The Academy motto was “Ex Astris, Scientia“, meaning “from the stars, knowledge” in Latin. The ancient Romans usually did not use verbs in inscriptions, so it would be better translated as “knowledge comes from the stars”. According to Michael Okuda, the motto was inspired by a quotation on the Apollo 13 mission patch, “Ex luna, scientia“. The Apollo 13 quote was itself a paraphrase of the motto of the United States Naval Academy, “Ex Scientia Tridens,” which means “from knowledge, seapower.”
The initially shown version “Ex Astra, Scientia” was grammatically wrong, as the preposition “ex” requires the ablative case “Astris“. According to Michael Okuda, six months after “The First Duty” aired he received a letter from a Latin Professor at Brown University pointing this out. This error was corrected for later appearances and also in the remastered Blu-ray edition of TNG.
There have been three versions of the Academy logo. The first version being the original logo seen in “The First Duty”, with the Latin grammar error. The second was seen only briefly behind Harry Kim’s interrogation in the alternate timeline in “Non Sequitur”. This logo had the updated Latin phrase and the rays of light had been inverted when compared to the original, with the outer border changing to a shade of pink rather than a red. The third logo debuted shortly thereafter with an updated Starfleet insignia and the border returning to a shade of red.memory-alpha.fandom.com