**Guo Shoujing** was a Chinese astronomer who worked on spherical trigonometry and the calendar.

- We know that at age 14 Guo did construct such a clock.
- By the age of sixteen Guo was studying mathematics.
- This then was the situation in the northern part of China where Guo was growing up.
- By the age of twenty Guo was working as an hydraulic engineer.
- This was a river in Guo's home province of Hebei and the bridge he renovated was a little way north of the town of Xinzhou.
- Zhang Wenqian, who was a friend of Guo and like him was a central government official, was sent by Kublai Khan in 1260 to Daming where unrest had been reported in the local population.
- Guo accompanied Zhang on his mission.
- Guo was not only interested in engineering, but he was also an expert astronomer.
- Zhang advised Kublai Khan that his friend Guo was a leading expert in hydraulic engineering.
- Kublai knew the importance of water management, for irrigation, transport of grain, and flood control, and he asked Guo to look at these aspects in the area between Dadu (now Beijing or Peking) and the Yellow River.
- To provide Dadu with a new supply of water, Guo found the Baifu spring in the Shenshan Mountain and had a 30 km channel built to bring the water to Dadu.
- This pleased Kublai Khan and led to Guo being asked to undertake similar projects in other parts of the country.
- Guo travelled extensively along with his friend Zhang taking notes of the work which needed to be done to unblock damaged parts of the system and to make improvements to its efficiency.
- Guo, together was Wang Xun, were asked to set up a special bureau to undertake the necessary research and to make proposals for the new calendar.
- An accurate calendar depends on the quality of the astronomical data used to support it and Guo's first move was to built seventeen new astronomical instruments so that accurate data could be collected.
- Let us briefly describe an improvement Guo made in constructing one such instrument.
- To make it easier to determine these points Guo fixed a crossbar to the top of a gnomon and used the principle of the pinhole camera to cast its shadow onto a measuring scale.
- Building began in March of that year and, following a design proposed by Guo, the work was completed in two months.
- Guo's friend Zhang Wenqian was appointed as director and Guo together with his colleague Wang Xun were the two co-directors.
- Making sense of the data gathered from the instruments required a knowledge of spherical trigonometry and Guo devised some remarkable formulae.
- The work was completed by 1280, Guo having calculated the length of the year correct to within 26 seconds, and in the following year Kublai Khan introduced the use of this extremely accurate calendar.
- Zhang Wenqian died in 1283 and Guo was promoted to be director of the Observatory in Beijing.
- As always he met with success and even after the death of Kublai Khan, although Guo was by this time an old man, his advice continued to be sought by Kublai's successor.
- We should now look at the rather remarkable work which Guo did on spherical trigonometry and solving equations.
- These formulae are approximate ones, but Guo was well aware of this.
- In the diagram ddd is the diameter of the circle, aaa is the length of the arc ABABAB and xxx is the length of NBNBNB which Guo wanted to calculate.
- To solve this equation Guo used a numerical method similar to Horner's method.
- The equation has two real roots, the smaller being the solution to the problem while the other, being numerically larger than the length of the arc, was rightly discarded by Guo.
- Guo looked at the accumulated difference, namely the difference in degrees moved by the sun in a day compared with the expected degrees moved if the motion was constant.

Born 1231, Xingtai, Hebei province, China. Died 1316, China.

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Ancient Chinese, Astronomy, Chinese, Origin China

Epochs: 1

**Oâ€™Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive