate advocate of the links between mathematics and animal form. Thompson was ungrateful. His letters to Huxley show

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In the Beat of a Heart: Life, Energy, and the Unity of Nature that he thought he had anticipated this work in On Growth and Form—“I never thought it necessary to coin a phrase for it,” he wrote—and he took issue with Huxley’s use of logarithms and power laws, arguing that simple arithmetic would have been just as good at describing growth. Most biologists, however, followed Huxley’s lead. Noah’s Scales Across the Atlantic, Kleiber was not challenging the surface law single-handedly. Brody too had lost faith, coming to the conclusion that surface area was impossible to measure accurately and bore no relationship to metabolism even if it was. In Missouri his team measured the metabolic rate of more and more species of a broadening range of sizes. In 1934 they published a graph that stretched from a mouse to an elephant. The gradient of this was 0.73, statistically indistinguishable from Kleiber’s result. Both, however, were significantly different from the value of 0.67 predicted by the surface law. By the late 1930s even DuBois was wavering: “The metabolism per square meter shows a pronounced tendency to be increased with the increasing size of the animal,” he wrote. “The puzzling question [is] why small animals, like mice, have a lower metabolism per unit of surface [area] than large animals, like horses.” As for the answer to this puzzle, DuBois was showing signs of fatalism. “I do not know where or when the various species of animals were given their basal metabolism,” he wrote. “Perhaps Noah did it when they left the Ark. I suppose that the reason he could not do a uniform job with the animals was because he did not have scales small enough for the dwarf mice and large enough for the bull and the elephant.” Yet surface area approximations, including the DuBois brothers’ formula and updated versions of it, are still used in medicine today, although weight is used more often. Calculations of body surface area are used to prescribe drug doses in chemotherapy treatments. Patients’ fluid and energy requirements are often calculated the same way. And some of the body’s workings are still most commonly expressed in terms of its surface area. These include the rate at which the kidneys filter waste from body fluids, and the cardiac index, a measure of how

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In the Beat of a Heart: Life, Energy, and the Unity of Nature hard the heart is working, which is defined as liters of blood pumped per minute divided by surface area. If surface area is hard to define, nigh on impossible to measure, and not a particularly good proxy for metabolic rate anyway, why is it still used? The main reason is probably the medical profession’s inertia or, more politely, tradition. Medical practice is driven by what works, rather than the best current scientific knowledge. And in truth there is only a small difference in the answers given by calculations based on body area and those based on body weight. Combined with the human body’s ability to stabilize itself, adjust to different feeding and fluid regimes, and process different quantities of drugs, the medical use of body surface area calculations, although arguably irrational, does little or no damage. In the same way, the different doses recommended for adults and children on over-the-counter drug labels are not precise calculations, but the approximation should be harmless. Anyway, the issue is becoming moot: In hospitals, direct measurements of patients’ metabolic weight will probably supercede calculations based on both body weight and area—provided doctors can be persuaded to give up their time-honored mathematical spells. DuBois’s doubts were in part the result of a long debate with another opponent of the surface law, the physiologist Francis Benedict. By 1930, Benedict had spent two decades measuring metabolic rate in hundreds of humans and other animals. He learned physiology from an American who had shared a lab with Rubner, and in the early twentieth century he extended Rubner’s work by showing that alcohol too could be burned to release energy. The work caused consternation in the temperance movement, especially as it was carried out at Wesleyan, a college in Middletown, Connecticut, run on teetotal Methodist lines. Benedict, himself an abstainer, repaid his debt to sobriety during the Prohibition years, when he turned in the chief bootlegger in his Maine hometown to the authorities. In 1907 he left Wesleyan to run the Carnegie Institution’s newly established Nutrition Laboratory in Boston. Benedict is remembered for his work on human metabolism. The Nutrition Laboratory built up a detailed picture of people’s energy requirements and how they changed with weight, height, age, and sex.

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In the Beat of a Heart: Life, Energy, and the Unity of Nature Along with Eugene DuBois’s investigations, the lab set the standard for measuring metabolism: subjects reported for their trial at 8 a.m., having fasted for 12 hours, and lay down for 30 minutes to calm themselves before measurement. Subjects ranged from newborns only two hours old to 93-year-olds who had lived through the Civil War, and from college athletes to vegetarians—who at the time were so rare that Benedict traveled to the Kellogg sanatorium in Michigan to find 10 of them to measure. In 1919 he and his colleague Arthur Harris used this impressive data set to derive equations to calculate metabolic rate from knowledge of height, mass, sex, and age, now called the Harris-Benedict equations. These equations are pretty accurate, although they tend to overestimate metabolic rate by about 5 percent—for me they predict a resting metabolic rate of 1,815 calories per day. They are still widely used to calculate approximate energy needs. The Nutrition Laboratory moved on from humans to probe the metabolisms of a menagerie of other animals. Besides domestic species such as sheep and cows, and laboratory favorites such as rats and dogs, Benedict persuaded zoos and circuses to lend him exotic beasts such as the cassowary, a flightless bird from Papua New Guinea; the marmot, an alpine rodent; macaque monkeys; and chimpanzees. Having all these measurements done in the same lab, under the direction of one person, gave them an internal consistency that made them the best metabolic comparisons between species done to that date. In 1938, a year after he retired, Benedict published a monograph, summing up his life’s work, titled Vital Energetics. The book included a figure showing the increasingly familiar logarithmic plot of metabolic rate against mass, with, as he wrote, “a most gratifying straight-line relationship between the total heat production and the body weight.” The gradient of this straight line was 0.73. Benedict, however, resisted gratification. He saw the pattern in his results as a mirage and warned others against the intoxicating effects of the search for a higher order: It is obvious that this apparent straight-line relationship is of no physiological significance, whatever its mathematical significance may be thought to be…. It seems illogical to make use of complicated mathematics in the attempt to unravel the end results of the pooled activities of millions of cells, each acting differently. [A]ll attempts by mathematical means to

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In the Beat of a Heart: Life, Energy, and the Unity of Nature secure a uniform expression of the basal metabolism findings on different animals species are utterly futile…. [N]o unifying principle in metabolism has been found to exist. It takes an unusual scientist to collect this much information and not see any trends in it—most, using the human gift for spotting patterns, will begin joining the dots as soon as they have two data points. And as Kleiber pointed out, measuring the metabolic rate of one animal is a unification, a summing-up of the pooled activities of millions of cells—does that make it a meaningless quantity? “If this is the way Benedict feels,” he retorted, “one cannot help but wonder how he ever became interested in conducting a respiration trial.” The dispute between lumpers such as Kleiber and splitters such as Benedict about how we should compare different measurements of metabolic rate and what such comparisons reveal continues to this day. But both Kleiber and Benedict agreed that the surface law could no longer stand. To misquote Thomas Huxley, the surface law was a beautiful theory destroyed by beautiful facts. But Rubner’s law was one of those wrong ideas that, because of all the thought, arguments, and experiments it stimulated, proved a lot more useful and influential than many a correct notion. In Rubner’s time, biological knowledge, with the giant exception of Darwin’s theory, was a pile of facts. Biologists accumulated information about the living world, but there was little attempt to put it into context, to see if any larger structures emerged from the mass of details. In the years between the two world wars, the debate on whether the surface law held or not was one of the most active in biology. Without Rubner’s search for laws of metabolism, Kleiber might never have been stimulated to find general trends in the data. Even today, biological principles that make firm predictions across a wide range of species are extremely rare. Kleiber’s rule is one of the few examples and one of the most precise. If Bergmann’s rule was like Kleiber’s, rather than being a general trend with many exceptions, we could point to a spot on the map and say with reasonable confidence what size animals lived there. Yet Kleiber’s discovery only made things more puzzling. To believe that metabolic rate corresponds to the relationship between body mass and surface area, or mass raised to the power of 2/3, is intuitively

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In the Beat of a Heart: Life, Energy, and the Unity of Nature satisfying and makes good mathematical sense. There is no reason to expect metabolic rate to correspond to body weight raised to the power of 3/4. It’s like trying to understand the ultimate question of life, the universe, and everything: If 3/4 (or 42) is the answer, what’s the question?

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