"luth.org" sounds like a great resource. However, the square root of two is NOT 17.817! WOW.
"Prs Mods too" is right, the "Rule of 18" or 17.817 is the mathematical constant that gets us closest to the theoretical location of the next fret using a scale length equation. Here's how you use it:
The nut is the ZERO FRET. Using a PRS Custom 22 with a scale length of 25 inches (from ZERO FRET to SADDLE), you divide 25 inches by 17.817, which is 25/17.817 = 1.403 inches. This 1.403 " measurement is the length between the ZERO FRET and the FIRST FRET (the next fret).
To calculate the SECOND FRET, you must calculate the distance between the FIRST FRET and the SADDLE (25 - 1.403 = 23.597") like it were its own scale length. Now repeat the same calculation and divide 23.597 by 17.817 and you get 1.324 inches. This is the distance between the FIRST FRET and the SECOND FRET, or the length of the SECOND FRET.
Find the third fret position by again calculating the distance between fret 2 and the saddle, divide by 17.817 and you get the length of the third. Wash, rinse repeat.
When you fret a note, you change the distance of the surrounding notes because the octave is always the midpoint of the open vibrating string. Fret the 12th fret and now your open vibrating scale length is half what it was, and ITS octave is now the midpoint between the 12-fret and the saddle, or the 24th fret. It's a sliding scale.
There are minute changes in saddle position, or COMPENSATION, to overcome the increase in string length that happens when you change the string from floating to its fretted position by fingering a note. The string lowers RADIALLY from the nut, and therefore, being the hypotenuse of a triangle, is LONGER than the rule of 18 calculated length WHEN FRETTED. This compensation explains why your bridge saddles are generally arranged in two separate slants on the bridge to compensate for different string gauges down the bridge.
Back in the day, most lutes had moveable frets like bands to allow for proper tunings, also because strings back then were made of extruded gut, and were anything BUT consistent in thickness. I have no idea when the Rule of 18 was discovered, but it's pretty cool. It of course applies to the string primarily, as this rule applies to fretless instruments as well--although fretless technique requires you to place your finger pad ON the fret location and not in between.
Great question.
EDIT: If you want to really blow your mind, check out the approach of using "True Temperment"--a system that overcomes the shortcomings of the Rule of 18 approach. SPAGHETTI!
https://guitargearfinder.com/faq/true-temperament-frets/
