Take it to the Limit One More Time – A History of Calculus IV
Today, I will slightly diverge from the syllabus and discuss a topic that did not have a direct influence on the evolution of calculus. There is a common inclination to focus on individual figures making discoveries as if they occurred in isolation. Nonetheless, there are notable instances in the annals of mathematics and science where two or even multiple individuals arrived at the same conclusion independently, such as Darwin and Wallace regarding the theory of evolution through natural selection. In earlier times, these simultaneous discoveries were deemed unusual; however, as the story of science unfolded, it has become evident that such occurrences are quite prevalent. In my previous writings, I have explored the method of exhaustion, tracing its application and advancement in Greek geometry from Antiphon, through Bryson of Heraclea, Eudoxus, and Archimedes. Remarkably, this mathematical procedure was also independently formulated in China and employed in a nearly identical manner.
Like many ancient civilizations, the Chinese initially estimated 3 as a value for pi. The earliest known effort to establish a precise value was made by the astronomer Liu Xin (c. 46 BCE–23 CE) between the years 1 BCE and 5 CE. He constructed a measuring vessel that suggested a pi value of 3.1547, although the methodology he used to achieve this figure is unknown. In 130 CE, the polymath Zhang Hen made two assessments of pi. The first was calculated by comparing the celestial sphere to the Earth’s diameter, attributing a value of 736 to the former and 232 to the latter, thereby deriving pi as 3.1724. His second estimate was made while attempting to refine the ratio of the area of a square to that of the inscribed circle, leading to a value of pi equivalent to the square root of 10, roughly 3.162. The astronomer and mathematician Wang Fan (228–266 CE) applied a pi value of 142/45 = 3.155.
The contributor who clarified the Chinese computation of pi was the mathematician Liu Hui (fl. 3rd century CE).
The Nine Chapters on the Mathematical Art is a Chinese mathematical collection compiled between the 10th and 2nd centuries BCE, reaching its definitive version in the 1st century CE. In the 3rd century CE, Liu Hui authored a comprehensive commentary on The Nine Chapters, which includes his technique for determining pi, a variation of the method of exhaustion referred to as Liu Hui’s π algorithm.
Liu Hui outlined a step-by-step iterative algorithm to compute π to any desired level of precision via the bisection of polygons:
Liu Hui stated:
“Multiply one side of a hexagon by the radius (of its circumcircle), then multiply this by three, resulting in the area of a dodecagon; if we divide a hexagon into a dodecagon, multiply its side by its radius, then again multiply by six, we derive the area of a 24-gon; the more divisions we make, the less the discrepancy regarding the area of a circle, thus with further subdivisions, the area of the created polygon will coincide and merge with the circle; there will be no loss.”
In addition, Liu Hui demonstrated that the area of a circle is half of its circumference multiplied by its radius. He asserted:
“Between a polygon and a circle, there exists an excess radius. Multiply the excess radius by a side of the polygon. The resulting area surpasses the limit of the circle.”
In the illustration, d = excess radius. Multiplying d by one side yields oblong ABCD, which exceeds the boundary of the circle. If a side of the polygon is minimal (i.e., there is an exceedingly large number of sides), then the excess radius will be minimal, and thus the excess area will also be minimal.
As indicated in the diagram, when N → ∞, d → 0, and ABCD → 0.
“Multiply the side of a polygon by its radius, and the area is doubled; therefore, multiply half the circumference by the radius to produce the area of the circle.”
As N → ∞, half the circumference of the N-gon approaches a semicircle, thus half a circumference of a circle multiplied by its radius equals the area of the circle. Liu Hui did not elaborate extensively on this reasoning. However, it is apparent by utilizing Liu Hui’s “in-out complement principle” provided elsewhere in The Nine Chapters on the Mathematical Art: Divide a geometric figure into components, rearrange the components to create another shape, and the areas of both shapes will remain equal.
Thus, rearranging the six green triangles, three blue triangles, and three red triangles into a rectangle with width = 3L, and height R illustrates that the area of the d