Squaring of the Circle

...[produces] a straight line equal to the line drawn in a circle, just as appears in the following figures.

A horizontal line segment labeled A at the left and F at the right. It is divided into segments by vertical tick marks. Points B, C, and D are marked above the line. Above the segment A-B is the label "diameter.1.3a.p8." and above the segment C-D is the label "22a.p8."

¶ On the proportion of the circle to the diameter according to Ptolemy and Archimedes.

A circle with a vertical diameter drawn through it. The diameter is marked with seven equidistant horizontal tick marks, illustrating the ratio of the diameter (7 parts) to the circumference (22 parts).

Ptolemy, the prince of astronomers, in the sixth book of his Mathematical Composition original: "mathematicæ constitutionis"; better known by its Arabic title, the Almagest. demonstrated that a circle has a proportion to its own diameter which is 3 plus 8/60 and 30/3600 to 1. For 3, 8, and 34 parts The author is using sexagesimal (base-60) fractions, which was the standard for astronomy. In modern decimals, Ptolemy's value is approximately 3.1416. to one is nearly triple plus one-seventh. Similarly, 3, 8, and 27 to one is triple with the addition of ten-seventy-firsts, between which lies the proportion of 3, 8, and 30 to one.

Archimedes of Syracuse, however—as [Giorgio] Valla An Italian humanist whose posthumous work De expetendis et fugiendis rebus (1501) contained many translations of Greek mathematics. says, and as will be clear in his third book On the Squaring of the Circle—strove to demonstrate through helical and curved lines that a circle has a proportion to its diameter less than triple plus one-seventh 3 and 1/7, or 22/7 but greater than triple plus ten-seventy-firsts 3 and 10/71. This is more than three times the diameter by 10 parts out of 71. Or, you might say, less than 22 to 7, and greater than 10 parts of 71 units. He also intended for the circular line to have a common proportion of 11 to 14 compared to the square that is formed from the diameter. This refers to the ratio between the area of a circle and the area of its circumscribed square.

¶ To construct a circle as well as a square from a given straight line.