In previous posts, I pointed out how logarithms took the hardship out of arithmetic in the days before electronic calculators. However, the use of logs hinged crucially on access to an accurate and usable set of logarithm tables. This is probably why logarithms make their appearance only after Gutenburg’s invention of the printing press in Europe. This invention meant that useful log tables could (eventually) be printed cheaply and in bulk, and so instead of being just another special functions, logarithms found a place in the bookshelves of mathematicians, engineers and scientists across the world.

But how did people use these tables? It’s actually a small skill in and of itself. In this post, I’ll outline how log tables were used for most of the 20th Century, having by then evolved into their most streamlined form. I’ll use a set of four figure tables for these examples to keep things simple, but tables with higher precision were available, along with more sophisticated correction methods. This will probably only serve as a refresher for log table veterans.

We’ll stick with the basics for now. Below is a twin set of four-figure log and anti-log tables, in the common base of $10$, listing values $\log_{10}x$ for $x$ from $1.0$ to $10$ and $10^m$ for $m$ from $0.0$ to $1.0$. The values are given to 4 places of decimals, hence the term, four-figure tables. Each table consists of two pages, lying facing on another.

So, how were these tables read. Let’s focus on the log tables for now. Consider the following crop of the first page of the $\log$ table, showing the top left of the first page.

On the leftmost column you can see values of $x$ being listed, in differences of $0.1$, from $1.0$ down to $2.6$. On the top row, additional second place digits for $x$ are listed, along with another list to the right for third place digits — but more on these later.

Returning to the values of $x$ in the first column, the second column lists the corresponding values of $\log_{10} x$, minus a leading zero to the left of the decimal point. So for example: To calculate the value of $\log_{10}(1.4)$, simply travel down the leftmost column to $1.4$ and read off the digits next to this to obtain $\log_{10}(1.4)=0.1461$, which is the value correct to 4 decimal places. The image below illustrates this process, starting from the solid red block.

So far so good, but what if a more finely graded table in $x$ is needed? Suppose we want to calculate the log of $2.05$. Here’s where the columns at the top come in. These represent additional digits of $x$.

To find $\log_{10} (2.05)$, we navigate using the digits of $x$. First travel down to $2.0$ on the leftmost column. Now, staying on the same row as $2.0$, travel right along the columns until the first $5$ column is reached. The set of four digits at the intersection of the $2.0$ row and big $5$ column are $3118$. These represent the digits to the right of the (unprinted) decimal point. So $\log_{10}(2.05) = 0.3118$.

But it’s possible to obtain an even finer grading in $x$ by using the mean differences columns, with only a slight loss in accuracy. Suppose we want to find $\log_{10}(1.836)$. As before, we navigate by digits. First, travel down the leftmost column to $1.8$. Staying on this row, travel across to the first $3$ column. The digits there read, $2625$. But we’re not done.

With these four digits, and still on the same row, travel across to the second set of columns representing third place digits, and go to the smaller $6$ column. The digits listed are $14$; and this is the mean difference, which must be added to $2625$ to obtain

\[ 2625 + 14 = 2639\]

These are the estimated four digits of the result, and so $\log(1.836) = 0.2639$. The second column set of mean differences has allowed a table with a three digit grading of $x$ from $1.0$ to $10$, on only two facing pages. The log tables can be left open and 4-figure logs of numbers to $3$ significant digits can be looked up without having to turn a page.

The anti-log tables worked similarly. For example, suppose we want to find the log of $10^{0.5678}$. The tables are giving us four significant figures of a result lying between $10^0=1$ and $10^1=10$, but otherwise we navigate by digits as before.

Travelling down the left most column, we pass on to the second page until reaching the $5.6$ row. Then we travel across to the large $7$ column and read off $3690$, and then travel to the $8$ column in the mean differences to read $7$. Adding these two numbers gives $3690+7=3697$. So $10^{0.5678}=3.697$, as the result lies between $1$ and $10$.

So that’s how logs and exponents were found in the days before digital calculators. More finely graded values could be interpolated at a pinch. More advanced sets of tables also included higher differences to facilitate this, though most everyday tables stuck to just mean first differences.

Typically, there were also more than just logs and anti-logs in a set of tables. All kinds of commonly used functions could be listed. The image below shows the tables being used to calculate $\sin (11^\circ 34′) = 0.2005$.

All of the tables above are taken from the old Irish state mathematical tables — since discontinued. A more prolific set of four figure tables were the Godfrey and Siddons Four-Figure Tables, printed by Cambridge University Press, and which was apparently one of the few publications there to rival the Bible in terms of numbers sold. Tables like these were printed and reprinted in the hundreds of thousands and were in essentially universal use. The tables allowed look-ups of values of many special functions like: $\cos$, $\sin$, $\tan$, $x^2$, $\sqrt{x}$, $1/x$, $\ln x$, $e^x$, and of course common logs and anti-logs $\log_{10} x$ and $10^x$. By shelling out a little more, you could also get sets of 5 figure tables, which offered more precision, more functions, more data, and more mathematical formulas — in effect, the graphing calculators of their day.

All in the past now. Modern tables still include mathematical formulas, which is quite a good thing, as tables only get bigger and better the father you go in mathematics, and learning that they **should** be used is a useful lesson to learn in and of itself. But there will likely be no more **tables** in mathematics tables any-more, that role now being fulfilled by calculators and computers. Still, as with slide rules, old tables linger on in the hands of a few enthusiasts. And who knows, the old tables may yet have a few tricks to teach us.