Wow, Dr. Shapiro. I'm gonna need a hint
Are you serious
Yes
Hint
These words are also unrelated to a word defined as "a strong, regular, repeated pattern of movement or sound".
Thanks to Kate for suggesting the midday addition of an Al Gore reference (the former candidate)! The Hint was referencing rhythm.
Solution
The words are logarithm and algorithm. The roots of logarithm can also be seen in the words logic and arithmetic; meanwhile, algorithm is a Latinized form of Al-Khwarizmi, the mathematician–astronomer who taught the Western world how to calculate using Hindu–Arabic numerals.
Congratulations to yesterday's solvers Zachary Z., Atticus, Anna K., Leo S., Jacob C., Yana, Charlie, Jessica, Kate, and Graham. Thanks to everybody who made a guess!
Ten years later, Dr. Shapiro revived Puzzle of the Day at Proof School, writing each day's puzzle on a name tag. After 600 puzzles or so, he was just starting to feel normal about students reading his chest all the time when campus closed and the puzzle, like the rest of our lives, moved online. New puzzles are posted daily on school days.
Want to catch up on old PotDs? There's an archive currently containing puzzles from March to June 2020.