  Because of the formal success of his logical theory, Aristotle also considers most mathematical proofs as having the form of a universal affirmative syllogism, namely Barbara. (See the section on The Syllogistic in the entry on  Aristotle's logic.)  This means that most mathematical theorems are one thing A said of another C and that every mathematical demonstration has a middle term B which explains the connection between A and C. Aristotle provides several examples of such triads of terms in mathematics, e.g., two right angles-angles about a point-triangle, or right angle-half two right angles-angle in a semicircle. It has long been noted by commentators that mathematical proofs work with a particular case through universal instantiation (ekthesis) and then universalize to the general claim, and that not all propositions have the form: A is said of B, e.g., Elements 1 1, “To construct an equilateral triangle on a given line.” A more modern objection is that the formal theory of the syllogism as presented in Prior Analytics 1 1, 3-7 is woefully inadequate to express a theory involving conditionals and many-many relations, as is the case with all ancient mathematics. Nonetheless, Aristotle does think that most mathematical proofs actually do have this form. Those that wouldn't would certainly be negative propositions and possibly existential propositions. (We simply do not know enough about how Aristotle conceived of the logical form of existential propositions.) From a careful reading of the rest of the Prior Analytics, it becomes clear that Aristotle has a flexible notion of “one thing said of another” and that he regards standard mathematical proofs as really being in a universal form, which we express for purposes of comprehension as particular. 