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# John Napier Discovers Logarithms

## Overview

Logarithms are of fundamental importance to an incredibly wide array of fields, including much of mathematics, physics, engineering, statistics, chemistry, and any areas using these disciplines. However, until the early seventeenth century, they were unknown. Invented by a Scottish amateur mathematician named John Napier (1550-1617) after 20 years of work, they were met with almost immediate acceptance by mathematicians and scientists alike. In the intervening centuries, logarithms and their converse, exponents, have proven to be among the most useful mathematical tools of all time.

## Background

Arithmetic (addition, subtraction, multiplication, and division) dates back to human prehistory. Of these most basic operations, addition and subtraction are relatively easy while multiplication and division are much more difficult to master. Until the Renaissance, however, mathematics and the sciences were not very dependent on mathematical calculation, and these difficulties, while vexing, were not insurmountable.

However, the flurry of discoveries in physics and mathematics that began in the fifteenth century gave rise to more and more need for calculation, and what had been inconvenient now became drudgery. Unfortunately, in those days before computers (or even slide rules), no other options existed, so scientists and their assistants plugged away, laboriously multiplying and dividing by hand.

Towards the end of the seventeenth century, John Napier, a Scottish laird, began to look for an easier way to undertake these operations. One observation he made dealt with multiplying powers of numbers. For example, 4 × 8 = 32. If we write these as powers of two, we have 22 × 23 = 25. Napier observed that the exponents on the left side of the equation add up to equal the exponent in the answer (that is, 2 + 3 = 5). This meant that, instead of multiplying two numbers together, one could simply add the exponents together, and then calculate the final answer using exponents. Or, in terms of our example,

This series of operations has a few drawbacks. First, it is far too much work for such a simple problem. Second, in this form, it will only work for integer multiples of integers—it is just too limited in scope to be of general use. Napier's breakthrough came in realizing that this method could be extended in such a way as to make it more generally useful. After this breakthrough, he spent the next 20 years calculating the first table of logarithms, publishing his results in 1614. It is worth noting that a Swiss mathematician, Joost Bürgi (1552-1632) apparently invented logarithms some time before Napier, but for some inexplicable reason, neglected to publish his results and their utility until 1620. Because of this, he lost his claim to scientific priority and is not given credit for his work.

There are, of course, some differences between what Napier called logarithms and our current definition. These differences are, however, not fundamental, and all of Napier's work easily translates to current usage with only minor adjustments. In general, it is safe to say that the fundamental concepts of Napier's system remain intact, even after nearly 400 years—a remarkable achievement. What is even more remarkable is that Napier performed this work in intellectual near-isolation. Or, as Lord Moulton said in a 1914 tribute marking the 300th anniversary of Napier's paper,

The invention of logarithms came on the world as a bolt from the blue. No previous work had led up to it, foreshadowed it, or heralded its arrival. It stands isolated, breaking in upon human thought abruptly without borrowing from the work of other intellects or following known lines of mathematical thought.

## Impact

As mentioned above, Napier's work was greeted with instant enthusiasm by virtually all mathematicians who read it. The primary reason for this is because his tables of logarithms vastly simplified computation. Along these same lines, the principles upon which the slide rule works are dependent on the addition of logarithms, a fact that helped speed up computation in precalculator days. Mathematicians also quickly found other uses for logarithms, and invented other related concepts such as fractional exponents, the number e, and similar mathematical tools. Although these were not of great importance to the average person of the seventeenth century, Napier's invention was of such a tremendous boon that, directly or indirectly, it has affected virtually everyone in some way.

## Simplifying computation

As mentioned above, the invention of logarithms greatly simplified mathematical operations. While this sounds relatively straightforward, its importance may not be obvious. Consider, however, the fate of an astronomer or physicist before and after 1614:

To determine a planetary orbit, an astronomer needed to make a number of relatively sophisticated calculations, many of which used rather large numbers and several operations of multiplication and division. To multiply, for example, two five-digit numbers required several minutes, five sets of multiplication, and adding the results of each of these sets. At any point of this laborious process a mistake could be made, so each calculation had to be checked for accuracy. This could take many minutes for each multiplication, and division was much more difficult. With Napier's system, on the other hand, this operation took just a few minutes. First, the astronomer would look up the logarithms of each factor. Next, he would add these logarithms together, and then would find in the tables the number for which this sum was the logarithm (called the antilogarithm).

Saving five minutes of calculation time is not significant—but saving five minutes of calculation time for each of several thousand calculations makes a dramatic difference. This single mathematical invention was responsible for an incredible decrease in mathematical drudgery while simultaneously increasing computational accuracy and scientific output. In fact, Pierre-Simon Laplace (1749-1827) commented that Napier had "by shortening labors doubled the life of the astronomer." The next such dramatic increase in computational efficiency would be the invention of the slide rule in the 1620s, very shortly after publication of Napier's original paper. By removing the necessity to look up values of logarithms and antilogarithms, the slide rule sped up computation even more, while further reducing the chance of error. In fact, the slide rule would remain virtually unchanged for nearly four hundred years, an indispensable tool for anyone performing computations. The only device to simplify calculations even further has been the electronic calculator, which has replaced the slide rule in virtually every office in the world.

## Mathematical spin-offs

Logarithms not only led directly to the invention of slide rules, but also produced other intellectual spin-offs. Exponents, for example, were already well known in Napier's time, but were of limited utility because of the contemporary insistence of using only integers as exponents for other integers. For example, one could write the term 23, but not 22.5. This limitation placed an unnecessary constraint on the use of exponents. However, shortly after publication of Napier's paper, mathematicians realized that logarithms were simply exponents. Since logarithms were also written in decimal notation, this opened the door to a wider use of fractions and decimals as exponents, again simplifying mathematical computation. Similarly, mathematicians realized that they could use exponents with fractions and decimal numbers as the base. Today, neither of these seems a revelation, but in the early 1600s, this was a major breakthrough.

Another spin-off of Napier's work was the realization that, when examined mathematically, his work produced the number e, the base for the natural logarithms. Like π, e is a transcendental number that will never terminate or repeat; it has also, like π, proven itself to be an incredibly versatile number that pops up in calculations performed in just about every field that uses mathematics. Compound interest, radioactive decay, the growth and decline of bacterial populations, astronomical calculations, and any number of engineering problems all make use of e, and solutions to many of these problems also require the use of logarithms. In the eighteenth century, the brilliant mathematician, Leonhard Euler (1707-1783) would help give logarithms and exponential functions an important place in higher mathematics and the calculus.

## The effects of logarithms on society

While logarithms and exponents are basic mathematical concepts, they are rather esoteric for those who do not work with them regularly. This was especially true for the average person in the seventeenth century, who was certainly had no appreciation for any mathematics beyond simple arithmetic. Even today the average person is likely to have only a passing familiarity with these concepts. Logarithms, however, are so fundamental to the mathematics of many disciplines and are so ubiquitous in mathematical problem solving that most people's lives have been affected in some way by them. For example, the mathematics by which aircraft, internal combustion engines, electrical generators, and petroleum refineries are designed depend intimately on the use of these concepts. Compound interest calculations, important to anyone with either a bank account or a loan, are made simpler and more accurate by use of e. The mathematics upon which television and radio broadcast and reception is based also depend on these mathematical tools. Except for very isolated, primitive tribes, it is likely that everyone has been exposed to at least one of these categories of objects, or to the fruits of one.

Put simply, John Napier, laboring on his own, created a mathematical concept that has proven extraordinarily useful to mathematicians, scientists, financiers, and many others. Because of the incredible utility of logarithms and exponents, technical and statistical calculations were made much simpler and more reliable, and all of society has benefited as a result.

P. ANDREW KARAM