You Could Look It Up (20 page)

As handy as these were, though, they answer only very specific kinds of questions. Of course no book could provide tables to answer
every
calculation someone might need to perform, but with the discovery of the mathematical function known as the logarithm, one could compile tables to help with almost any calculation—huge tables, to be sure, but finite, and eminently useful.

Merriam-Webster’s definition of
logarithm
is as good as any brief definition can be: “the exponent that indicates the power to which a base number is raised to produce a given number.” Alexander John Thompson is more expansive in
Logarithmetica Britannica
, the standard modern work on the subject:
3

The logarithm of a number
N
, to any base
a
, is defined here as the power
y
to which the base
a
must be raised to produce the number
N
; that is, if

N
=
a
y
,

y
is the logarithm of
N
to base
a
, or

y
= log
a
N
.

The mathphobic will welcome a clearer explanation still, which begins with exponents. Exponentiation is simply repeated multiplication. We use superscript numbers to indicate the number of times a number (called the
base
) is multiplied by itself: 2
3
= 2 × 2 × 2 = 8; 6
4
= 6 × 6 × 6 × 6 = 1,296. More generally,
a
n
means
a
×
a
×
a
… with
a
appearing a total of
n
times.

Because our counting system is known as base 10, powers of 10 are especially significant, and 10 is the most common base for logarithms. The number 10 can also be written as 10
1
. The number 100 can be written as 10 × 10, or 10
2
; 1,000 can be written as 10 × 10 × 10, or 10
3
; and so on. A logarithm performs exponentiation in reverse.
4
Since 10
1
= 10, it follows that log (10) = 1. Since 10
2
= 100, it follows that log (100) = 2; since 10
3
= 1,000, it follows that log (1,000) = 3; and so on. Logarithms need not be whole numbers. It is easy to see that log (1,000) = 3 because 10
3
= 1,000, but it is less obvious that log (25) is 1.39794, because 10
1.39794
= 25, or log (2,000) is 3.30103, because 10
3.30103
= 2,000.

Logarithms fascinate mathematicians. They speed the calculation of compound interest and the half-lives of radioisotopes. They are linked in unexpected ways to prime numbers and trigonometrical functions. The constant
e
—the base of the “natural” log, roughly 2.71828—makes a surprise appearance in one of the most remarkable equations in history,
e
π i
+ 1 = 0, which brings together the five most basic constants in mathematics. But logarithms might have remained little more than mathematical curiosities were it not for one further remarkable property: the logarithm of a product is the sum of the logarithm of the two factors—in a formula, log (
a
×
b
) = log (
a
) + log (
b
). To put it another way, if
a
×
b
=
c
, then log (
a
) + log (
b
) = log (
c
). We can check it with an example: since 10 × 100 = 1,000, it follows that log (10) + log (100) = log (1,000): log (10) = 1; log (100) = 2; log (1,000) = 3.

This seemingly abstruse insight has a practical application: it lets us express multiplication in terms of addition, and division in terms of
subtraction. Since addition is easier than multiplication, especially for large numbers, and subtraction is much easier than division, calculations can be speeded up by reducing complicated multiplications and divisions to much simpler additions and subtractions, with the help of a list of logarithms. To multiply any two numbers—say,
a
×
b
—look up log (
a
) and log (
b
) in a table; manually add those two numbers together; then look up that sum in another column of the table and find the number that has that sum as its logarithm—that gives the product. So, for instance, to multiply 2,384 by 1,635, look in the table for the logarithm of 2,384—it is 3.377306—and the logarithm of 1,653—it is 3.213517—and add them together: the sum is 6.590823. Turn to the table again and find the number that has for its logarithm 6.590823: it is 3,897,840, the product of the numbers. Division works in much the same way. It can be expressed in terms of subtraction: log (
a
÷
b
) = log (
a
) – log (
b
). To divide 9,910,233 by 4,387, look for the logarithm of 9,910,233—it is 6.996084—and the logarithm of 4,387—it is 3.642165—and subtract one from the other: the difference is 3.353919. Then find the number that has for its logarithm 3.353919: it is 2,259, the quotient of the numbers. Best of all, this requires no mathematical sophistication, because you can do it without knowing the first thing about logarithms and exponents. It is simply a procedure: look up
a
, look up
b
, add or subtract them, and look up the result.

Logarithms make many other mathematical operations simpler. If the concept sounds complicated, that is because we live in the computer age, when dividing 9,910,233 by 4,387 requires no more effort than typing the numbers into a spreadsheet, pocket calculator, or mobile phone. In the age of long division, though, it could take half an hour to perform a division like that, and with so many steps, error was always a possibility. How much easier it was when logarithms were introduced to look up two numbers in a table, perform one simple subtraction, and then look up another number in a table.
5

These properties of logarithms were discovered in the early sixteenth century by a Scottish baron named John Napier. Born in 1550 in Merchiston Castle, not far from Edinburgh, Napier studied at St. Andrews University.
6
He was notoriously argumentative and a vindictive neighbor, but he also did much to improve society, such as developing new fertilizers and water
pumps for coal pits. He worked on plans to repel a Spanish invasion of England with gigantic mirrors that could direct the sun’s rays and set enemy ships on fire, and he drew up plans for new guns and even prototypes of twentieth-century tanks.
7
This master tinkerer also made a number of advances in mathematics; the decimal point was his invention.
8

He was working on logarithms as early as 1594. Napier complained that “there is nothing that is so troublesome to mathematical practice, nor that doth more molest and hinder calculations, than the multiplications, divisions, square and cubical extractions of great numbers,” and he “began therefore to consider in my mind by what certain and ready art I might remove those hindrances.”
9
His book,
Mirifici logarithmorum canonis descriptio
, appeared in Latin in 1614, laying out the principles of logarithms. Early support from the astronomer Johannes Kepler gave Napier the cachet he needed. Soon logarithms were the talk of all the mathematicians in Europe, and people began discovering practical uses for them in navigation, making them of interest to commercial interests such as the East India Company.
10

Calculating logarithms for individual applications is, however, unimaginably time-consuming. Long division is child’s play next to calculating a logarithm by hand. What was needed was a table of logarithms to facilitate any real-world mathematical problem. Preparing that list required a preposterous effort, but that was the contribution of Henry Briggs. This English mathematician became interested in logarithms in 1615, and he wrote to a friend, “Naper [
sic
], lord of Markinston, hath set my head and hands a work with his new and admirable logarithms. I hope to see him this summer, if it please God, for I never saw book, which pleased me better, and made me more wonder.”
11
After visiting Napier in 1616, Briggs resolved to calculate as many logarithms as possible, to allow his readers to perform all manner of calculations more easily.

The first fruits of his labor appeared right after Napier’s death in 1617, a pamphlet called
Logarithmorum chilias prima
, the first printed table of logarithms. The preface explained the purpose:

Here is the first thousand logarithms which the author has had printed, not with the intention of becoming public property, but
partly to satisfy on a private basis the wish of some of his own intimate friends: partly so that with its help he might more conveniently solve not only several following thousands but also the integral table of Logarithms used for the calculation of all triangles… .

In a slight volume neither the enjoyment nor the toil has been slight.
12

The pamphlet was a start, but Briggs’s greatest work appeared in 1624 under the title
Arithmetica logarithmica
, a table of the base-10 logarithms of all the whole numbers from 1 to 20,000 and from 90,000 to 100,000.

His precision was formidable: fourteen decimal places for each of his thirty thousand entries. His second edition, published in 1628, was supplemented by the Dutch publisher Adriaan Vlacq, who covered 20,001–89,999, giving the world a complete set of the first hundred thousand base-10 logarithms. Complicated calculation would never be the same, and Briggs’s book was the foundation of every table of logarithms published for centuries. No one bothered with new calculations until the end of the eighteenth century.

This work in many ways made the scientific revolution possible. The engineering projects of the Industrial Revolution could never have come to fruition without logarithms, and scientific astronomy would still be in its infancy. Pierre-Simon Laplace, the nineteenth-century French mathematician and astronomer, marveled at this “admirable artifice which, by reducing to a few days the labour of many months, doubles the life of the astronomer, and spares him the errors and disgust inseparable from long calculations.”
13

The logarithm made possible an ingenious invention, the slide rule—a convenient way of approximating logarithms without a table. The numbers on a ruler are spaced according to their logarithms, so the physical positioning of the rulers takes the place of looking up numbers in a book. The earliest slide rule was devised by Edmund Gunter around the year 1620, but a great leap forward came from William Oughtred in the 1620s and ’30s. The result “would be the faithful companion of every scientist and engineer for the next 350 years, proudly given by parents to their sons and daughters upon graduation from college”
14
—a tradition brought to an end only by the invention of the pocket calculator in the
1970s. What had served the world as a calculator was actually a reference book reduced to a few notched sticks.