Great Calculations: A Surprising Look Behind 50 Scientific Inquiries

by Colin Pask

  Figure 4.1. Geometry for the sun's rays striking the earth at Alexandria (A) and Syênê (S), which are a distance d apart. Figure created by Annabelle Boag.

  We assume Alexandria and Syênê are a distance d apart on the same north-south meridian, and then, by a simple proportion argument, we get

  The measurement technique for obtaining θ is shown in figure 4.2. Measuring the shadow cast by a vertical rod gives the required angle through the fact that the shadow length is proportional to tan(θ). If the rod is at the center of a spherical dish then the length of the shadow in the dish is directly proportional to the angle θ.

  Figure 4.2. Measurement of angle θ using the shadow cast by a vertical rod of length p on a plane or a spherical surface. Figure created by Annabelle Boag.

  Eratosthenes found that θ was one fiftieth of a whole circle (or 7.2º), and the distance d between the cities was 5,040 stades. Equation 4.1 gives the earth's circumference C as 50 × 5,040 or 252,000 stades. This may be translated as 39,690 kilometers, which is remarkably close to the modern figure for the polar circumference: 40,009 kilometers. The mathematics may be trivially simple, but this calculation at last gave a believable estimate for the size of the earth. It is my calculation 8, Eratosthenes measures the earth.

  4.1.2 Analysis

  Because this is such a simple and transparent calculation, it provides a good example for learning about various factors that should be considered when a calculation is used to describe a physical situation. Perhaps surprisingly, there are many issues involved.

  Any physical theory will rest on certain basic assumptions. In this case, Eratosthenes assumes a spherical Earth (or at least a great circle for the circumference under consideration). Newton later showed that the earth is not a sphere, and that is confirmed by accurate surveying. However, little error is introduced here. Eratosthenes must also assume that the rays of the sun are parallel as they strike the earth, which is not too unreasonable. Finally he must assume that Alexandria and Syênê do actually lie north-south on a great circle passing through the poles. In fact, their longitudes differ by about 3º, so a small error is made.

  Any calculation of a physical quantity will need input data, which may be a mixture of physical constants and data from measurements. In this case, values for θ and d are required. We cannot be certain exactly how θ was measured, and the fact that the sun is of finite size means there would be some fuzziness in the shadow being observed. According to Dutka (see bibliography) θ should be 1/50.6, not 1/50, times the value 360 for the full circle. We do not know the precision with which Eratosthenes lined up the rod so that it was vertical and pointing to the center of the earth as assumed in figure 4.2. An error is thus introduced, but we should note that the magnitude that Eratosthenes calculates for C is still substantially correct. The other piece of information required is the value of d. There is much speculation about how d might have been measured—see the references to Nicastro and Dutka. Again some error was obviously introduced, but it is not sufficient to invalidate Eratosthenes's conclusion.

  For any calculation, we need to consider the interpretation of the result. In this case it is clear that a meaningful estimate of the earth's circumference has been obtained. The real problem here is to be sure what the ancient unit of a stade actually referred to. This is a complicated matter, and I refer you to Nicastro and Dutka for a very detailed analysis of the possibilities.

  In a modern calculation, there is often an assessment of likely error given alongside the final result. In this way, the various uncertainties in the method and the input data can be taken into account to give error bounds for the answer.

  Despite all of the above points, I believe that Eratosthenes came up with a very good value for the size of the earth, and his work is worthy of inclusion on any possible list of great calculations.

  4.2 WHAT IS THE MASS THE EARTH?

  If we know the size of the earth and make some assessment of its density, we may find an approximate value for the mass of the earth. This approach is linked to knowledge of the constitution of the earth which will be considered in section 4.4. I will return to the mass of the earth in the next chapter where a quite different approach will be discussed.

  4.3 HOW OLD IS THE EARTH?

  All cultures have their myths and creation stories, some of which suggest how long ago the earth was created. For some people, including Aristotle, no age was involved; the world was eternal. In the nineteenth century, particularly in Britain, a conflict arose as a number of groups of people were interested in estimating the age of the earth, and diverse figures were being suggested. At that time, the question was still quite open: Was the earth a few thousand years old? Hundreds of thousands of years old? A few million years old? Or even many millions of years old? This was a time of great progress and upheaval in the sciences, and knowing the age of the earth was of major importance. (The article by Badash gives a good brief history of the subject.)

  For Christians, the Bible may be taken as the book giving all necessary knowledge; the Bible may be assumed to give the literal truth, a belief still held by many people even today. The creation of the earth is described in Genesis, and the chronology of later events and generations may be used to work back to a creation date. Several people worked on the details (see chapter 2 in the book by Weintraub), but the most famous is Bishop James Ussher (1581–1656). He worked with great precision to come to the view that

  In the beginning God created Heaven and Earth, Genesis. 1, v. 1, which beginning of time, according to our chronologie, fell upon the entrance of the night preceding the twenty third day of October in the year of the Julian Calendar, 710 [equivalent to 4004 BCE].1

  Thus the fundamental religious view suggests an age of thousands of years for the earth. Although many people would dismiss Ussher's estimate as ludicrous, we should note that it actually involved a very long and detailed analysis of the events described in the Bible, and an intricate calculation is required to trace back through the generations to the first persons placed by God upon the earth. A Christian fundamentalist might even suggest this to be an innovative approach worthy of consideration as a great calculation. It should also be noted that figures like Kepler and Newton also considered a few thousands of years to be a valid estimate for the age of the earth.

  The nineteenth century saw tremendous progress in geology and biology, the two merging in the study of fossils found in different strata of rocks. Sir Charles Lyell's great Principles of Geology, An Attempt to Explain the Former Changes of the Earth's Surface by References to Causes Now in Operation was published in three volumes in 1830–1833. Charles Darwin took the first volume of Lyell's treatise on his round-the-world voyage on the Beagle, and it played a part in his work leading to his Origin of Species in 1859. This was a period of theorizing on a grand scale. Some geologists were convinced that the earth was many hundreds of millions of years old. Darwin himself, in his Origin, came up with a figure of about 300 million years for the time taken for water to have cut out the valley of the Weald near his home. (This calculation was not endorsed by many people, and Darwin removed it from the second edition of his book.) He also required a very great age for his theory of evolution based on natural selection to be valid. The scene was set for a major conflict with one of the great physicists of the period.

  4.3.1 Enter Lord Kelvin

  William Thomson (1824–1907) was, together with James Clerk Maxwell, the leading figure in nineteenth-century British physics. Thomson was made Lord Kelvin in 1892 and is now universally known just as Kelvin. His contributions to science and life in Britain were enormous, and it is no surprise to find Smith and Wise needing over 800 pages for their biography of him (see bibliography).

  Kelvin was one of the founders of thermodynamics and explained its influence on the classical theory of heat and mechanical processes. In particular, he understood the conservation of energy and the manner in which all physical actions involve friction and processes by which available ener
gy gradually runs down. Thus it was natural that he turned his attention to our source of energy for life, the sun. He tried to understand how the sun produced energy and calculate a possible lifetime for it, suggesting in 1862 that the sun could not have illuminated the earth for more than 100,000,000 years, or 500,000,000 at the outside. (I will return to this topic in chapter 11.) Clearly this had implications for the age of the earth.

  In 1862, Kelvin published his famous paper On the Secular Cooling of the Earth. (This paper is freely available today and is a joy to read.) He begins his paper with this challenge to geologists and their trust in “uniformitarianism”:

  For eighteen years it has pressed on my mind, that essential principles of Thermo-dynamics have been overlooked by those geologists who uncompromisingly oppose all paroxysmal hypotheses, and maintain not only that we have examples now before us, on the earth, of all the different actions by which its crust has been modified in geological history, but that these actions have never, or have not on the whole, been more violent in past time than they are at present.2

  Kelvin suggests that the earth has been cooling from some initial state, and the theory of heat as developed by Fourier may be used to calculate how long the cooling has been taking place to give the present conditions. Others before Kelvin had wondered about the cooling of the earth (Newton mentions a cooling ball of iron in his Principia), but Kelvin had the theoretical tools to produce meaningful time estimates.

  4.3.2 Kelvin's Calculation

  Kelvin assumes that “the earth is a warm chemically inert body cooling” from molten rock assumed “to be at 7000º Fahr.” To carry out the calculations, he uses Fourier's equation for the temperature T at time t and position x:

  Fourier's theory tell us that if we are given the temperature at points on an extended body at time t = 0, we may solve equation (4.2) to find the temperature T(x, t) at any future (or past) time as heat conduction takes place.

  Kelvin studied cooling in a planar geometry (arguing later that it may be used to give suitable estimates for the spherical case), so that only the one spatial coordinate x is used. The parameter κ denotes the conductivity of the cooling material. Kelvin uses a solution of equation (4.2) which gives the state of a cooling system as a function of time by taking a boundary condition making the surface held at a constant temperature. He gives an age for the earth of 98,000,000 years but states that “the consolidation cannot have taken place less than 20,000,000 years ago, or we should have more underground heat than we actually have, nor more than 400,000,000 years ago.”

  Thus Kelvin satisfies no one: the biblical estimates are made to look ridiculous, and the times required by the theories in geology and biology appear to be far too large. Kelvin himself was a practicing Christian. Ivan Ruddock describes how, in his vote of thanks at a somewhat anti-Darwin lecture, and in subsequent correspondence in the Times (London), Kelvin revealed his belief that “science positively affirmed Creative Power.”3 Perhaps the fact that his age for the earth did not cover the period required by evolution was not such a worry for Kelvin.

  Kelvin's calculation was an application of physical principles on a grand scale, and it stirred up an extensive debate among the various types of scientists of his time. There is no doubt that Kelvin's calculation led to an examination of the underlying assumptions and methods in many areas of science, and thereby it changed the direction of science in a major way. For that reason I choose it as calculation 9, Kelvin and the age of the earth.

  4.3.3 Analyzing Kelvin's Calculation

  Kelvin's work on the cooling of the earth is what today we would call a model calculation: the details of a physical situation are simplified or approximated in such a way that a manageable mathematical theory may be applied. Kelvin, in his 1862 paper, makes it quite clear that “the solution thus expressed and illustrated applies, for a certain time, without sensible error, to the case of a solid sphere, primitively heated to a uniform temperature, and suddenly exposed to any superficial action, which for ever after keeps the surface at some constant temperature.” He is also clear that he needs a value for the conductivity parameter κ, and he discusses how he chooses possibilities from certain experimental evidence. He does also discuss some limitations of his model (more of which in a moment).

  These circumstances were appreciated by those engaged in the long debate about physics, geology, evolution, and the state of the earth. A wonderful example is provided by T. H. Huxley (“Darwin's bulldog”) when he said:

  Mathematics may be compared to a mill of exquisite workmanship, which grinds you stuff of any degree of fineness; but, nevertheless, what you get out depends upon what you put in; and as the grandest mill in the world will not extract wheat-flour from peascod, so pages of formulae will not get a definite result out of loose data.4

  Huxley is setting out general ideas which should be kept in mind when considering any of the calculations described in this book.

  Kelvin's model simplifications were analyzed by John Perry, who was at one time Kelvin's assistant before going on to professorships in Tokyo and London. Perry published several papers in Nature (see bibliography) commenting on Kelvin's assumptions and what they implied for his calculations. In his January and February 1895 papers, he pointed out that κ would vary in the extreme conditions inside the earth, and he offered some sample calculations showing that the ages given by Kelvin could be increased by a factor of more than a hundred. Kelvin responded in an 1895 Nature paper defending an age of 24 million years. Perry reviewed that again in his April paper where he also discussed the fact that Kelvin's model ignores other processes like convection in the interior of the earth. Perry concluded that Kelvin's ages were much too short. We can only conclude that Kelvin made far too sweeping simplifications and it is not easy to model the extremely complex body that is our earth, something we return to below in section 4.4.5. (For a full discussion of debates around Kelvin's work see the paper by England, Molnar, and Richter and the book by Smith and Wise.)

  Kelvin may not have got to the 4.6 billion years accepted today as the age of the earth, but through his calculation, he did produce a revolution in thinking about science and the ways it is applied on those large scales becoming so prominent in nineteenth-century science.

  4.3.4 Rutherford's Wonderful Story

  It is impossible to leave this topic without touching on one of science's great stories. Kelvin assumed that there was no source of heat within the earth (recall that it was “a warm chemically inert body cooling”) and so only a calculation of cooling from some initial state was required. However, it was in Kelvin's lifetime that radioactivity was discovered, and thus a source of heat within the earth was identified. In 1904, Ernest Rutherford gave an address at the Royal Institution, and there in the audience was the eighty-year-old Lord Kelvin! Here is Rutherford's famous description of the event:

  I came into the room, which was half dark, and presently spotted Lord Kelvin in the audience and realized that I was in for trouble in the last part of the speech dealing with the age of the Earth, where my views conflicted with his. To my relief he fell fast asleep but as I came to the important point, I saw the old bird sit up, open an eye and cock a baleful glance at me! Then sudden inspiration came, and I said Lord Kelvin had limited the age of the Earth, provided no new source of heat was discovered. That prophetic utterance refers to what we are now considering tonight, radium! Behold! The old boy beamed at me!5

  Heat generated by radioactivity inside the earth represents another flaw in Kelvin's calculations. However, he did not believe radioactivity would greatly affect his calculations (see Smith and Wise, chapter 17), and indeed it does not appear that such a source of heat is too important (see the England, Molnar, and Richter paper). Ironically, it is methods based on the physics of radioactivity that are used to give good estimates for the age of Earth.

  4.4 WHAT IS INSIDE THE EARTH?

  We may readily explore the surface of the earth, climb up mountains, and go down into valley
s, gorges, and mines. That does not take us very far inside the earth and the deepest hole drilled so far is around 12 km deep compared with the earth's radius of 6,360 km. Naturally people have wondered what is inside the earth, and it has been a subject for myths and science fiction. It might seem to be an impossible quest, but in fact we do know a great deal about the inside of our earth. In the broadest terms, there is a crust (which we can explore to a certain extent) on top of the mantle, which extends down to 2,890 km; there is an outer core in the 2,890–5,150 km region; and finally there is a largely iron inner core.

  In the nineteenth century, scientists could use mathematics to investigate the connection between the properties of the interior of the earth and its shape, rotation, orbit, and tidal variations. These were and remain very indirect methods needing many factors and physical phenomena to be taken into account. This was the realm of the great classical physicists like Lord Kelvin. But as the twentieth century dawned, a new and potentially more accurate method was coming onto the scene.

  4.4.1 Earthquakes and Their Detection

  An earthquake sends out disturbances in the earth that may be detected all over the globe. In the nineteenth century, many instruments (seismographs) were developed for recording the effects of an earthquake at sites remote from its center (see Oldroyd, chapter 10 for details). As more and more observations were made, it was inferred that the disturbances traveled as waves, and since they moved through the earth, perhaps those observed waves held the clue to the constitution of the interior.

  Two types of waves propagate through a solid body like the earth. P-waves involve compressions and dilations; they are pressure waves (like sound waves), and they describe traveling longitudinal displacements in the material through which they are propagating. S-waves involve displacements perpendicular to the direction of motion (like the waves on an elastic string); they are shear waves. The speeds of these waves depend on the elastic properties of the transmission medium, and a comprehensive mathematical theory has been built up over the last two centuries (see Bullen and Bolt for examples).