Great Calculations: A Surprising Look Behind 50 Scientific Inquiries

by Colin Pask

  ALSO BY COLIN PASK

  Magnificent Principia

  Math for the Frightened

  Published 2015 by Prometheus Books

  Great Calculations: A Surprising Look Behind 50 Scientific Inquiries. Copyright © 2015 by Colin Pask. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, digital, electronic, mechanical, photocopying, recording, or otherwise, or conveyed via the Internet or a website without prior written permission of the publisher, except in the case of brief quotations embodied in critical articles and reviews.

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  The Library of Congress has cataloged the printed edition as follows:

  Pask, Colin, 1943-

  Great calculations : a surprising look behind 50 scientific inquiries / by Colin Pask.

  pages cm

  Includes bibliographical references and index.

  ISBN 978-1-63388-028-3 (pbk.) — ISBN 978-1-63388-029-0 (e-book)

  1. Mathematics—Miscellanea. I. Title.

  QA99.P385 2015

  510—dc23

  2015001967

  Printed in the United States of America

  Acknowledgments

  Preface

  Fifty Questions for the Calculations to Answer

  Chapter 1: Introduction

  Chapter 2: Ancient Mathematics

  Chapter 3: Steps into Modern Mathematics

  Chapter 4: Our World

  Chapter 5: The Solar System: The First Mathematical Models

  Chapter 6: The Solar System: Into the Modern Era

  Chapter 7: The Universe

  Chapter 8: About Us

  Chapter 9: Light

  Chapter 10: Building Blocks

  Chapter 11: Nuclear and Particle Physics

  Chapter 12: Methods and Motion

  Chapter 13: Evaluation

  Notes

  Bibliography

  Index of Names

  Index of Subjects and Terms

  My interest in the calculations discussed in this book goes back fifty years, and over that time a great many colleagues have helped and inspired me. My doctoral-thesis supervisor, John M. Blatt, was a pioneer in the use of computers for tackling physical problems, and his enthusiasm and wisdom set me off on my career. Barry Ninham met my wife and me when our boat arrived in Australia fifty years ago, and he has been a friend and supporter ever since. As a highly skilled applied mathematician, he will recognize his influence on me with the appearance of Bessel functions in this book. The wonderful Annabelle Boag has produced the magnificent figures that grace this book; I would be lost without her cheerful support and highly professional assistance. Peter McIntyre is a meticulous reader of manuscripts, and I thank him for much help and advice. Connie Wintergem has supported me in much of the research for this book. My beautiful wife Johanna has shared life with me for fifty years, and this book owes more than I can say to her love and support.

  Science combines observation, experiment, and theory. The observational and experimental sides are probably better known and more readily appreciated. The spectacular observations of astronomers, for instance, are often in the news. Many people have read about or have seen television reports on the 2012 experiment at the Large Hadron Collider in which particles were smashed together to reveal the existence of the Higgs boson. But it is likely that few of these people understood why the experiment was carried out and what the result means for the future of science. It is the interplay between experiment and theory that gives sciences its coherence, integrity, and power. This book aims to improve the understanding of the role played by theory in science. It complements the many books attempting to identify science's most influential experiments by describing a selection of its great calculations.

  The calculations described in this book are mostly relatively simple (at least in principle, if not always in practice) but still of major significance. I believe they are accessible to the ordinary reader; occasionally the more technical details are given in separate sections for those wishing to see them. In any case, it is usually the context which is of major importance: Why was the calculation made? What was the result? What was the effect of the calculation on the progress of science? Did the result have broader implications for society and the future of mankind? Answers to questions such as these reveal how calculations are both important and interesting—and also sometimes controversial, as the predictions for climate change illustrate.

  The calculations are grouped under broad subject areas such as the solar system or the building blocks of matter. In each group they are presented in chronological order, and in this way readers may learn something of the history of science (although the different parts of the book may be read in any order and dipped into as desired). Similarly, I illustrate the ways in which the styles and techniques used in calculations have evolved, something I review in the final chapter and which may be of particular interest to those readers who have grown up entirely in the computer age. We shall also meet some of the great characters involved, from Archimedes to Einstein, and discover the intriguing stories surrounding certain calculations.

  By reading about the various calculations I hope that you will come to better appreciate the vital link between experiment and theory.

  It will be obvious that the choice of calculations to be described reflects my personal interests, background, and views of science. It was not easy to keep to such a restricted number of examples (and some extras have briefly sneaked in as “honorable mentions”). I expect some readers may be upset not to find their personal favorites on my list. Even more provocative may be my final choice of the ten calculations most worthy of the label “great.” However, that is part of the fun. I stand ready to be informed and educated about the superior qualities of other worthy candidates I have excluded.

  How did a parson warn us about the problems of overpopulation?

  How mathematically advanced were the ancient Babylonians?

  What could have been another “eureka moment” for Archimedes?

  What did Fibonacci do apart from showing how rabbit populations get out of control?

  How do you double the life span of an astronomer?

  Can you add up an infinite number of things?

  How did Gauss find a mysterious formula for prime numbers?

  How big is the earth?

  How old is the earth?

  What is inside the earth?

  How did Galileo mathematically shoot a cannon?

  Which nineteenth-century calculation used an analogue computer that was then used in planning the allies’ D-Day landings in France?

  Which masterpiece dominated astronomy for a thousand years?

  How did Kepler battle with Mars?

  Why is the link between a falling apple and a falling moon so important?

  How do you weigh the planets?

  How did Halley predict the return of his comet?

  How do you find a new planet without using a telescope?

  How far away is the sun?

  Which calculation led Einstein to say that “for a few days, I was beside myself with j
oyous excitement”?

  Why is the sky dark at night?

  How old is the universe?

  How does carbon-based life on Earth depend on a quirk of nuclear physics?

  How do we know there is lots of “dark matter” in the universe if we can't see it?

  How did a parson's dark stars turn into black holes?

  How did a little calculation of pumping by the heart overturn over a thousand years of medical practice?

  What links a famous comet and the cost of an annuity?

  How did a pure mathematician who claimed to have never done anything “useful” give his name to one of the first major results in genetics?

  Why is the CT scan unfairly named?

  What links a mouse to an elephant?

  How can Jupiter's moons tell you the speed of light?

  How did Newton upset the poet John Keats?

  What limits the details you can see?

  How did Maxwell see the light?

  What is light?

  Could Einstein's light bending prove his “Dear Lord” wrong?

  How do you know atoms really exist?

  How did a Swiss schoolteacher find patterns in light?

  How did a French prince explain the schoolmaster's patterns?

  How can you measure the mass of a neutron?

  What is the most accurate calculation of all?

  How did a threat to the conservation of energy law lead to a John Updike poem?

  Can you stay younger by going faster?

  Why should you believe in quarks?

  Why does the sun shine?

  How do you plan to make the biggest bomb of all?

  Who discovered a mathematical poem for violin strings?

  How were those amazing mathematical tables made?

  How do you ask a computer about a string of oscillators?

  How easy is it to find chaos?

  in which I review the complementary roles of calculation and experiment in science; outline some general aspects of calculations and see them in operation in a first example; and explain the structure of the chapters to follow.

  According to Einstein,

  science is the attempt to make the chaotic diversity of our sense experience correspond to a logically uniform system of thought. In this system single experiences must be correlated with the theoretic structure in such a way that the resulting coordination is unique and convincing.1

  In science, we investigate sense experiences in a systematic way. This may involve planned suites of observations in astronomy or in zoology, or the arrangement of particular physical entities and conditions in experiments. The importance of the use of both theory and experiment was neatly summarized long ago by philosopher Immanuel Kant (1724–1804):

  Experience [experiment] without theory is blind, but theory without experience is mere intellectual play.2

  (By the way, we must not be too disparaging about intellectual play because it might be said to cover pure mathematics, where the formalism itself rather than its applications is of interest; we shall see some famous calculations within this category, too, before we move on to applying mathematics in science.)

  The development of this viewpoint by people like Sir Francis Bacon, Galileo, John Locke, Sir Isaac Newton, and Kant was central to the scientific revolution underway by the seventeenth century. Nevertheless, its recognition was still in question for many years as is apparent from this excerpt from an 1861 letter from Charles Darwin to Henry Fawcett:

  How profoundly ignorant B. must be of the very soul of observation! About thirty years ago there was much talk that geologists ought only to observe and not theorize; and I well remember someone saying that at this rate a man might as well go into a gravel-pit and count the pebbles and describe the colors. How odd it is that anyone should not see that all observation must be for or against some view if it is to be of any service!3

  Today, it is generally accepted that we need theory to discover order in the mass of data revealed by experiments and also when planning those experiments (in Einstein's words, theory tells us what to measure4). However, while there are many books about the outstanding, or most important, experiments (see those by Crease, Johnson, and Shamos for examples), very few deal with the actual calculations behind such experiments. There are books about the equations of physics (see Crease, for example), but equations are the tools of theory—much like the telescope, cyclotron, and oscilloscope are tools in experimental physics. Here I am concerned with the most influential results of using these theoretical tools.

  One reason for the theory-experiment imbalance is that to treat the contribution of theory in the physical sciences often means coming to terms with a mathematical formalism. Mathematics has proved itself to be the tool required by scientists; for Galileo there is a mathematical language for the universe.5 Mathematics allows us to use and interpret observations to “go where we cannot go”—to explore the outer reaches of the universe, the interior of our planet, and the subatomic world—and even reveals what is inside our own heads.

  Kant's idea that theory without experience is mere intellectual play is at the heart of a debate about the value of string theory in present-day fundamental theoretical physics.6 There is a danger that the search for ever more fundamental theories loses contact with experiment and the actual physical universe as we know it; for science to prosper there must be calculations and a direct comparison with observations.

  I contend that many calculations in science may be appreciated without a deep knowledge of the mathematics involved, just as the output from an experiment may be appreciated without understanding the intricacies of apparatus design and manipulation. This book gives a discussion of calculations to go along with those books that describe the experimental side.

  1.1 ABOUT CALCULATIONS

  To put the mathematical aspects into perspective, I suggest that four questions should guide the discussions:

  Why was the calculation made?

  What was calculated?

  What was the result of the calculation?

  What impact did the calculation have?

  The details of exactly how the calculation was made may be of secondary importance, although in some cases the approach and techniques used are so innovative or revolutionary that they do merit discussion. Mostly, if I do give technical details, I will put them in separate sections for those who wish to see them, and always in the bibliography I give references for readers who wish to study the original papers reporting the calculations.

  We shall see how calculations help to turn data into information and thus fit them into the patterns of science. Sometimes single numbers are involved. Sometimes whole data sets, perhaps very extensive in nature, are to be treated. The calculation may result in a formula or parameters to be used in a descriptive formula. This often reduces experimental results to a form that requires explanation and lends itself to interpretation; one calculation may inspire another. Sometimes the formula is already suggested by theory, and then the calculation will be validating or discrediting that theory.

  Remembering Einstein's theory tells us what to measure, we shall also see examples where the calculation produces a prediction for experimental verification. As mentioned above, this is the key step in real science: put the theory in a form that can be tested by experiment. When Einstein's general theory of relativity was published, it contained one calculation explaining a mysterious anomaly in planetary motion measurements and another predicting an optical effect. For many people, it was the prediction that was so impressive because there is always a lingering feeling that a theory may be manipulated to produce calculations of already-known effects. (Of course there is then the related question of whether data may be selected or doctored to fit a prediction.7)

  An example will help to clarify some of these matters.

  1.2 EXAMPLE: MALTHUS GIVES US CALCULATION 1

  I begin with a very simple calculation: generate two sequences of numbers, each starting with the numbe
r 1. In one case, go from one term to the next by doubling the term; in the other, just add 1 to the term. We obtain

  1 2 4 8 16 32 64…

  1 2 3 4 5 6 7…

  The first one is an example of a geometric sequence (each term is just a multiple of the previous one and here the multiplying factor is 2). The second one is an example of an arithmetical sequence (each term is a sum of the previous one and a constant factor, and here that factor is 1). Not a very profound result, but we shall see more exciting numerical examples in the next chapter.

  The interest in these two sequences increases dramatically when we note their use in Parson Thomas Malthus's Essay on the Principle of Population as It Affects the Future Improvement of Society (1798). Malthus begins with two postulata:

  First, that food is necessary to the existence of man.

  Second, that the passion between the sexes is necessary, and will remain nearly in its present state.8

  He goes on to deduce that population, when unchecked, increases in a geometrical ratio. Subsistence increases only in an arithmetical ratio. He then points out: a slight acquaintance with numbers will shew the immensity of the first power in comparison of the second. The numbers he uses as examples are just those given above. To make things clear, I define time periods (with zero indicating the initial situation), and the population and the available resources in those periods. Those resources can provide for a certain population, and I calculate that supportable population as three times the resources value. You might think of something like a number of years (Malthus assumed 25 years), millions of people, and thousands of acres of farming land. Malthus's results then appear as follows:

  time period 0 1 2 3 4 5 6…

  population 1 2 4 8 16 32 64…

  resources 1 2 3 4 5 6 7…

  supportable population 3 6 9 12 15 18 21…

  Clearly life is good initially; the resources are more than adequate for supporting the population. But by period 4 the population is barely making ends meet. Life becomes very tough in period 5 (perhaps surviving by using stored food excesses from previous periods), and by period 6 there is complete disaster with the actual population more than three times the supportable one. In Malthus's own words: