My logic book is making me memorize the mnemonic that lists all these Latin names in order to remember valid syllogism forms, as well as how to reduce them into the first figure. My question is, do I have to? Is there a way to know how to reduce a syllogism without using this list of names that are impossible to easily memorize?
The answer is, "if your teacher requires it."
Given how far logic has advanced, this assignment clearly falls in the realm of "history of logic." Even solving syllogisms themselves is largely a relic of earlier times. So you're unlikely to win this case by the argument that the rhyme is archaic and outdated, because the entire topic is.
Harry J. Gensler begins his chapter, "Syllogistic Logic", as follows:
Syllogistic logic studies arguments whose validity depends on "all," "no," "some," and similar notions. This branch of logic, which goes back to Aristotle, was the first to be developed. It provides a fine preliminary to modern symbolic logic, which we begin in the next chapter.
Gensler does not mention the mnemonic. He does mention his star test for validity, Venn diagrams as a second validity test and translating natural language arguments into syllogistic form. These topics would likely be more valuable than memorizing the mnemonic, but knowing that the names in the mnemonic exist and how they were formed adds historical interest.
One can also view names such as Barbara as AAA or AaB, BaC, AaC.
Gensler, H. J. (2012). Introduction to logic. Routledge.