Minggatu

Minggatu (* around 1692 in Xilin Gol ; † 1764 ) was a Mongolian mathematician, cartographer and astronomer at the court of the Qing Dynasty , who was one of the first in China to investigate infinite series.

It is also spelled Myangat or Ming Antu. Its Chinese name suffix was Jing An.

Minggatu came from the Mongol tribe of the Sharaid. He first appears in official records in 1713 during the reign of Emperor Kangxi as a state scholarship holder at the Imperial Office for Astronomy (quintianjian). He helped with the publication of astronomy books (in particular the compendium of astronomy Lixiang kaocheng, published in 1742) and in surveying (for example in 1756 in the newly conquered areas in the west, the Xinjiang ). From 1724 to 1759 he worked at the imperial observatory. In 1759 he became director of the Astronomy Office.

At the time, Jesuit missionaries were working as astronomers in China, who were also responsible for calendar reform. The Jesuit Pierre Jartoux (1669–1720, called Du Demei by the Chinese) came to China in 1701 and introduced the Chinese to infinite ranks, including those of Isaac Newton and James Gregory such as

{\ displaystyle \ pi = 3 \ left (1 + {\ frac {1} {4 \ cdot 3!}} + {\ frac {3 ^ {2}} {4 ^ {2} \ cdot 5!}} + {\ frac {3 ^ {2} \ cdot 5 ^ {2}} {4 ^ {3} \ cdot 7!}} + \ dotsb \ right)}

{\ displaystyle \ sin x = x - {\ frac {x ^ {3}} {3!}} + {\ frac {x ^ {5}} {5!}} - {\ frac {x ^ {7} } {7!}} + \ Dotsb}

{\ displaystyle 1- \ cos x = {\ frac {x ^ {2}} {2!}} - {\ frac {x ^ {4}} {4!}} + {\ frac {x ^ {6} } {6!}} + \ Dotsb.}

This impressed the Chinese because, for example, in the formula for pi, no rooting was necessary as in Liu Hui's known method . But Jartoux did not pass on the methods of analysis that were behind it, and Minggatu set out to find a method to derive them himself. He started from regular polygons of ever higher order that divide the circle and derived recursion relations. Most of his knowledge came from European sources through the mediation of the Jesuits. He is said to have worked on his work for more than thirty years.

He wrote down his results in his major work A Quick Method to Get the Exact Division Ratio of the Circle (Ge Yuan Mi Lu Jie Fa). The manuscript was unfinished at his death and was completed by his student Chen Jixin, but did not appear in China until 1839 (although it was previously in circulation with various Chinese mathematicians). In it he found over ten new infinite series in China that were not previously known from European sources, developed methods for manipulating infinite series and was the first to discover Catalan numbers in the 1730s: ${\ displaystyle C_ {n}}$

{\ displaystyle \ sin (2x) = 2 \ sin x- \ sum _ {n = 1} ^ {\ infty} C_ {n} {\ frac {{(\ sin x)} ^ {2n + 1}} { 4 ^ {n-1}}}}

Catalan figures at the Minggatu plant

Like other Chinese mathematicians of the 18th century and well into the 19th century, he did not know the methods of analysis, but used various ad hoc methods from algebra, geometry and trigonometry.

literature

Jean-Claude Martzloff: A history of chinese mathematics. Springer, 2006, p. 357.
Jianjin Luo: A Modern Chinese Translation of Ming Antu's Geyuan Milv Jifa, translated and annotated by Luo Jianjin, Inner Mongolia Education Press 1998
Jianjin Luo: Ming Antu and His Power Series Expansions, in: Knobloch u. a., Seki, Founder of Modern Mathematics in Japan, Springer 2013

Web links

Luo Jianjin: Ming'antu and His Power Series Expansions. (Abstract, PDF).

Individual evidence

^ Date of death according to Martzloff: History of chinese mathematics. Springer, 2006.
↑ The first publication in China on infinite series also referred to Jartoux and was in a book by Mei Juecheng (1681–1763) published in 1759, in which he mentions the three series of Jartoux. No reference has been made to the original Chinese manuscript by Jartoux. It is therefore not known exactly what Jartoux passed on and whether he did not give any hints about the methods behind it. There were also many mathematical works in the library of the Jesuits in Beijing (Beitang library with around 4100 books), but as far as is known, no Chinese mathematician of the time spoke the languages in which the books were written. Martzloff, p. 355 f.

personal data
SURNAME	Minggatu
BRIEF DESCRIPTION	Mongolian mathematician
DATE OF BIRTH	around 1692
PLACE OF BIRTH	Xilin Gol
DATE OF DEATH	1764

[1] Date of death according to Martzloff: History of chinese mathematics. Springer, 2006.

[2] The first publication in China on infinite series also referred to Jartoux and was in a book by Mei Juecheng (1681–1763) published in 1759, in which he mentions the three series of Jartoux. No reference has been made to the original Chinese manuscript by Jartoux. It is therefore not known exactly what Jartoux passed on and whether he did not give any hints about the methods behind it. There were also many mathematical works in the library of the Jesuits in Beijing (Beitang library with around 4100 books), but as far as is known, no Chinese mathematician of the time spoke the languages in which the books were written. Martzloff, p. 355 f.