What are these numbers in a logarithmic table?

Question

Below is an image from a table of logarithms. As an example, one sees that $\log(661.3) = 2.82\color{red}{040}$. In this logarithmic table there are some numbers to the right. My question is: What is these numbers?

Answer 1

They're for linear interpolation, if you need a geater precision than the precision directly given by the table.

Example: you see from this table that $\log(6.612)=0.82033$, $\log(6.613)=0.82040$ ($\Delta\log(x)=0.00007$). Hence, to have, say, $\log 6.6124$, you look at the right hand side and take $\log(6.6124)\approx 0.82033+0.000028=0.820358\approx 0.82036$ (rounded to $5$ decimals).

Conversely, if a number $x$ is such that, say, $\log x=0.82025$, you know first that $0.6610<x<0.6611$, and from the right-hand side, you know that, more precisely: $x\approx 0.66107$.

On a given page, you'll find the same series of numbers for all values of $\Delta\log(x)$ that appear on the page.

Answer 2

Those are intended to allow an additional digit of accuracy to be obtained. In a region where the values are approximately linear, these values show the amount that, appropriately scaled, should be added to the table's result for each digit appended to the input value.