{"id":376083,"date":"2026-09-09T09:16:04","date_gmt":"2026-09-09T09:16:04","guid":{"rendered":"https:\/\/wolfscientific.com\/?p=376083"},"modified":"2026-09-09T09:16:04","modified_gmt":"2026-09-09T09:16:04","slug":"the-extensive-chronicle-of-calculus-expanding-horizons-in-calculus-iv","status":"publish","type":"post","link":"https:\/\/wolfscientific.com\/?p=376083","title":{"rendered":"“The Extensive Chronicle of Calculus: Expanding Horizons in Calculus IV”"},"content":{"rendered":"

Push the Limits Yet Again: A Journey Through Calculus IV<\/p>\n

Today, I\u2019m shifting gears a bit and discussing a topic that didn\u2019t actually influence the evolution of calculus. There is a common perception that innovations are attributed to individual discoverers and that discoveries happen just once. However, there are notable instances in mathematical and scientific history where multiple individuals uncovered the same principles independently, such as Darwin and Wallace regarding the theory of evolution via natural selection. There was a time when these parallel discoveries were deemed uncommon; nevertheless, as the history of science has evolved, it’s become evident that such occurrences are indeed quite frequent. In earlier writings, I have examined the method of exhaustion, tracing its application and progression in Greek geometry starting from Antiphon, and including Bryson of Heraclea, Eudoxus, and Archimedes. This mathematical technique was also wholly developed independently in China and applied in a remarkably similar fashion.<\/p>\n

Similar to many ancient societies, the Chinese originally approximated pi as 3. The earliest known effort to determine a more accurate value was by the astronomer Liu Xin (c. 46 BCE\u201323 CE) sometime between 1 BCE and 5 CE. He crafted a measuring vessel that suggested a pi value of 3.1547, though the method he used to derive this value remains unknown. In 130 CE, the polymath Zhang Hen made two calculations of pi. The first was ascertained by relating the celestial sphere to the Earth’s diameter, assigning values of 736 and 232 to the two respectively, which resulted in a pi estimate of 3.1724. His second calculation aimed to refine the ratio between the area of a square and that of the inscribed circle, leading him to a pi approximation of the square root of 10, around 3.162. The astronomer and mathematician Wang Fan (228\u2013266 CE) utilized a pi value of 142\/45 = 3.155.<\/p>\n

The individual who enhanced the precision of the Chinese calculation of pi was the mathematician Liu Hui (fl. 3rd<\/sup> century CE).<\/p>\n

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<\/a>
Liu Hui Source: Wikimedia Commons<\/figcaption><\/figure>\n<\/div>\n

The Nine Chapters on the Mathematical Art<\/em> is a Chinese mathematical anthology composed between the 10th<\/sup> and 2nd<\/sup> centuries BCE, reaching its final version in the 1st<\/sup> century CE. In the 3rd<\/sup> century CE, Liu Hui penned a comprehensive commentary on The Nine Chapters<\/em>, which included his method for calculating the value of pi, a variant of the method of exhaustion, referred to as Liu Hui\u2019s \u03c0 algorithm.<\/p>\n

<\/a>
A page from The Nine Chapters on the Mathematical Art<\/em> 1820 ed. Source: Wikimedia Commons<\/figcaption><\/figure>\n

Liu Hui outlined a step-by-step iterative algorithm to compute \u03c0<\/em> to the desired level of accuracy through the process of bisecting polygons:<\/p>\n

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Liu Hui contended:<\/p>\n

\u201cMultiply one side of a hexagon by the radius (of its circumcircle), then multiply this by three, yielding the area of a dodecagon; if we divide a hexagon into a dodecagon, multiply its side by its radius, then multiply by six, we obtain the area of a 24-gon; the finer we divide, the lesser the discrepancy in comparison to the area of the circle, hence with continued divisions, the area of the resultant polygon will converge and merge with the circle; there will be no loss.\u201d<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"

Push the Limits Yet Again: A Journey Through Calculus IV Today, I\u2019m shifting gears a bit and discussing a topic that didn\u2019t actually influence the evolution of calculus. There is a common perception that innovations are attributed to individual discoverers and that discoveries happen just once. However, there are notable instances in mathematical and scientific […]<\/p>\n","protected":false},"author":1,"featured_media":376084,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"Default","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[175],"class_list":["post-376083","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-uncategorized","tag-source-thonyc-wordpress-com"],"_links":{"self":[{"href":"https:\/\/wolfscientific.com\/index.php?rest_route=\/wp\/v2\/posts\/376083","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wolfscientific.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wolfscientific.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wolfscientific.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/wolfscientific.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=376083"}],"version-history":[{"count":0,"href":"https:\/\/wolfscientific.com\/index.php?rest_route=\/wp\/v2\/posts\/376083\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/wolfscientific.com\/index.php?rest_route=\/wp\/v2\/media\/376084"}],"wp:attachment":[{"href":"https:\/\/wolfscientific.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=376083"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wolfscientific.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=376083"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wolfscientific.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=376083"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}