Wackerbarth's Logarithm Tables

Logarithm tables.Already at age 13 or 14 I had developed a well-deserved reputation for being an eccentric - even in my own family. One of the reasons was that I had put a book with logarithm tables on my Christmas wish list. An even better reason was that after I got it, I used to bring it with me when we went for a walk.

My interest must have been triggered by a book on the history of mathematics, probably Bell's "Men of Mathematics". I came to realise what heroic efforts astronomers and navigators must have put into sheer number "crunching" before logarithms were discovered, and what a revolution logarithms must have meant to their professions. I was especially impressed with the work of Johannes Kepler, who divided his time between saving his mother from being burned as a witch (or heretic?) and calculating the orbits of the planets and deriving some simple laws for their motion from Tycho Brahe's long lists of observations.

The magic of logarithms was that they transformed a multiplication into an addition, and a division into a subtraction. It all stems from a basic rule in algebra:

am * an = am+n

The trick is this: If we want to multiply two numbers X and Y without going through the drudgery of actually performing the calculation, which can be considerable if we are dealing with, say, two 10-digit numbers, we just need to express X and Y in the form am and an, respectively. Then we calculate m + n and look up which number am+n corresponds to.

Originally, am just meant "multiply a with itself m times" (or m-1 times if you want to be pedantic), so that for instance 53 = 5 * 5 * 5 = 125. But mathematicians soon discovered that the "exponents" m and n did not necessarily have to be integers. For instance, if m = ½ and n = ½, then the formula above says that a½*a½ = a. But that by definition means that a½ is the square root of a. Further extensions can be made so that m or n in the formula above can be any real number (actually any complex number, but let us not go into that).

Logarithm tables commonly use 10 as a base, corresponding to the number a above. To calculate, say, 2*3 with the means of logarithms (this is just for the purpose of the discussion - I know the result!), you look up the logarithms of 2 and 3 to base 10 in the table: 0.30103 and 0.47712. You add these and obtain 0.77815. In the table, that corresponds to the number 6. - Voilá, you have multiplied two numbers without actually performing the calculation!

When I learned this, I thought: Great! I will memorize logarithms and become a master arithmetician! Actually, what I was proposing to do, without realizing it, was to memorize the slide rule! My enthusiasm soon waned, when I discovered that I would either have to learn a very large number of logarithms by heart or be prepared to do a lot of interpolation between logarithms in a table with large holes in it.

As an aside, I really was considered a wizard with numbers in school. Once, my math teacher needed the square of 155 and, intending it as a joke, turned to me: "Zenker, how much is that. You mean you don't know?" And I immediately replied: "Sure, it's 24025." He responded: "You are from Mars!" - (The "secret" was that I knew that I needed 15*15 + 15 and then had to append 25 at the end. - All squares of numbers ending in 5 are like that: (10n + 5)2 = 100n2 + 100n + 25 = (n2 + n)*100 + 25. In fact, I could even have used it to find 15*15 = 225, if I had not already memorized that.)

When logarithms were discovered (some would say "invented") by John Napier in 1614, the discovery was already "in the air", so to speak. Trigonometric identities had been used for several decades to transform division into subtraction. A Swiss mathematician and clockmaker, Joost Bürgi, discovered logarithms independently of Napier but did not publish his results until 1620. - Interestingly, logarithms were not recognized as exponents of base numbers until much later in the century.

A key figure in the development of logarithm tables was Henry Briggs. He quickly recognized the significance of Napier's work and compiled a table to the base of 10 as early as 1617. This led to the rapid spread of the use of logarithms in Europe in the following years among navigators, land surveyors and astronomers. Of particular interest, perhaps, Kepler's discovery of the Third Law of Planetary Motion was undoubtedly related to the discovery of logarithms, and Kepler dedicated his Ephemerides in 1620 to Napier, who had died in 1617.

One can imagine the joy of the likes of Kepler, when the necessity to perform complicated calculations by hand was reduced by an order of magnitude. At the same time, one has to admire the determination and perseverance of the mathematicians who compiled the first tables of logarithms, also by hand. The drudgery of the work must have been incredible, while errors could hardly be tolerated. Today it is fashionable among scientists to call themselves "workers", but in those days the term carried real meaning!

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    Last edited or checked January 8, 2010.  

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