Great Calculations: A Surprising Look Behind 50 Scientific Inquiries

by Colin Pask

  Equation (8.9) is used to investigate the scaling properties of animals, and y might be, for example, lifespan, lung volume, brain size, or skeleton size.

  There is an important mathematical step that is taken to make the scaling easier to handle and interpret. We make use of logarithms (to base 10 here) as introduced in chapter 3. If we take the log of equation (8.9) we get

  If we plot the logarithm of y against the logarithm of M, equation (8.10) tells us that we will get a straight line with slope b. Straight lines are very easy to deal with, and the comparison curves are illustrated in figure 8.4. Notice also that if M varies from 1 to 10 to 100 to 1,000, the log (to base ten) varies from 0 to 1 to 2 to 3, meaning that the enormous range from 1 to 1,000 is simply 0 to 3 on the log scale. This is usually called plotting on a log-log scale. Notice also that it is very easy to read off the slope of a line and hence find the exponent b.

  (Readers wishing to see a more expansive introduction to the subject should seek out the classic Scaling: Why Is Animal Size So Important? by Knut Schmidt-Nielsen, or introductory biology books such as those by Burton, Schmidt-Nielsen (1972), and Vogel.)

  Figure 8.4. Examples of how curves of the type in equation (8.9) translate into straight lines when plotted on a log-log scale. Figure created by Annabelle Boag.

  8.5.2 A First Example

  The calculations we are considering take data about some quantity and investigate whether it follows a scaling law of the form described by equations (8.9) and (8.10). As an example, I take the mass of mammalian skeletons as studied by H. D. Prange and others (see also Schmidt-Nielsen, chapter 5). They took the data, converted it to logarithmic form, and produced graphs like the simplified version shown in figure 8.5.

  The first thing we gain from figure 8.5 is that there does appear to be a relationship between the mass of the skeletons of animals and their total mass. Of course, there is some scatter, as there always will be in biological data. However, the trend, or regularity, is very clear; remarkably, it covers around five orders of magnitude in the masses involved, from animals weighing a few grams to those weighing hundreds of kilograms.

  As a first guess, we might expect the skeleton mass Mskel to increase in exactly the same way as the total mass, but this is not what is revealed in figure 8.5:

  expected Mskel = aM according to the data Mskel = aM1.09.

  According to the calculation made by Prange and his colleagues, the scaling exponent is 1.09 rather than 1; the graph in figure 8.5 does not have slope equal to one. This implies that if the mass M increases by a factor of ten, we expect that the mass of the skeleton increases according to the factor 101.09 = 12.3 times.

  This is an example of the importance of the exponent in scaling calculations. We now need to ask why the exponent takes on its particular value. I have chosen this case because it is easy to think about bone sizes or cross-sections, the weight to be supported by the bones without them being crushed, and the different lifestyles of the animals involved. Figure 8.6 shows the skeletons of a cat and an elephant drawn to the same size. It is quite clear that the cat is much finer-boned; we know that it has relatively little weight to support, and it readily jumps and prances around in ways that an elephant does not. Interestingly, Galileo was already writing about these variations in bone sizes in his 1638 Two New Sciences. We should expect heavier animals to have more massive skeletons to support them, and we can argue for scaling exponents greater than one.

  Figure 8.5. Skeletal mass versus body mass for mammals plotted on a log-log scale. Reprinted with permission, © Cambridge University Press, from K. Schmidt-Nielsen, How Animals Work (Cambridge: Cambridge University Press, 1972).

  This example illustrates four values or roles of scaling calculations:

  Organizing data

  Exhibiting trends and regularities

  Providing a parameter to precisely characterize those regularities

  Suggesting the theories that must be developed to understand the origin of scaling exponents

  Figure 8.6. Skeletons of a cat (left) and an elephant (right) drawn to the same scale. Reprinted with permission of Princeton University Press, from Steve Vogel, Life's Devices: The Physical World of Animals and Plants (Princeton: Princeton University Press, 1989).

  8.5.3 Regularity in the Fire of Life

  Animals use the same chemical processes to maintain life, and surely the most remarkable (and certainly the most debated) scaling result concerns the variations in how these processes operate as a function of the size of the animals involved. Animals use chemical reactions to produce energy and a variety of products that the body needs, and it is the process of metabolism that I now consider. (To describe the definition and technical details would take much space, and I refer you to the books by Schmidt-Nielsen for excellent introductions. For animals, it is the use of oxygen that is readily measured.) We know that animals operate in vastly different ways depending on their size; tiny shrews eat more than their body weight of rich insect food each day whereas elephants consume around 3 percent of their weight and that as vegetation. Is there a regular pattern to be found in the metabolism of very different animals?

  Studies of metabolic rate have a long history (see the books by Schmidt-Nielsen and the full story as given by Whitfield). The most famous results are those given in 1932 by Max Kleiber (1893–1976) in his paper Body Size and Metabolism. An enormous number of follow-up studies have been conducted since then. Kleiber was born in Switzerland, and, after a colorful career (see Whitfield), he finished up in the Animal Husbandry Department at the University of California, Davis. Animal nutrition and needs was an important research topic. The essential results are shown in figure 8.7. The result is sometimes called the mouse-to-elephant curve, and it displays a truly remarkable regularity over about five orders of magnitude. The calculations reveal that

  I have used an exponent of 0.74, but it is commonly referred to as the three-quarters exponent. It was Kleiber who first analyzed the data to produce a three-quarters exponent. He thus overturned the two-thirds law that had become dogma ever since Max Rubner did his famous measurements on dogs and other animals over many years before he retired in 1924.

  The scaling results have been extended to cover a variety of other organisms. (The papers in the report edited by Brown and West give references to a large number of papers on this topic and on other scaling processes in biology.) A great deal more information has now been collected since Kleiber's pioneering work, but what has not changed for many is the strong belief in Kleiber's law: the exponent for metabolic scaling is three-quarters. But what has also not changed is the controversy around this area of biology. There are arguments about the validity of the calculations producing the three-quarters exponent, about the exponent's universality, about its accuracy, about its relevance to metabolic rates under different conditions (resting or exercising, for example), and so on. Kleiber published his book The Fire of Life in 1961, and now, over fifty years later, the debate about metabolism and scaling is still very much alive.

  It is accepted that for many cases, metabolic rates do scale with animal size as measured by mass, and the scaling exponent may be taken as three-quarters. It is the value of the exponent, the three-quarters, that causes the problems. We might expect that the rate must scale with all of the active units in the body, so it should be proportional to mass and the exponent should be one. Or we may consider the heat generated and the mechanisms that animals use to keep temperatures constant, in which case concepts about cooling lead to metabolic rates depending on surface areas and thus to a two-thirds scaling exponent as in Rubner's old theory. In special circumstances, varying exponents may be arrived at, but it does seem that they lie between ⅔ and one, and in many cases, Kleiber's law does appear to hold. The data processing and exponent calculation by Kleiber led to the identification of an amazing regularity that few would expect to find in biology.

  Figure 8.7. The dependence of metabolic rate on body mass for a range of animals. R
eprinted with permission, © Cambridge University Press, from K. Schmidt-Nielsen, Why Is Animal Size So Important? (Cambridge: Cambridge University Press, 1984).

  8.5.4 The Origin of Scaling Exponents

  If the scaling process involved is simply geometric (so quantities scale, like lengths, areas, and volumes) the exponents will be ⅓, ⅔, and 1 as shown in equation (8.8). Many people accept that the calculations for metabolic rates produce exponents with confidence intervals excluding those values but centering on ¾. It also appears that exponents involving quarters rather than thirds are found in many other scaling results for different areas of biology. We now come then to the fourth role of scaling theories and must ask: Why do those particular exponents occur in nature?

  The calculation of scaling exponents involving quarters has opened up a whole new research area in biology, and this might be the perfect example of George Bernard Shaw's complaint: “Science is always wrong. It never solves a problem without creating ten more.”13 A complaint for Shaw maybe, but joy for the curious scientist.

  A new approach was begun by Geoffrey West and James Brown (see their 2004 Life's Universal Scaling Laws). They suggested that the answer lies in the networks (such as those carrying blood) that are found in biological systems and in the ways these networks are constructed and optimized. West and Brown introduced three constraints:

  Networks service all local biologically active regions in both mature and growing biological systems. Such networks are called space filling. [Ideas from fractal mathematics are used here.]

  The network's terminal units [capillaries, for example] are invariant within a class or taxon.

  Organisms evolve toward an optimal state in which the energy required for resource distribution is minimized.14

  Calculations by West and Brown (and other colleagues) have shown that concentrating on networks servicing biological systems leads to scaling exponents that involve quarters, and specifically they calculate an exponent of three-quarters for metabolic rate scaling.

  The calculations, connected with biological scaling laws, have led to the discovery of biological regularities and have stimulated further research and calculations aimed at understanding the origin of these regularities. The range over which metabolic rates may be scaled is quite breathtaking, and it has been extended since Max Kleiber did his original work. I add calculation 30, scaling from mice to elephants as a worthy member of my list of important calculations.

  (This is a very active area of research in biology, and one involving many debates and controversies. To get a flavor of the field, I have given references to some recent papers by Banavar, Glazier, Savage, and Spence as well as references for Brown and West.)

  8.6 WHAT HAS BEEN LEFT OUT

  As I stated at the start of this chapter, I have included only a very small sample of calculations made by scientists exploring the natural world. However, I hope this sampling has indicated the range of subjects involved and the key role that calculations can play. Maybe there should have been more about DNA and its discovery. Biophysicists will wish that I had found space for the Nobel Prize–winning work of Alan Lloyd Hodgkin and Andrew Huxley on signal transmission in nervous systems, and mathematical biologists might feel that Alan Turing's mathematically elegant work on pattern formation merited inclusion. I agree, but unfortunately there had to be some cut-off point.

  in which we examine six calculations trying to unravel the properties and mysterious nature of light.

  “And God said: Let there be light. And there was light.” Whichever religious, mythological, or scientific version of the origins of the universe appeals to you, one thing is indisputable: one of the strangest components of our world is light. Dr. Samuel Johnson, compiler of the first dictionary, told his biographer: “We all know what light is, but it is not easy to tell what it is.”1 Johnson confronts this problem in his 1755 A Dictionary of the English Language, writing:

  light: that quality of action of the medium by which we see.

  Light is propagated from luminous bodies in time, and spends about seven or eight minutes of an hour in passing from the Sun to the Earth.2

  Johnson seems to identify two aspects of light: its nature and its properties. The former is difficult to explain, and Johnson resorts to quoting some of the latter in his dictionary definition.

  It is still difficult to give a definitive answer to the question about the nature of light, but the choice of theoretical concept will decide the methods used to explain its properties. In this chapter, we will see calculations relevant to both aspects, and we will conclude with a statement by Einstein that would surely have met with Johnson's approval.

  9.1 THE SPEED OF LIGHT

  It is clear from echoes and the rumble of distant thunder that sounds travel with a finite speed. But what of light? For many centuries, the general opinion was that the speed of light was infinite, though there were, of course, other opinions. In ancient Greece, Empedocles suggested a finite speed, but mostly authorities like Aristotle and Heron dismissed this opinion. Toward more modern times, both Kepler and Descartes believed that the speed of light was infinite, although the respected Arab physicist Alhazen (965–1040) was in the finite-speed camp, as was Christiaan Huygens. Galileo suggested that an experiment was needed; a first observer would uncover his lantern to send light to a distant second observer who, on seeing it, would uncover his lantern and send a return light to the first observer. The delay in receiving the returned signal would give the speed of light. Galileo tried it with a one-mile separation and (obviously) failed to make a satisfactory measurement. Clearly, a new approach was required.

  9.1.1 Using the Moons of Jupiter

  A method of determining longitude was eagerly sought in the sixteenth and seventeenth centuries. Galileo had observed moons around Jupiter, and many measurements were made of their orbits and periods in the hope that variations could be used for this illusive method. Ole Roemer (1644–1710) realized that Jupiter's moons—more precisely, its innermost moon—had another use and that was to measure the speed of light. This is another excellent example of how observations may be used in a simple calculation to give a result of major significance.

  Ole Roemer was born in Aarhus on the coast of Denmark. (His name is variously spelled as Romer, Römer, and Rømer.) Although little known today, he was one of the most talented and productive men of his time. He was a scientist and mathematician of note, and he was involved in many important astronomical researches in France. If there was any justice in the world, we would celebrate Roemer rather than Fahrenheit as a pioneer designer for thermometers. He was an engineer (including in maritime matters) and studied cardioids in his quest for the optimum shape for gears. He also held civic positions in Copenhagen, such as inspector of naval architecture, purveyor of pyrotechnics and ballistics, first magistrate, mayor, and chief tax assessor. (The article by Cohen is a fascinating account of Roemer and his work. Also see Daukantas.)

  Figure 9.1. Roemer's diagram for the motions of the earth E around the sun A, and a satellite around Jupiter B. From Ole Roemer, “A Demonstration Concerning the Motion of Light” (1676).

  Roemer's work on the speed of light can be readily appreciated using his original 1676 paper “A Demonstration Concerning the Motion of Light.” Figure 9.1 is his diagram for explaining his method. Roemer writes:

  Let A be the Sun, B Jupiter, C the first satellite of Jupiter, which enters into the shadow of Jupiter, to come out of it at D; and let EFGHLK be the Earth placed at divers distances from Jupiter.

  Now, suppose the Earth being at L towards the second Quadrature of Jupiter, hath seen the first satellite at the time of its emersion or issuing out of the shadow in D; and that about 42½ hours after, (vid. after one revolution of this satellite) the Earth being at K, do see it returned in D; it is manifest that if the Light require time to transverse the interval LK, the satellite will be returned later in D than it would have been if the Earth had remained at L, so that the revolution of this satellit
e being thus observed by the Emersions, will be retarded by so much time, as the Light shall have taken in passing from L to K, and that on the contrary, in the other Quadrature FG, where the Earth by approaching goes to meet the Light, the revolutions of the Immersions will appear to be shortened by so much, as those of the Emersions had appeared to be lengthened.3

  Roemer is saying that the innermost moon of Jupiter has a period that appears to be different depending on whether the earth is approaching or moving away from the planet, and this difference is linked to the finite speed of light.

  Roemer notes that the effect is not great if observed over one moon orbit, but

  that what was not sensible in two revolutions, became very considerable in many being taken together, and that, for example, forty revolutions observed on the side F, might be sensibly shorter, than forty others observed in any place of the Zodiack where Jupiter may be met with; this is in proportion to 22 minutes for the whole interval of HE, which is double of the interval that is from hence to the Sun.

  Roemer has found that the speed of light is such that it takes 11 minutes to travel from the sun to the earth. He confirms that recorded observations at the Paris Observatory fit his theory. He also makes a calculation and reports on an observation made confirming his prediction:

  It hath been lately confirmed by the Emersion of the first satellite observed at Paris the 9th of November last at 5 a Clock, 35’ 45” at Night, 10 minutes later than it was to be expected, by deducing it from those that had been observed in the month of August, when the Earth was much nearer to Jupiter: Which M. Roemer had predicted to the said Academy from the beginning of September.

  Roemer goes on to say that none of the properties of the satellite's orbit could explain the observed effect. Nothing impresses like a confirmed prediction, and from this time on, the finite speed of light was accepted. The conversion of times into speeds requires accurate values for the distances, which were not available in Roemer's time, and it is not claimed that Roemer accurately determined the speed of light—but Roemer did show how the speed of light can be measured and demonstrated the viability of his approach. (Boyer gives an entertaining discussion of the various values calculated and used by many observers in that early period of science.)