# John Napier: Life, Logarithms, and Legacy

John Napier (1550-1617) is a Scottish scientist who is probably best known for his invention of the logarithms. He also made a meticulous analysis of the Apocalypse and he propagated the use of salt as a fertilizer in farming. In his time he had the reputation of an alchemist and a magician having bonds with the devil. But that was a common trait attributed to scientists in those days.

Born in a well respected family, he became the 8th Laird of Merchiston. Being a devote Protestant, he found in the The Book of Revelation the proof that Catholicism was the quintessence of evil and that the Pope was the Antichrist. This must also be seen in the context of the Spanish Armada's attempt to invade England in 1588 and Spain was the Catholic enemy `par excellence'.

In a first chapter, Havil gives a survey of Napier's life, to switch in the second chapter to a discussion *A plain discovery of the whole revelation of St. John* (1593), the first book by Napier. The apocalyptic last book of the New Testament is a collection of psychedelic visions that allow for several different interpretations. Napier believed it was a true account of historical facts and that it contained prophesies, i.e., predictions of the near future. Napier's book consists of propositions in which symbols of Revelation are linked to historical facts and persons. Proofs and demonstrations are given to make these statements acceptable. The seven seals border seven year periods spanning the period from Jesus Christ (29 CE) till the destruction of Jerusalem in 71 CE and the seven trumpets border subsequent spans of 245 years till 1541 CE with the start of the Reformation. He calculated the end of the world (like many others did with varying results) to happen in 1688 or 1700. Because his book was written in English (and not in Latin) it had great impact and certainly after its translation in French, Dutch, and German, Napier's reputation was established, not only in the British Isles, but on the Continent as well.

Chapter three is about Napier's second book: *Mirifici Logarithmorum Canonis Descriptio* (1614). This contains the first logarithm tables with their definitions and an explanation on how to use them. These tables consist of "artificial numbers" which Napier called logarithms (literally "ratio numbers") they are used to transform multiplication into addition and division into subtraction. Originally the ratios referred to the ratios of lengths of triangles appearing in the definition of trigonometric functions in a goniometric circle, but Napier later realized that this was an unnecessary restriction. Napier chose the radius of the circle to be 107 and the logarithm of that number was 0. If we denote Napier's logarithm as NapLog(*x*) then this corresponds to what we would now recognize as 107ln(107/*x*)=107log1/*e*(*x*/107). Thus not exactly in basis e but in basis 1/e. Havil explains how Napier came to this concept and how the tables can be used. The importance of this computational tool was quickly realized and used widely. Henry Briggs, lecturing in London, adopted the tables with enthusiasm and became a good friend. He also solved the annoying problem that multiplying with a power of 10 did not show easily in the NapLog by shifting the digits or by adding a power of 10. Of course, that is the germ of what we now recognize as the Briggsian logarithms that work in base 10. Note that Napier never considered the log function. The tables were just tools for goniometric and other computations.

It was Napier's second son Robert (from his second marriage) who published the *Mirifici Logarithmorum Canonis Constructio* (1619) two years after his father's death in April 1617. This is the subject of chapter four. Havil does here an excellent job in explaining in a way that it is understandable for our 20th century knowledge how the tables were constructed and how they are linked. In a later chapter about Napier's legacy, he also shows how Napier touches unwittingly upon calculus and the relation with the natural logarithm mentioned above is also revealed there.

There are other computational inventions by Napier that were important in his time, but whose importance, unlike the logarithms, has faded today: the *Napier rods* also known as *Napier bones* (they were made of ivory) and the *Promptuary*. These tools, and how to use them was explained in Napier's last publication *Rabdologia seu numerationes per virgulas libri duo* (1917). Napier was inspired by the *gelosia*. That is a tool that helps multiplying two integers. For example to multiply 72 x 35 = 2520, one constructed.

$$

\begin{array}{c|c|c|c}

&{\color{red}7}&{\color{red}2}&\\\hline

& 2/\ & 0/\ & \\

{\color{blue}2} & \ /1 & \ /6&{\color{red}3}\\\hline

& 3/\ & 1/\ & \\

{\color{blue}5} & \ /5 & \ /0&{\color{red}5}\\\hline

& {\color{blue}2} & {\color{blue}0} &

\end{array}

$$

The 72 goes on top, the 35 on the right. Fill the (in this case 2 x 2) table with the products of the digits, separating tens and units by the upward sloping diagonal line. Finally add the digits in the upward sloping diagonals, starting from bottom-right to top-left, and use carries. This gives 0, then 5 + 1 + 6 = 12, write 2, carry 1, then 3 + 1 + 0 + 1 = 5, and finally 2. The product 2520 can now be read off. This is just a mechanization of our familiar way of long multiplication. The drawback is that a new table had to be constructed for every multiplication. The idea of the Napier bones is to use a rod for the number 7 not with the two multiples needed for this example, but with *all* multiples of 7 listed from top to bottom, and similarly for all the other digits. Placing the rods for 7 and 2 next to each other, one had to select rows 3 and 5 to make the previous product. To economize on the hardware, 4 different sets of multiples were placed on the 4 sides of the rod.

Napier's *Promptuary* went a step further and turned this in an actual analog computer. Each square on the rod for digit 7 is replaced by an identical copy of a 3 x 3 block that contained all the 9 possible multiples of 7 (exclude 0 and separate tens and units above and below the main diagonal of the block in a particular way). Similarly for the digit 2. Place these two strips next to each other. Select in the (1,2) block the pattern 0/6 from 2 x 3 = 6 and mask the rest in that block. Then select in the (2,2) block the pattern 1/0 from 2 x 5 = 10, and mask the rest in that block. Identical masks are repeated in every block row. Then summing the unmasked numbers in the block diagonals as before gives the product. The ideas become much clearer with graphics like for example here. Of course all this generalizes to products of integers with arbitrary lengths.

A final contribution from the *Rabdologia* was a discussion of "local arithmetic". Multiplication, division, square roots were all possible if the numbers are represented in basis 2. That is binary computation several centuries before the digital computer! All entries in the tables become 0 or 1 and that obviously simplifies things considerably.

Some unpublished papers were passed on in the Napier clan, and were only published much later by the historian Mark Napier as *De Arte Logitica* (1839). Havil discusses them in his 6th chapter. They deal with all sorts of subjects like decimal notation, negative numbers, irrationals, long division and multiplication, the rule of three, notations for nth roots etc.

In a last chapter, Havil surveys in what ways Napier's findings have influenced his successors. His rods and Promptuary have inspired many to develop analog computing devises, and of course there is obviously the slide rule as a consequence of his logarithms. On a mathematical level he triggered the decimal Briggsian logarithms, and the natural logarithms as they are known today. Also later came the link with the area under the hyperbola, the number e (known as Napier's or Euler's constant, but the e refers to Euler), the exponential and logarithmic functions, etc.

In an extensive set of appendices, Havil provides additional historical, religious, and mostly mathematical background.

Julian Havil has published several other books on popular mathematical subjects that were well received. I am sure this books will be the next in the sequence of successes. It is a "general mathematics" book, and secundary school mathematics will allow to understand everything in this book. However, some affinity with mathematics will increase the appreciation of the reader. The treatise on the Book of Revelation comes as a surprising side product. For most readers, Napier will stand somewhat in the shadow of other mathematical giants, not being the brightest star on the mathematical firmament, but I'm sure after reading this entertaining and enjoyable book, Napier will climb some rungs on your ladder of famous mathematicians.

**Submitted by Adhemar Bultheel |

**24 / Nov / 2014