Zu Chongzhi

Zu Chongzhi the Chinese mathematician and astronomer of the Liu Song and Southern Qi periods (429-500) who calculated the value of pi to remarkable accuracy and compiled a calendar of great precision. He is remembered as one of the most distinguished scientists of ancient China.

Early Life and Learning

Zu Chongzhi was born in 429, during the period of division between the northern and southern dynasties, into a family with a tradition of scholarship in astronomy and mathematics. His father and grandfather had served as officials and had worked on calendrical problems, and the boy grew up in an environment of scientific study. The family's background gave him an early grounding in the exact sciences.

From his youth he showed exceptional aptitude for calculation and observation. He studied the classical mathematical texts, including the Nine Chapters on the Mathematical Art, and he mastered the astronomical and calendrical learning of his time. His contemporaries noted his precision and his refusal to accept traditional results without proof.

He entered government service and held a series of posts at the southern court, where his learning brought him to the attention of the emperor. He was employed in the bureau charged with astronomy and the calendar, the field in which he would make his greatest contributions, and he was also given administrative responsibilities in the provinces.

The Value of Pi

Zu Chongzhi is most famous for his determination of the value of pi, the ratio of a circle's circumference to its diameter. He showed that the value lies between 3.1415926 and 3.1415927, a result correct to seven decimal places, and he proposed the simple approximation of 355 divided by 113, known in Chinese as the close ratio, as well as the rougher value of 22 divided by 7.

The accuracy of this determination was extraordinary. It was not surpassed in Europe for nearly a thousand years, and it remained the most precise value known in China for centuries. His result was achieved before the development of the infinite series and other techniques that later made such calculations routine, and it speaks to the sophistication of ancient Chinese mathematics.

The method by which he reached the result is not fully preserved, but it is thought to have involved the inscription of polygons with ever greater numbers of sides within a circle, a procedure already used by earlier Chinese mathematicians and culminating in a figure with thousands of sides. His work on pi was recorded in a mathematical text, the Zhui Shu, or method of interpolation, which was later lost.

The Daming Calendar

Zu Chongzhi's chief work in astronomy was the Daming calendar, which he submitted to the throne in 462. The calendar introduced a number of improvements, including a more accurate reckoning of the length of the tropical year and of the motion of the moon, and it took account of the precession of the equinoxes, the slow shifting of the point where the sun crosses the celestial equator.

He determined the length of the tropical year as about 365.2428 days, a figure very close to the modern value, and he calculated the period of the moon's revolution with similar precision. He also observed and predicted eclipses, and he used his calculations to test the accuracy of older calendars, exposing their accumulated errors.

The new calendar was not adopted at once. Its introduction was resisted by conservative officials, and it was only after a public test, in which the predictions of the rival calendars were compared with the observed heavens, that its superiority was demonstrated. It was finally promulgated in 510, after Zu Chongzhi's death, and it marked a significant advance in Chinese astronomy.

Other Scientific Achievements

Beyond mathematics and astronomy Zu Chongzhi made contributions to mechanics and engineering. He is credited with the construction of a water-driven mill, a paddle boat propelled by wheels, and other devices, and with improvements in the technology of the time. The accounts of his inventions show a practical as well as a theoretical turn of mind.

He worked also on the measurement of distances and on problems of surveying, and he wrote on the calendar and on the theory of numbers. His son Zu Geng continued his work in mathematics, and together the two are associated with the method for finding the volume of a sphere, an important result in the history of Chinese geometry.

His reputation rests on the union of theory and practice. He was not content to speculate; he measured, computed, built and tested, and he insisted that the results of astronomy be checked against the sky. This empirical temper, joined to his mathematical skill, made him a model of the scientist in the Chinese tradition.

Influence and Legacy

Zu Chongzhi's value for pi stood as the most accurate known for centuries, and his name is remembered in the history of mathematics for this achievement. Later Chinese mathematicians honoured him, and modern historians of science have recognised the sophistication of his methods and the accuracy of his results.

His calendar reforms improved the reckoning of time in China and influenced later astronomical work. The principle of testing competing calendars against observed phenomena, which his case helped to establish, became important in the development of Chinese astronomy and calendrical science.

He is remembered as one of the greatest scientists of ancient China. A crater on the moon and an asteroid have been named in his honour, and his measurement of pi is taught as a landmark in the history of mathematics. He stands for the rigorous, empirical study of the natural world that flourished in China alongside its philosophical traditions.

See also

• Pi
• Chinese mathematics
• Chinese astronomy
• Daming calendar