If it is true that the air in Scotland hangs heavy with history, that same atmosphere may feel even heavier at the end of a reading of this book. I found it quite heavy-weather.

Julian Havil has written one of the few books to deal with the details of John Napier’s life and work at such length and in such depth. An amateur historian of mathematics can expect to find much of value about the invention of logarithms in this book, and well-presented at that. However, the story of Napier’s work is long and involved and much more is written here than will ever be found in a short *précis* of that work (as is often presented in textbooks). Therefore, weary reader, gird up your loins as if for battle and take the sharp sword of mathematics in hand if you wish to win this battle!

The opening chapters of this book deal with the history of the Napier family (in his life John Napier was known as the “Baron of Merchiston”), some possible name origins and background history of Scotland of his times. There is brief (but dismissive) mention of Napier’s reputation as a warlock and none of the florid stories on which this junior-level history of mathematics buff was raised. The first chapter is followed by 26 pages of densely-written prose spelling out the argument of Napier’s “great” work on the Book of Revelations and the Apocalypse. The ground of this argument features the blood and guts of the Reformation, anti-papacy and chronologies of the Apocalypse. Spoiler warning: for those of you who may be concerned, Napier’s upper estimate is 1786. It would appear that we are truly living in post-apocalyptic times! White horses, Seven Seals and trumpet blows aside I found this chapter rather heavy-going. In my impatience I wanted to bring on the mathematics! Little did I know what was in store…

The discussion of Napier’s logarithms begins in Chapter 3 “a new tool for calculation”. Introduced to the world in quarto-sized book of 147 pages, Napier’s work was of great importance for its time. No less a figure than Kepler pleads for the improved version of the Rudolphine tables (predecessors of Napier’s work):

Do not sentence me completely to the treadmill of mathematical calculations-leave me time for philosophical speculations, my sole delight.

The use of Napier’s tables is outlined in some depth in this chapter. Napier’s logarithms are ratio numbers for computations with sines of angles (where the reference circle has a radius of \(10^7\) parts): \[\mathrm{NapLog}(10^7\sin\theta)=10^7\ln\left(\frac{10^7}{10^7\sin\theta}\right),\] in more modern notation. The nature of this equivalence is hidden, however, and must await the following chapter which reproduces Napier’s motion analogy. If the reader is a fan of computational algorithms then they may well enjoy this chapter. It is full of computations showing how the tables were used along with a curious method of interpolation which will likely be unrecognizable to those who grew up with textbook linear interpolation. The tables themselves are constructed with maximal efficiency for presentation of the ratio numbers and this humble reader soon found himself sinking under the pounding surf of line after line of computation. An attempt at motivation occurs late in this chapter with a (too) short explanation of spherical trigonometry and the *Penta Mirificum*. For those new to these fascinating bits of spherical trigonometry, my advice is to look elsewhere for more detail on the celestial sphere and the various angles (there are 6) of a spherical triangle.

Chapter 4 “Constructing the Canon” goes into Napier’s last publication (finished by his son and Henry Briggs) *Mirifici logarithmorum canonis constructio* which makes plain Napier’s method for constructing his tables. The story begins with a comparison of motion along two line segments (one of length\(10^7\), the other unbounded). On the first line segment, a point P travels from endpoint A to endpoint B in such a way that its speed is proportional to the distance PB. On the second a point Y travels from X to Z at constant speed. Napier’s critical definition is then \( \mathrm{NapLog}(PB)=XY.\)

In order to further normalize these speeds, Napier chose the starting speed of the first point to be \(10^7\) and the second point has the same speed \(10^7\) throughout. Notice that, somewhat surprisingly, \(\mathrm{NapLog}(10^7)=0\). The function \(\mathrm{NapLog}\) is a *decreasing* function of its argument! From here Havil follows Napier through five closely reasoned pages of segment arithmetic in order to show that the logarithm of the geometric mean of two numbers is the arithmetic mean of their logarithms. This becomes the essential property that Napier uses to bound values of his logarithm in terms of the distances AB and PB and this takes us up to the construction. And what a construction it is. Napier generates some 69 tables repeatedly using a sequence of geometric sequences that function as “reference points” for further computation. I will leave it to the industrious reader to take on the mission of following these densely written pages — it was a challenge I found myself not nearly brave enough to face.

The remaining chapters are somewhat easier going and address the issues of computation in Napier’s time and before. There is a delightful connection between the gelosia (or grating) method of multiplication and Napier’s bones (which may be of interest especially to students of the Everyday Mathematics Curriculum). Counting boards and various other logistic methods are also explained in brief and the last part tackles the computation of roots and algebraic expressions. The book culminates in a lengthy discussion of the legacy of Napier’s work and Briggs’ improvements of Napier’s logarithms. Twelve (!) distinct appendices (which are really delightful to read) add content about Napier, the Scottish Science Hall of Fame, the Reformation in Scotland, Napier’s inequalities, the Rule of Three, Mercator’s Map and a possible Swiss contestant to Napier’s title of inventor of logarithms.

This book’s flyleaf makes the claim that this is “the first contemporary biography to take an in-depth look at the multiple facets of Napier’s story”. It’s hard to contest that but the casual reader will, I think, find the demands upon her readership uneven as the chapters move from biography to theology to mathematics. Nor is there much of an emotional tint to the story. Napier is portrayed as an intellectual hermit and the story itself presented about as coolly as the dry formula-following-formula presentation that constitutes the core of the book. This book sets out to be a serious presentation of ideas and techniques and succeeds at this level. Read it if you want all the gory details behind what Gauss described as “the poetry that lies behind the calculation of a table of logarithms”.

Jeff Ibbotson is the Smith Teaching Chair at Phillips Exeter Academy. He spends much of his time reading, playing ping pong and raising beagles.