Squaring of the Circle

...five parts and a half in three half-parts, that is, in three halved parts. Therefore, from the preceding fifth proposition (the major premise) and the following sixth minor premise, the final conclusion follows in the first mode of the first figure A term from formal logic referring to a "Barbara" syllogism, the most direct form of deductive reasoning., namely: that every circle is equal to a square whose side is exceeded by the diameter of that same circle precisely by three half-parts.

A singular proof of this matter may be made in this way: let a circle of any size be established, and let its diameter be divided into seven equal parts according to the teaching handed down in the third conclusion. Then, let an equilateral square equilateral square quadratum equilaterum be established through the art of the fourth conclusion; the side of this square should precisely contain five and a half parts of the aforementioned diameter. Thus, it is clear that if the premises are diligently inspected, a circle of this kind will be equal to this square—just as a circle of such a size relates to a square of such a size.

Addition.

decorative woodcut initial 'H' showing figures or patterns Here begins the declaration of this sixth and final minor proposition. Our Campanus Campanus of Novara, the 13th-century mathematician famous for his edition of Euclid. says this is contained in the fourth conclusion. For if, as was said above and according to what many of the finest mathematicians have written, a circle's circumference is divided into twenty-two parts, and if only one part is removed from these, the third part of the remainder—namely seven—will result in the diameter of the circle. This refers to the Archimedean approximation of Pi as 22/7. If the circumference is 22, the diameter is 7.

But it must be noted that one quarter of the circle, as was also said above, contains only five parts and a half. 22 divided by 4 equals 5.5. In that case, the aforementioned diameter of the circle—namely, seven—will exceed the quarter of the circle (the five parts and a half) precisely by three half-parts, that is, by three halves. 7 minus 5.5 equals 1.5, or three-halves. Therefore, from the preceding fifth proposition...