An introduction to the life and work of John Napier while introducing students to logarithms will bring the “dry” material to life.

Napier was a Scottish mathematician who lived from 1550 to 1617. He worked for more than twenty years to develop his theory and tables of what he called logarithms, a word he derived from two Greek roots: *logos*, meaning word, or study, or reasoning, or in Napier’s use, “reckoning”, and *arithmos*, meaning “number”. Much of our mathematical terminology, and indeed our English vocabulary, derives from Greek and Latin roots. It is a useful exercise to take a few moments when new terms are introduced to explore the etymology of the word and to have the class try to name other words also deriving from these roots. For example, you might ask “Where else have you seen a word derived from *arithmos*?

If etymology is not your strong suit, you will find *The Words of Mathematics* by Steven Schwartzman an excellent resource. It is published by the MAA.

Napier chose the name “logarithms” because he thought of them as “reckoning numbers”. Their use could save computational time, especially the time of beleaguered astronomers. These men had to carry out computations involving very large numbers. Any simplifying devices were welcomed with joy. In fact, the French mathematician Pierre Laplace (1749-1833) said that Napier’s new tool “doubled the life of the astronomer.” Come back to this idea after students have seen that logarithms are exponents and after they have learned the rules for working with logarithms. Then the students will be able to appreciate the computational improvements – especially when the lack of computers and calculators is borne in mind!

An introduction to the life and work of John Napier while introducing students to logarithms will bring the “dry” material to life.

Napier was a Scottish mathematician who lived from 1550 to 1617. He worked for more than twenty years to develop his theory and tables of what he called logarithms, a word he derived from two Greek roots: *logos*, meaning word, or study, or reasoning, or in Napier’s use, “reckoning”, and *arithmos*, meaning “number”. Much of our mathematical terminology, and indeed our English vocabulary, derives from Greek and Latin roots. It is a useful exercise to take a few moments when new terms are introduced to explore the etymology of the word and to have the class try to name other words also deriving from these roots. For example, you might ask “Where else have you seen a word derived from *arithmos*?

If etymology is not your strong suit, you will find *The Words of Mathematics* by Steven Schwartzman an excellent resource. It is published by the MAA.

Napier chose the name “logarithms” because he thought of them as “reckoning numbers”. Their use could save computational time, especially the time of beleaguered astronomers. These men had to carry out computations involving very large numbers. Any simplifying devices were welcomed with joy. In fact, the French mathematician Pierre Laplace (1749-1833) said that Napier’s new tool “doubled the life of the astronomer.” Come back to this idea after students have seen that logarithms are exponents and after they have learned the rules for working with logarithms. Then the students will be able to appreciate the computational improvements – especially when the lack of computers and calculators is borne in mind!

Sharing some interesting highlights will help bring Napier to life for the students. My students have enjoyed hearing of some of Napier’s famous predictions. For example, he wrote of a future machine that would “clear a field of four miles circumference of all living creatures exceeding a foot in height”, a description calling to mind the feared weapon of the World Wars (and today), the machine gun. He also wrote of a chariot with a “living mouth of mettle” (sic) that would “scatter destruction on all sides”. This is an apt description of the modern tank.

Another incident my students have appreciated is the story of Napier’s dispute with a neighbor over the neighbor’s pigeons. It seems that the birds were eating Napier’s grain. Despite his repeated protests, the neighbor was unable or unwilling to stop the birds from the ongoing thefts. Finally, Napier threatened to impound the creatures. Evidently secure in the knowledge that this would be impossible, since the birds would simply fly off when approached, the neighbor assured Napier that he was welcome to impound the birds – if he could. The next day the neighbor was shocked to see his birds staggering in the field while Napier walked around, plucking them up one by one, and dropping them into a sack. Napier had soaked a batch of peas in brandy and then scattered them in his field. The intoxicated pigeons were easy targets!

It takes relatively little time to share with the class this biographical material and the etymological material that precedes it. The pay off is that it goes a long way toward bringing the material to life, which in turn helps to engage the students in the concepts.

Finally, a hands-on experience will almost certainly surprise the students while also sharing another aspect of Napier’s life. He would take objects that were relatively straight, such as sticks or bones, and inscribe the multiples of one or the other of the single digits (up to the product of the digit and nine). For example, he might have the following for “6”:

These objects came to be known as Napier’s bones (or Napier’s rods). You can use old Popsicle sticks to create your own or buy new ones from a craft store. You will need many sticks for each digit, not only so that several students can do this at the same time, but also because you need one stick for each occurrence of a digit in a factor. When digits are repeated, so are "bones."

The bones were used to expedite the multiplication process as can be illustrated in the examples depicted in this *Wikipedia* entry.

Despite the fact that the essential features of this procedure are the same as the ones our students learned when they learned multiplication in elementary school, they are generally surprised and delighted to find the “bones” working as they do. Once you have done the hard work of creating lots of these bones, either physically or through an animation like the one offered by Wolfram, you will be able to delight your students as well.

Be sure to stress to your students that, although we are surely multiplying here in an unusual context, we are doing nothing but the ordinary multiplication algorithm. A multiplication problem thus becomes an addition problem through the use of Napier’s bones.

The following links provide additional information on John Napier and his “bones”:

John Napier < http://www-history.mcs.st-and.ac.uk/Biographies/Napier.html > is a biography of Napier from the very useful MacTutor History of Mathematics Archive maintained by the University of St Andrews in Scotland. It has links to other references.

The Science Museum in London has several sets of "Napier's bones" in its collection.

Three images of a boxed set of Napier’s bones (circa 1690):

https://collection.sciencemuseumgroup.org.uk/objects/co60130/napiers-bones-napiers-bones

Six images of two sets of cylindrical "Napier’s rods" (circa 1671–1700):

https://collection.sciencemuseumgroup.org.uk/objects/co60059/napiers-bones-napiers-bones

Brass Napier’s bones (17th century):

https://collection.sciencemuseumgroup.org.uk/objects/co60079/napiers-bones-napiers-bones

Ivory Napier’s bones (18th century):

https://collection.sciencemuseumgroup.org.uk/objects/co60408/set-of-napiers-bones-18th-century-napiers-bones

Napier Bones in Various Bases < http://www.cut-the-knot.org/blue/Napier.shtml >, by Alexander Bogomolny, describes an extension of Napier’s bones to number bases other than base ten.

Finally, be aware that slide rules have sometimes been referred to as Napier’s bones. An internet search will sometimes lead you to slide rule information instead of the tool described in this article.

Thanks to Wayne Anderson of Gannon University’s Center for Excellence in Teaching and Learning for his help with the Napier’s bones Flash animation at < http://ww2.gannon.edu/cetl/caulfield/NapierBones.html >. [Editors’ Update: Adobe ceased supporting Flash on December 30, 2020, so the animation is no longer functional. For animations on other websites, see the links on the preceding page.]