Take it to the edge one more time – a chronicle of calculus IV
Today, I’m going to slightly shift from the course and discuss something that didn’t actually influence the evolution of calculus. There exists a propensity to highlight individual figures as the sole discoverers of concepts and to imply that ideas are discovered once. Nevertheless, there are notable instances in the annals of mathematics and science where two or more individuals arrived at the same conclusion independently, such as Darwin and Wallace with the theory of evolution via natural selection. In earlier times, such simultaneous discoveries were deemed uncommon, but as the narrative of science has unfolded, it has become apparent that these occurrences are quite prevalent. In past entries, I have explored the method of exhaustion and traced its application and evolution in Greek geometry from Antiphon through Bryson of Heraclea, Eudoxus, and Archimedes. This mathematical technique was also independently formulated in China and utilized in a remarkably similar manner.
Like various ancient civilizations, the Chinese initially approximated pi as 3. The earliest recorded endeavor for a more accurate value was undertaken by the astronomer Liu Xin (c. 46 BCE–23 CE) at some point between 1 BCE and 5 CE. He devised a measuring vessel that suggested a pi value of 3.1547, though the specifics of his calculation remain unknown. In 130 CE, the polymath Zhang Hen arrived at two calculations of pi. The first was deduced by relating the celestial sphere to the Earth’s diameter, assigning a value of 736 to the former and 232 to the latter, thus deriving pi as 3.1724. His second determination stemmed from his efforts to enhance the ratio between the area of a square and that of the inscribed circle. His computations resulted in a pi value of the square root of 10, approximately 3.162. The astronomer and mathematician Wang Fan (228–266 CE) utilized a pi value of 142/45 = 3.155.
The mathematician Liu Hui (fl. 3rd century CE) is credited with introducing precise calculations for the Chinese value of pi.
The Nine Chapters on the Mathematical Art is a Chinese mathematical anthology compiled between the 10th and 2nd centuries BCE and finalized in the 1st century CE. In the 3rd century CE, Liu Hui penned a detailed commentary on The Nine Chapters, which includes his technique for calculating pi, a variant of the method of exhaustion termed Liu Hui’s π algorithm.
Liu Hui presented a stepwise iterative algorithm to ascertain π to any desired accuracy through polygon bisection:
Liu Hui posited:
“Multiply one edge of a hexagon by the radius (of its circumcircle), then multiply this by three, to yield the area of a dodecagon; if we transform a hexagon into a dodecagon, multiply its edge by its radius, then multiply by six again, forming the area of a 24-gon; the finer the subdivisions, the lesser the discrepancy regarding the area of the circle, hence with continued subdivision, the area of the resulting polygon will align and unify with the circle; there will be no loss.”
Additionally, Liu Hui demonstrated that the area of a circle equals half its circumference times its radius. He stated:
“Between a polygon and a circle, there is an excess radius. Multiply the excess radius by a polygon side. The resulting area surpasses the circle’s boundary.”
In the illustration, d = excess radius. Multiplying d by one side produces oblong ABCD which extends beyond the circle’s boundary. If a polygon’s side is diminutive (i.e., there are numerous sides), the excess radius will be minimal, consequently leading to a small excess area.
As depicted in the illustration, when N → ∞, d → 0, and ABCD → 0.
“Multiply the polygon’s side by its radius, and the area doubles; thus, multiply half the circumference by the radius for the circle’s area.”
As N → ∞, half the circumference of the N-gon approximates a semicircle, therefore half the circumference of a circle multiplied by its radius results in the circle’s area. Liu Hui did not elaborate on this reasoning extensively; nonetheless, it is evident through Liu Hui’s “in-out complement principle” which he detailed elsewhere in The Nine Chapters on the Mathematical Art: Divide a geometric figure into segments, rearrange them to form a different shape; the areas of both shapes will be equal.
Thus, by reorganizing the six green triangles, three blue triangles, and three red triangles into a rectangle with width = 3L, and height R, it demonstrates that the area of the dodecagon =