pag

The root pag comes from Latin pangere, "to fasten, fix in place, drive in" — think of hammering a stake into the ground so it stays put. From that single image of fixing something firmly, two very different word families grew.

The first family is about a fixed surface. Latin pāgina originally meant a row of vines fastened to a trellis, then a column of writing fixed onto a sheet — and finally just page, one side of a leaf in a book. The connection isn't obvious until you remember that early text was laid out in fixed columns, like vines trained on a frame.

The second family is about driving things forward and outward. Add pro- (forward) to the 'fasten/plant' sense and you get propāgāre, "to fix slips of a plant forward into new soil" — that is, to multiply a plant by cuttings. English kept this literal gardening meaning in propagate (you can still propagate a houseplant from cuttings), and then extended it to ideas: to propagate a rumor is to plant it in new minds so it spreads. Propagation is the spreading itself — of plants, of ideas, even of waves and signals through space. And propaganda is, literally, "things that must be propagated": the word began as the name of a Catholic committee for propagating the faith (Congregatio de Propaganda Fide) before it took on its modern, loaded political sense.

A more violent cousin keeps the raw 'drive in / strike against' meaning. Add in- (against) to pangere and you get impingere, "to drive against, strike into" — English impinge, to hit up against or encroach on something (your rights, your free time).

So from one act — fixing something firmly — the family splits: fix it flat and you get a page; drive it forward and you get propagate; strike against and you get impinge.