Since Lukasiewicz (1951) and Patzig (1969) reconstructed the syllogism as formal implication within an axiomatic system, thereby replacing the prevailing non-formal interpretations of the 19th century, there have been numerous attempts to represent the direct, indirect, and ekthetic proofs of the syllogism within a calculus of natural deduction. Both approaches are examined in the first part of the book for logical consistency and philological proximity, incorporating ancient commentaries. A method developed here determines validity and invalidity while also providing the possible line proofs of a valid conclusion form. The second part offers a detailed commentary on the non-modal syllogistics of the first seven chapters of the Analytica priora.