Note that in a categorical syllogism each premise and the conclusion have to be in the form of a general categorical statement, of which there are four kinds (A, E, I, O).
Since each figure (each of the columns above is called a "figure") has two premises and a conclusion, each figure has 43 = 64 possible syllogistic forms. With four figures, that means 64 x 4 = 256 possible syllogistic forms.
By following certain syllogistic rules, we can rule out most of the syllogistic forms as invalid, resulting in 24 valid syllogistic forms (in the traditional interpretation -- in the modern interpretation, this is further pared down to 15). They are:
First Second Third Fourth
figure Figure Figure Figure
AAA AEE AAI AAI
AII AOO AII AEE
EAE EAE EAO EAO
EIO EIO EIO EIO
AAI* AEO* IAI IAI
EAO* EAO* OAO AEO*
The forms with the asterisk are called weakened forms, since a stronger, more universal conclusion can be obtained. For example, the AAI form in the first figure:
All dogs are mammals
All poodles are dogs
therefore Some poodles are mammals
The conclusion, while valid (again, in the traditional interpretation -- it is invalid in the modern interpretation), can be strengthened to say "All poodles are mammals" (AAA).
Further, all valid forms can be reduced to forms in the first figure, by applying immediate inference rules. Because of the difficult task of remembering all these rules, medieval scholars devised a clever mnemonic device that started with Barbara Celarent... to remember all these.
It was much later that diagrams were used to represent these rules. Euler was the first to use diagrams. Lewis Carrol had his own diagrammatic representation, but it was the Venn Diagram that came into popular use.
Harlan Miller, course notes, Philo 5505