Why can A propositions be converted only per accidens?

In Aristotle's syllogistic logic, to convert a proposition is to interchange its subject and predicate. While some forms can be converted simply (simpliciter), an A proposition (universal affirmative, "Every $\beta$ is an $\alpha$") cannot be converted this way because it does not follow that every $\alpha$ is a $\beta$.

An A proposition can only be converted per accidens (accidentally) by weakening the universal quantifier to a particular quantifier. This results in the valid inference that "some $\alpha$ is a $\beta$." This process is necessary because the original universal claim does not guarantee that the entire scope of the predicate is contained within the subject, only that the subject is contained within the predicate.1