Cosmic Numbers is a collection of thirteen essays on important numbers in physics and chemistry. Each essay focuses on a single number, and that number is either a fundamental physical constant or a functional relationship depending on fundamental physical constants. The idea is an intriguing one, and was perhaps inspired by the other current books focused, for example, on e, π, i, or γ. The essays are rambling and discursive, sometimes witty, sometimes annoying.
What would the hypothetical reader who picks up this book want to know? Maybe some context, an explanation of why the number is important. Then we’d like to know what the number is, its numerical value in some units, how precisely it’s known, how it is measured or calculated, and what other numbers it may be connected to. The author does a decent job on the background of the numbers, but he is amazingly indirect on the values of some of the numbers.
For example, the first chapter deals with the gravitational constant G. Yet nowhere in the chapter does the author tell us what its value is. He does tell us that — surprisingly for such a cosmologically significant number — the value is known only to five significant digits. Later, in the chapter on the comparable electric proportionality constant k, the author does a calculation where the value of G is used to compare the strengths of the gravitational and electric forces, but even that is done in a way that obscures the value of G. This might be only mildly annoying had not the author spent much of the first chapter discussing Kepler’s laws. As he points out, those laws require no knowledge of G whatsoever.
Besides the gravitational constant G and the electric proportionality constant k from Coulomb’s law, the author’s other cosmic numbers are: the speed of light, the ideal gas constant, absolute zero, Avogadro’s number, Boltzmann’s constant, Planck’s constant, the Schwarzschild radius, the efficiency of hydrogen fusion, the Chandrasekhar limit, the Hubble constant and Ω, the ratio of the density of matter in the universe to the critical density that determines the overall geometry of the universe.
The author is a mathematician with sufficient background in chemistry and physics to write very credibly for a popular audience. The urge to teach mathematics comes through: in the essay on Planck’s constant, he can’t resist the urge to show how to sum a geometric series. Other parts of this chapter don’t go so well. Although the occasional formulas throughout the book are mostly typeset well, the formula for the infinite series of the exponential in a critical spot is messed up. Then the author’s concluding quotation is mistakenly attributed to Eddington instead of Haldane. More materially, he misses the opportunity to discuss how Planck’s constant gives us the opportunity to define fundamental units of space and time.
In the essay on the efficiency of hydrogen fusion — how much of the mass of hydrogen is converted to energy in the process whereby hydrogen atoms fuse to form helium — the author emphasizes how critical this number (about 0.007) is. At 0.006, deuterium would never form in the nuclear reaction leading to helium (and the universe would consist only of hydrogen), but at 0.008, bonding would be too easy, and atomic hydrogen would become sparse and thus so would water. Yet the reader is left to wonder how sensitive hydrogen fusion really is to this number — how precisely the value is known and whether a smaller change of 1, 5 or 10% would have significant consequences. There are several other places throughout the book where the author frustrates the reader because he’s a bit too casual about the values of these cosmic numbers.
Overall this collection is a mixed bag. Fundamental physical constants make for a fascinating subject, but the author disappoints. He is informative and entertaining when he stays focused, but he tends to wander off topic too easily and in so doing misses some good opportunities.
Bill Satzer (firstname.lastname@example.org) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.