'Trace' is Latin for 'a repeated drawing' — from 'trahere' (to pull). Drawing a line and following one.
A mark, indication, or evidence of the former presence or passage of something (noun); to follow the course or trail of; to copy by drawing over lines visible through translucent paper (verb).
From Old French tracier (to trace, to track, to draw a path), from Vulgar Latin *tractiare (to drag, to draw a line), from Latin trahere (to drag, to pull, to draw), from PIE *tragh- (to drag, to draw along a surface). The root *tragh- meant the physical act of dragging something across a surface, which leaves a line behind it — a trace. This gives the primary senses of the English
The phrase 'without a trace' literally means 'without a line drawn' — no mark left behind to follow. The word 'trace' descends from Latin 'trahere' (to draw) through a Vulgar Latin form meaning 'to draw a line,' so a trace is fundamentally a line that was drawn by someone or something passing through.