marginalia The word marginalia comes from the Latin margo (margin). Scholars filled manuscript margins with commentary, corrections, and doodles — a parallel conversation alongside the main text. This is that.A note on curvature. Let 𝛾:[0,1]→ℝ2 be a smooth planar curve. The signed curvature is:𝜅=𝑥′𝑦″−𝑦′𝑥″(𝑥′2+𝑦′2)32For a circle of radius 𝑟 — parameterized as 𝛾(𝑡)=(𝑟cos𝑡,𝑟sin𝑡) — direct computation gives the constant 𝜅=1/𝑟. Smaller circle, sharper turn.A circle of radius 𝑟 has constant curvature 𝜅=1/𝑟.Taylor’s theorem gives a local polynomial approximation of any smooth 𝑓:ℝ→ℝ around 𝑥0:𝑓(𝑥)=∑𝑁𝑛=0𝑓(𝑛)(𝑥0)𝑛!(𝑥−𝑥0)𝑛+𝑅𝑁(𝑥)where 𝑅𝑁(𝑥)=𝑂((𝑥−𝑥0)𝑁+1) as 𝑥→𝑥0. The curvature formula above is a geometric instance of this: the osculating circle is the second-order fit to the curve at any point.