How do the medieval names Barbara and Celarent describe syllogistic moods?

In medieval logic, names like Barbara and Celarent serve as mnemonic devices to classify the valid moods of a syllogism.12

These names use a sequence of three vowels to indicate the logical form of the major premise, the minor premise, and the conclusion, respectively. The four standard forms of categorical propositions are represented by the vowels A (universal affirmative), E (universal negative), I (particular affirmative), and O (particular negative).132

• Barbara: Contains three A-form propositions (AAA), signifying a universal affirmative major premise, a universal affirmative minor premise, and a universal affirmative conclusion.13
• Celarent: Contains the vowels E, A, and E (EAE), signifying a universal negative major premise, a universal affirmative minor premise, and a universal negative conclusion.13

Beyond identifying the structure, these mnemonic names encode instructions for "reducing" valid moods from the second, third, or fourth figures to the "axiomatic" first-figure moods, such as Barbara or Celarent, to demonstrate their logical validity.13