does dividere mean to “vide” in two, such that we (should) have trividere &c?
No, di- simply gives the notion of separation, as in diripio, differo, dilacero, dilabor, which are “I loot, I scatter, I tear to shreds, I fall apart” respectively. It is clearly shown that this word has no notion of two in Caesar’s De Bello Gallico 1.1.1: Gallia est omnis divisa in partes tres, “Gaul is as a whole divided into three parts.”
Cur non et anglicè et latinè trividere pro dividere in partes tres? Quod superfluum est id dictum? Divisio est separatio quod per separationem duarum partium continuò explicatur. Eâ ratione, e viduare non videre verbo modo incerto conceptum esse postulo.
Why not also trivide in English? It’s superfluous, no doubt. Division is a separation that happens typically of one piece from another (therein a sense of two), and one cut after another. I could see the word more as deriving in some way from viduare before videre.
Salve, Sceptra Tenens. Jam inceperam cum epistulam tuam animadverti, exinde “duarum partium sensus” inter ancones addidi.