Interpolation (the Mean-Difference Method)

A four-figure log table lists mantissae directly only up to the third significant digit. The fourth significant digit is supplied by interpolation — the assumption that, over the tiny interval between two consecutive table entries, the logarithm changes by an (approximately) constant amount per unit change in the number. This is the standard principle of proportional parts, and the table's mean-difference columns pre-compute it for you.

How it works. Suppose the first three significant digits give a main-table mantissa, and the next tabulated value (for the third digit increased by 1) is a little larger. The gap between these two neighbouring mantissae is the difference for a full step. The fourth significant digit is a fraction of that step, so its contribution is that same fraction of the difference. The mean-difference column simply lists this contribution for each possible fourth digit 1–9, already rounded, so you only add it.

Worked illustration (the digit string 4567). From the log table:

The difference for one full step in the third-decimal position is $0.6599 - 0.6590 = 0.0009$, i.e. 9 units in the fourth mantissa place, spread across 10 units of the fourth significant digit — about $0.9$ of a unit each. For a fourth digit of 7, the contribution is $0.9 \times 7 = 6.3 \approx 6$ units. Hence

exactly the mean-difference value the table lists against digit 7 in that row. The same proportional-parts idea, run in reverse, is what the antilog table's mean-difference columns do when recovering a number from its mantissa. …