John Napier and the Invention of Logarithms

Background and Introduction

For this lesson, we are going to step back in history. Imagine you live at the turn of the 15th century. A lot of scientific discovery is going on! Galileo Galilei is inventing the telescope and working on his heliocentric theories; Johannes Kepler is observing and calculating the laws of planetary motion, and the goundwork is being laid for Newton and Leibniz to discover Calculus in a few decades! But we have a problem. All this mathemtical and scientific discovery is hindered by the fact that arithmetic operations on large numbers have to be done by hand – there are no computers, calculators or slide rules.

Enter John Napier:

“Seeing there is nothing that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers…I began therefore to consider in my mind by what certain and ready art I might remove those hindrances.” – John Napier, 1614

Napier’s invention of logarithms, along with his other discoveries, provided scientists and mathematicians with tools to make their lives much easier. A famous mathematician named Laplace said that “by shortening the labors, [Napier]doubled the life of the astronomer.”

So, what is a logarithm, anyway!? It is an exponent. A logarithm is an exponent. Don’t ever forget that! It is the exponent you raise a base to, to get another number. For example, log₁₀100 = 2 reads “the log base 10 of 100 is 2″ and means that you must raise 10 to the second to get 100. So the logarithm, or exponent, is 2. Similarly, log₁₀1000 = 3, or you raise 10 to the third to get 1000. So the logarithm tells us how many times we have to multiply the base by itself to get the number we are looking for.

Here’s where it gets interesting. What would you have to raise 10 to, to get… 537? Well, 10² = 100 and 10³ = 1000, so you would have to raise 10 to “2.something” to get 537. If we have a way to compute logarithms, we can figure out what the “2.something” is!

Go ahead and grab your calculator, enter log 537 on a calculator and you’ll get the answer. Base 10 logarithms are called common logarithms and are usually written without the base. So log 537 means the same thing as log₁₀537.

Next on the agenda is some reading and discussion about the development of logarithms. In our next lesson we’ll start graphing and doing computations with logarithms and exponents.

Assignment

1. Read and take notes on chapters 1 and 2 of “e: The Story of a Number”
2. Take the History of the Development of Logarithms online quiz (due by midnight Tuesday).
3. Answer each of the discussion questions on the History of the Development of Logarithms discussion board (go to Communication and then click on Discussion board) (due Wednesday).
4. Respond to your fellow students answers and/or post your own question. Points will be awarded based on the thoughtfulness of your discussion (I will be moderating the discussion board on a daily basis through Saturday).