-
2. The Mysterious Greek Letter Pi
- Baylor University Press
- Chapter
- Additional Information
15 THE MYSTERIOUS GREEK LETTER π II It is well known that the Greek letter π (pi) is the symbol indicating the ratio between the circumference and the diameter or radius of a circle.1 Its numerical value therefore solves the problem of the rectification of the circumference and the corresponding measurement of the squaring of a circle. This link between a straight line and a curved line, in a corresponding manner between a polygon and a circle, is already stated in the first of only three theorems demonstrated in Archimedes’ Measurement of a Circle: The area of any circle is equal to a right triangle that has a right-angle side equal to the radius and the other right-angle side equal to the circumference of the circle. [see fig. 4] The measurement of the lines corresponding to the curve of the circle in respect to its diameter or radius profoundly intrigued both ancient and modern mathematicians, and the ancients tried in various ways to move this complex matter 16 THE GREAT ARCHIMEDES II Figure 4 The beginning of Measurement of a Circle in a copy of the Latin translation of Jacopo of Cremona. The translation was rendered in approximately 1450 on the order of Pope Nicholas V and was used by, among others, Piero della Francesca and Leonardo da Vinci (Venice, Biblioteca Marciana, no. 327, f. 106v). [54.224.52.210] Project MUSE (2024-03-19 13:36 GMT) THE MYSTERIOUS GREEK LETTER Π 17 II toward a satisfying, if somewhat mysterious, agreement. The Bible suggests that the relationship between the radius and half of a circle’s circumference would be equal to three,2 and one finds that many in antiquity were willing to settle for a generic, inaccurate measurement. Among these was the Latin poet Marcus Manilius (fl. at the beginning of the first century AD), who dedicates three verses of his Astronomica to the topic: Wherever a circle is cut through the center, a third of its circumference is then created by dividing the full sum of the circumference by the slight difference. [1.545–47] Some ancient mathematicians tried to achieve the desired results by easily identifying curved lines with those broken into a large number of sides, and curved surfaces with a superficial polyhedron having many faces. But the transition from the broken line to the curved line or from the polyhedral surface to the true curve implies the leap from the finite to the infinite (only when the number of sides of the broken line becomes infinite can it be argued that it comes to be identified with the arc on which it is inscribed). As Geoffrey E. R. Lloyd has explained, Archimedes’ general approach is twofold: He uses both inscribed and circumscribed figures with a view to compressing them to a point where they coalesce with the curved figure to be measured.3 Specifically, Archimedes assumes, in his less-than-perfect calculation of π, the perimeter of a regular polygon inscribed within the circumference, and as the excess value, that of a regular polygon circumscribed around it. He separated the number of sides from the hexagons and then, one after another, doubled that number, considering the inscribed and circumscribed polygons on the circle respectively at 12, 24, 48, and 96 sides, recognizing that in this way he would 18 THE GREAT ARCHIMEDES II still be able to fix the progression. For Archimedes, the higher and lower limits between which π must be measured are respectively 3 1 /7 and 3 10 /71 : The circumference of each circle is triple the diameter and less than one-seventh of the diameter, and yet greater than ten seventy-firsts. [Measurement of a Circle, proposition 3] Therefore, using decimal numbers, he is proposing a measurement just between 3.14084507 and 3.142857142, a figure remarkably close to the value we give π today. Archimedes was the first Greek scientist to make use of fractions in the field of mathematics, since prior to Archimedes the Platonic concept of the indivisible unit held sway. Simply put, before Archimedes, fractions were not considered from a scientific point of view and fell only to merchants and their commerce. Archimedes’ short but dense work entitled Size of the Circle is without doubt one of antiquity’s most exciting scientific texts. Not without reason, the mathematician Gaetano Fichera has defined it as “the most representative of his works.”4 In reality, the circumference and the diameter of the circle have proportions equally difficult to measure, and their...