The lower (D) scale is fixed. The upper (C) scale rides the slide. To multiply a × b: drop the cursor on a on D, drag the slide so its left index (1 of C) sits under the hairline, then move the cursor to b on C and read the product on D directly below it. Division runs the same dance in reverse. If the product runs off the right edge of D, slide so the right index (10 of C) sits over a instead — the reading is shifted by a decimal place.
The compressed A and B scales span two decades (1–100) in the same horizontal space that C and D use for one decade (1–10). This compression encodes a square relationship: the A-scale reading at any cursor position equals the square of the D-scale reading directly below it. To extract a square root: position the hairline on your number on the A scale and read D. The left half of A (1–10) and the right half (10–100) each cover one decade; numbers between 1 and 10 are found in the left half, numbers between 10 and 100 in the right half.
The Pickett N3-T and the K&E 4081 — the standard mid-century engineering rules — were graduated to about this precision. The Apollo design margins were originally checked on instruments of this class. Most of the engineers who used them are now retired.