In other words, we do not at first know any relation between 'Cicero' and 'vanity'; but we know that these two terms are severally related to a third term, 'author,' hence called a Middle Term; and thus we perceive, by mediate evidence, that they are related to one another.

This sort of proof bears an obvious resemblance (though the relations involved are not the same) to the mathematical proof of equality between two quantities, that cannot be directly compared, by showing the equality of each of them to some third quantity: A = B = C .'.

A = C.Here B is a middle term.
We have to inquire, then, what conditions must be satisfied in order that a Syllogism may be formally conclusive or valid.

A specious Syllogism that is not really valid is called a Parasyllogism.
Sec.2.General Canons of the Syllogism.
(1) A Syllogism contains three, and no more, distinct propositions.
(2) A Syllogism contains three, and no more, distinct univocal terms.
These two Canons imply one another.

Three propositions with less than three terms can only be connected in some of the modes of Immediate Inference.