Triple prime

Third in a series, most often the third derivative.

In calculus, primes count derivatives: f′(x) is the first derivative, f″(x) the second and f‴(x) the third. You read it “f triple prime of x”. This is Lagrange's notation, and it rarely goes past three: the fourth derivative is usually written f⁽⁴⁾(x) instead.

In geometry and physics, primes label related versions of the same thing. If a point A is moved, reflected or rotated three times in turn, its images are often called A′, A″ and A‴. The marks carry no number of their own here; they just say “the third one after the original”.

Why is it also linked to angles and watches?

It is the third step of the old sixty-based angle ladder.

Angles are split by sixties: a degree ° has 60 arcminutes ′, an arcminute has 60 arcseconds ″. Older astronomical tables went one step further and wrote ‴ for a “third”, 1/60 of an arcsecond. Modern writing uses decimal seconds instead, such as 12.5″.

Watchmakers still use the mark for the ligne, a traditional French unit of about 2.256 mm, to state movement diameters: a calibre described as 11½‴ is about 26 mm across.