Great Calculations: A Surprising Look Behind 50 Scientific Inquiries

by Colin Pask

  In this book, I have been concerned with the interplay of theory and experiment, but the interplay of pure and applied mathematics has also played an important part in the advancement of science. Much of the mathematics used by scientists has been borrowed from pure mathematics, and in turn, science has thrown up new mathematical challenges and inspired mathematicians to develop whole new areas of interest.

  13.1.3 People

  It is no surprise to find the giants of science appearing as some of my chosen calculators: Archimedes [3], Newton [15, 16, 17, 20, and 32], Gauss [7], Maxwell [34], Einstein [20, 35, 36, and 43], and Bohr [39]. More of their calculations might have been included, and Archimedes, Gauss, and Maxwell in particular are underrepresented. Looking back, it does surprise me that Euler, Fourier, Laplace, Planck, and John von Neumann are not more prominent.

  The calculations reveal what a remarkable man Edmond Halley was. Many people are aware of the fact that Halley guided the publication of Isaac Newton's Principia, and, of course, he is famous for his work on comets [17]. He also appears in the calculation of the astronomical unit [19], in early actuarial work [27], and in discussion of the dark nighttime sky problem [21]. Halley was born in 1656 and died in 1743, and so he unfortunately became one of the people who were overshadowed by Newton (Robert Hooke was another). Halley was mathematician enough to be appointed Savilian Professor of Geometry at Oxford; he was an astronomer of note and first detected stellar motions in 1718 (incidentally using Ptolemy's tables); he became Astronomer Royal; he had interests in subjects as diverse as archaeology and biology; and he was practical enough to be commissioned as a naval captain and command a survey ship from 1698 to 1700. (See Ronan for further details.)

  The calculations also reveal how important Sir William Thomson, later Lord Kelvin, was for nineteenth century science. He too was just a little overshadowed—this time by James Clerk Maxwell. Kelvin is the central figure for calculations [9] and [12], but his influence is apparent in other cases, like [21] and [45]. He was one of the great classical physicists working across fields such as electromagnetism and thermodynamics. It was also Kelvin who championed the importance of numerical studies and showed how analogue devices could be constructed to carry out difficult and extensive calculations. His textbook written with Peter Guthrie Tait was highly influential for many years.

  For me, the calculations also identify other people who deserve to be better known for their work, which might not be seen as glamorous as some other cases, but which provided vital, key advances. Mathematically, I can point to Napier and Briggs for their work on logarithms, and to Daniel Bernoulli for founding the methods of linear mathematics [47]. Harvey's work on blood circulation [26] changed medicine forever. The Reverend John Michell conceived the experiment using attracting spheres to measure gravitational forces (now associated with Henry Cavendish since he first carried out the experiment). Michell speculated about the effects of gravitation on light leading to the possibility of dark stars (and now black holes) [25] and the bending of light near massive bodies [36].

  Today, many magazines and television programs tell us about the remote reaches of the universe (and often give wild speculations about its nature and origins), but how many people know the names Oldham, Cormack, and Hounsfield? They too tell us about regions that are difficult, if not impossible, to access, but their work is also of paramount practical importance for our actual survival. Richard Oldham made giant strides in seismology [10] so that finally we could know what is inside our Earth. Similarly, the work of Allan Cormack and Godfrey Hounsfield led to the CT scan [29] and a revolution in medical science allowing us to see inside the body without dissecting it.

  Looking over the calculations shows that it was not always the famous scientist working in the great centers of research who made remarkable discoveries. Andrija Mohorovičić (1857–1936), in a 1910 paper, described the earth's crust-mantle discontinuity [10], which is still of major interest to earth scientists and exploration engineers. His name lives on as this discontinuity is known as the “Moho,” although it would be interesting to know how many people actually understand the origin of this term. Who would have thought that someone (Max Kleiber, [30]) working in a department of animal husbandry would find a place in a book like this? Similarly, many people might know the term Balmer series for some of the spectral lines of the hydrogen atom [38], but not know that it comes from the name of an obscure Swiss schoolmaster, Johann Jakob Balmer (1825–1898), who brilliantly identified a pattern in those spectral lines. Anyone studying genetics will come across the Hardy-Weinberg law [28]; however, probably few of them will realize that, strangely, the Hardy referred to was not a biologist but actually the wonderfully interesting pure mathematician G. H. Hardy.

  Surveying the calculations, I find that sadly there are only three women playing central roles, perhaps because I have set and struggled with the fifty calculations limit. Inge Lehmann (1888–1993) made the calculations that led to our picture of the earth's interior core having a solid center with a liquid surround [10]. Vera Rubin was a pioneer for women in astronomy, and her work on matter rotations in galaxies [24] led to one of the great puzzles of modern science: What is dark matter? The experiences of both of these women show how difficult it was for them to gain acceptance in the male-dominated scientific-research world.

  The third woman is Lise Meitner (1878–1968), whose seminal work on nuclear fission [46] played a part in the progress toward nuclear weapons (which she regretted). Meitner was born in Vienna, Austria, and became the first woman to receive a doctorate in physics from the University of Vienna. She studied and worked in Germany until 1938 when she fled to Sweden to escape the Nazi regime. Lise Meitner was a remarkable woman who made a career in nuclear physics at a time when women were expected to keep to “traditional roles” as this statement by Max Planck illustrates: “Generally it cannot be emphasized enough that nature herself prescribed to the woman her function as mother and housewife, and the laws of nature cannot be ignored under any circumstances without grave danger.”1 Meitner never married. As a physicist she earned the support, respect, and friendship of the great scientists of her era: Bohr, Boltzmann, Einstein, Pauli, and Planck. (Despite the above quote, Planck employed Meitner and became personal friends with her; in turn, according to biographer Patricia Rife, Meitner greatly respected and admired Planck and enjoyed his company.) Lise Meitner worked extensively with Otto Hahn, her expertise in physics complementing his abilities as a chemist. In later years, Hahn did everything possible to downplay her influence and push her into the background. It is a mystery and a scandal that Otto Hahn received the Nobel Prize for chemistry in 1944; Lise Meitner was never given that honor despite being nominated by many people, including Niels Bohr in 1946, 1947, and 1948. She did receive many other awards, held senior positions, and element 109 is named Meitnerium in her honor. If you wish to know about the difficulties and injustices faced by pioneering women scientists, read the story of Lise Meitner—but I warn you, it may make you feel angry, depressed, astonished, and maybe, as a man, a little ashamed. (Much has been written about Lise Meitner, and there are books by Barron and comprehensive biographies by Rife and Sime.)

  One gets the feeling that women's contributions were often played down, like the work of Mary Tsingou in the FPU calculations [49]. Perhaps the best known example is the way Rosalind Franklin was treated by Crick and Watson in the scramble to gain the glory for the discovery of the structure of DNA. Even when they are successful, women can be belittled in awful ways; when Dorothy Crowfoot Hodgkin was awarded the 1964 Nobel Prize for her work on the structure of complex biological molecules, it was reported in the Daily Mail under the headline “Nobel Prize for British Wife.” (By the way, Margaret Thatcher, the British prime minister, was a pupil of Dorothy Crowfoot Hodgkin at Oxford University, although I am not sure how that helped the cause of women in science!) It is hard not to feel a sense of anger and shame when reading such things, and I am sorry that I have not includ
ed more women in my list of important calculators.

  In those hidden areas where women have made significant calculations, we must include the people making detailed numerical calculations and especially those in the teams of calculators whose work was vital in the two hundred years up to around 1950 when electronic computers became more available. We saw Madame Nicole-Reine Étable de Labrière Lepaute working alongside Clairaut and Lalande through 1758 to calculate the returning path of Halley's Comet [17]—and typically getting little credit for her contribution. (I highly recommend the writings of David Grier for anyone wishing to learn more about this fascinating period of mathematical work and the part played by women mathematicians.) In chapter 12, I mentioned the tables in the famous “Abramowitz and Stegun” Handbook of Mathematical Functions. It is not widely known that Milton Abramowitz died in 1958 and it was the female contributor, Irene Stegun (1919–2008) who saw the project through to publication in 1964. Let us hope that women are more widely and fairly represented in any future list of great calculations.

  13.2 ASSESSING THE CALCULATIONS

  In 1798, Emperor Napoleon Bonaparte led an army to Egypt. After one particular battle, he took the opportunity (as most visitors to Egypt do) to see the Great Pyramid. While some of his officers climbed to the top, the emperor (sensibly?) decided to stay at the bottom. To pass the time, Napoleon calculated that the stone used to construct the pyramid could be used to build a wall 3 meters tall and 0.3 meters wide around all of France. (His result is reasonably correct too.) You probably find the story surprising, but Napoleon did claim to be something of a mathematician and maintained that “the advancement and perfection of mathematics are intimately connected with the prosperity of the State.”2 Certainly it is a curious result, and it emphasizes just how much stone had to be moved to build the pyramid. But is it a significant or important calculation? Why did I not consider it when forming my list of important calculations?

  It is now time to reveal how I evaluated my list of calculations; then we will see if I can narrow it down to just ten great calculations. My criteria for including a calculation are based on five attributes that I label (a)–(e). These are my criteria, and I accept that a strong personal preference or bias is involved both in setting these criteria and in applying them.

  Criterion (a). The calculation marks a turning point in science, a breakthrough. Usually it opens up a new field for investigation.

  Example: Calculations [35] revealing that light may be described in terms of photons.

  Criterion (b). The calculation reveals something unexpected and surprising. It often brings gasps of admiration!

  Examples: Newton's calculation [16] of the masses of planets or the discovery of chaos [50].

  Criterion (c). The calculation is mathematically or computationally innovative and shows the way for future work.

  Examples: Archimedes's calculation of π [3] or the production of tables of logarithms [5].

  Criterion (d). It is a heroic calculation.

  Example: Kepler fitting data to an orbit for Mars [14].

  Criterion (e). The calculation had a major social or intellectual impact or an influence on national affairs.

  Example: Evaluating uranium fission processes for weapons production [46].

  Napoleon's calculation does not score very well using those criteria, although (b) might be suggested by some people.

  In the list of calculations given above, I have indicated how I have assigned these attributes to them, with a star indicating an extreme case. Now I can go through the list and pick out ten great calculations. Actually, that is not at all easy to do, and at this point I wonder whether I have set myself an impossible task—probably a silly one too, but it is fun to try! Here are my choices.

  13.2.1 Great Calculations

  3. Archimedes Bounds π

  Archimedes showed how a physical constant could be evaluated using mathematics and how an iterative method could be used to generate an answer to any required degree of accuracy. The idea of placing ever-tighter bounds on the value of π and the actual numerical work were brilliantly carried out.

  5. Production of Tables of Logarithms

  The idea that multiplications could be replaced by additions was not just brilliant, it was one that changed the nature of computational tasks for many centuries. Only later in the twentieth century, when calculating devices became widely available, did the value of logarithms dissipate. The calculations to provide tables of logarithms can truly be described as heroic.

  10. Seismic Rays Reveal the Earth's Interior

  Until these calculations were made, it would have been assumed that there was no way to find out what was deep inside the earth. Certainly volcanoes had been observed, and the destructive power of earthquakes was well documented. But the turning of knowledge of seismic waves into information about the earth's interior required a major leap in theory and extremely difficult calculations. This work solved an inverse problem and showed the way for a vast new exploration of the structure and properties of planet Earth that continues today.

  14. Kepler's Astronomical Calculations

  Kepler's dedication and persistence in his work on understanding the solar system inspire the utmost admiration. He was able to deduce his three laws that exhibit the underlying order in planetary motions. It was Kepler who made the monumental step involving the discrediting of the old Greek idea that planetary motions must involve circles, an idea that had dominated astronomy for centuries. If any calculation deserves the label heroic, it is surely Kepler's calculation of the elliptical orbit of Mars. A new standard was set for scientific working and the comparing of theory and observation.

  17. Predicting the Return of Halley's Comet

  Newton demystified the nature of comets and suggested that they were not harbingers of great events and tragedies, but regular physical bodies whose orbits might be understood like those of the planets. The difficulty is that only a tiny part of a comet's orbit may be observed and it then moves into the outer reaches of the solar system and thus disappears for many years. The prediction of the return date for a comet was a vital test for Newton's universal theory of gravitation. The calculations required for such a prediction must have seemed daunting in the extreme, but Clairaut and his assistants refined Halley's orbit prediction to give one of the most spectacular and public confirmations of the validity of gravitational theory. (An aside: the discovery of the planet Neptune [18] was a close rival for this spot.)

  20. Rotating Orbits

  Newton needed properties of motion under an inverse square law to compare with observations in order to confirm that gravity is a force of that type. He was able to prove that for an inverse square law force, the planetary orbits are ellipses, while for other force laws, the orbit might be thought of as an ellipse but it would also slowly rotate in space. Using innovative mathematics and perturbation theory, Newton produced a formula for the rotation rate showing how it increased as deviations from inverse square increased. Comparing planetary data, he was able to mount a strong argument for the mathematical form of his theory of gravity. Many astronomers calculated the variations in orbits due to perturbations, and a small error in the calculated orbit of Mercury stubbornly remained. It was a compelling result in Albert Einstein's general theory of relativity that explained the discrepancy in terms of an orbit rotation that may be calculated using Newton's formalism. Brilliant mathematics, brilliant physics, and a precise numerical calculation gave Einstein one of his greatest triumphs.

  26. Harvey Establishes Blood Circulation

  A simple combination of data about the heart—its rate of beating and its volume—allows us to estimate blood flows and the amount of blood in the human body. The conclusion drawn by William Harvey (although he placed less reliance on this calculation than he did on other evidence) was that blood is not continuously manufactured in amazing quantities in the liver, but that it circulates around the body. This overturned centuries of medical dogma and opened the way
for a whole new approach to medicine and an understanding of the physiology of animals. If ever a virtually trivial calculation could be said to have a profound impact, it is surely Harvey's on the pumping of blood by the heart.

  34. Light and Electromagnetism

  Establishing the nature of light has been one of the great crusades in physics. Describing light as a wave motion is enormously successful, but always leaves us with questions about the nature of the optical waves. The big step was made by Maxwell using his theory which combined, or unified, electric and magnetic phenomena. In the first part of the calculation, Maxwell showed how to manipulate the equations forming that theory to obtain a wave equation, which describes the propagation of electromagnetic waves in space and through media. He was able to use parameters obtained from experiments involving electricity and magnetism to calculate the speed of those waves and thus to show that the speed of electromagnetic waves is the same as the measured speed of light. Few steps in physics could be said to have such a profound effect on both physics and technology as Maxwell's discovery that light is an electromagnetic wave.

  39. The New Mechanics Explain Atoms

  Modern science is largely based on the atomic hypothesis and its use in explaining the properties of matter and the processes that govern the way our world works. A key step was the explanation of how light and matter interact, and understanding the spectral lines produced by different elements was a central challenge. The result was a new type of mechanics, quantum mechanics, that must be used at the atomic level. The use of the Schrödinger equation to accurately calculate the spectrum for hydrogen represents one of physics’ greatest triumphs.