During the Northern and Southern Dynasties, a very famous scientist emerged: Zu Chongzhi.
Zu Chongzhi was born into an official family. His grandfather served in the Liu Song court as the "Grand Master of Works," overseeing civil engineering projects, and his father was an official in charge of architecture. As a child, Zu Chongzhi disliked reading ancient texts, and his father forced him to study theThe AnalectsAfter two months, however, he could only recite ten lines. His father was furious, gave him a severe beating, and scolded him for being worthless. But his grandfather did not think so. His grandfather felt that he was very interested in astronomical phenomena and might achieve something in this field in the future, so he taught him to read some astronomy books every day. Sometimes, Zu Chongzhi would study astronomical knowledge together with his grandfather and father, which made his interest in astronomy grow stronger and stronger.
One day, Zu Chongzhi visited the home of an official well-versed in astronomy with his grandfather. When the official learned that Zu Chongzhi was deeply interested in astronomy, he asked, "Studying astronomy is extremely hard work, and it cannot bring you promotion or wealth. What exactly do you study it for?" Zu Chongzhi replied, "I have never thought about seeking promotion or wealth through studying astronomy. The universe is full of mysteries, and I simply want to unravel them." The official smiled upon hearing this and said, "You are truly a child with great ambition!"
From then on, Zu Chongzhi frequently observed the trajectories of celestial bodies and repeatedly sought astronomical knowledge from that official.
The working people of ancient China, through long-term observation and practice, summarized the basic patterns of the sun and moon's movements and created a calendar based on them. By the time of Zu Chongzhi, the calendar had already become quite accurate, but Zu Chongzhi believed it could be made even more precise.
One day, in his study, he flipped through calendar texts such as the Taichu Calendar, the Later Han Four-Quarter Calendar, the Spring and Autumn Four-Quarter Calendar, and the Yuanjia Calendar, carefully comparing these ancient works. After thorough analysis, he discovered that in the Yuanshi Calendar by Zhao Fei of the Northern Liang, a system of 221 intercalary months over 600 years was used for the first time, replacing the earlier method of 7 intercalary months over 19 years. The so-called 7 intercalary months over 19 years meant that within 19 years, there were 7 leap years, each with 13 months. This calendar system was the one commonly used during the era in which Zu Chongzhi lived.
After seeing Zhao Xie's innovation, Zu Chongzhi could not help but praise, "This bold attempt is truly wonderful!" Having said that, he took out his counting rods and began calculating very carefully. Through his calculations, he discovered a flaw in the calendar system of seven leap years in nineteen years: with seven leap years in every nineteen-year cycle, this calendar would be one day ahead of the actual time every two hundred years. He felt this calendar was not precise enough and resolved to improve its accuracy. After thinking for a long time, he concluded that to enhance the precision of the calendar, merely reading the calendar texts left by predecessors was far from sufficient; he had to rely on his own actual observations.
From then on, he set up a gnomon about eight feet high at the observatory to measure the length of the sun's shadow. He recorded the daily changes in the shadow length in an observation notebook, which soon became completely filled. Believing that merely measuring the shadow length was far from sufficient, he designed several clepsydras to keep time, recording not only the shadow length each day but also the precise times of sunrise and sunset. After years of accumulated records and calculations, he finally discovered that because the shadow length showed no significant change around the winter solstice and the clepsydras were not accurate enough, it was very difficult to measure the exact moment of the winter solstice. This problem became the biggest obstacle on his path forward. However, he did not become discouraged; instead, he summarized the lessons from his failures and devised another method: instead of observing the shadow length on the winter solstice day alone, he observed the shadow lengths for more than twenty days before and after the winter solstice, then took the average to calculate the date and time of the solstice. This method proved highly effective, greatly improving the precision of determining the winter solstice moment. Based on these new research results, Zu Chongzhi formulated a new calendar—the Daming Calendar. This calendar stipulated a year as 365.24281481 days, which differs from modern astronomical measurements by only 50 seconds.
In 462 AD, Zu Chongzhi presented his research findings to Emperor Xiaowu of the Liu Song Dynasty, requesting the abolition of the old calendar and the adoption of a new one. Dai Faxing, a minister favored by the emperor, argued that the old calendar had been in use for a long time and that Zu's unauthorized changes were an act of great disobedience, so he strongly opposed the new calendar. Zu Chongzhi presented his research data, refuting Dai Faxing until he was speechless. Unable to argue with Zu, Dai relied on the emperor's favor and said unreasonably, "The ancients established the calendar, and later generations should follow it. How can it be arbitrarily changed?" Zu Chongzhi calmly replied, "You say the ancients' calendar is good, but do you have any factual evidence? If you do, bring it forward for debate. Using empty words to intimidate others—what's the point?" Seeing Dai at a disadvantage, Emperor Xiaowu tried to save face for him by summoning some people knowledgeable about calendars to debate Zu. However, they were quickly refuted into silence. Despite this, the emperor did not immediately implement the new calendar. It was not until more than ten years after Zu Chongzhi's death that the calendar he created was finally put into use.
Compared to his achievements in astronomy, Zu Chongzhi made even greater accomplishments in mathematics, as he accurately calculated pi.
The so-called pi refers to the ratio of a circle's circumference to its diameter; it is a constant, remaining fixed regardless of whether the circle is large or small. Before Zu Chongzhi, people used the expression "a circumference of one measures three in diameter" to represent pi. When Zu Chongzhi was young, his teacher told him, "The circumference of a circle is three times its diameter." Zu Chongzhi, driven by intense curiosity, pondered his teacher's words repeatedly and felt they might not be correct. So, he found a rope and ran to the roadside at the edge of the village, waiting for a horse-drawn carriage.
After a short while, a carriage came toward him. He asked the driver to stop the carriage and said, "I want to measure the wheel of your carriage with a rope—would you allow me to do so?" The driver nodded. Zu Chongzhi was delighted and immediately took out a rope, measured the wheel, then folded the rope into three equal lengths to measure the wheel's diameter. He repeated this measurement several times, but each time he found that one-third of the wheel's circumference was slightly longer than the wheel's diameter. Zu Chongzhi measured several carriages in succession, and the result was always the same. Not understanding why this was so, he resolved to get to the bottom of the matter.
When Zu Chongzhi grew up, he began to study Liu Hui's "method of cutting the circle." This method involves inscribing a regular hexagon inside a circle, where each side of the hexagon equals the circle's radius, then drawing a regular dodecagon, a regular icositetragon, a regular tetracontakaioctagon, and so on. The circumference of the circle is equal to the sum of all the sides of these polygons. Using this method, Liu Hui calculated pi as 3.14. He also concluded that the accuracy of pi's value is directly proportional to the number of sides of the inscribed regular polygon.
The achievements of Liu Hui provided the theoretical foundation for Zu Chongzhi to further calculate pi. Following Liu Hui's theory, Zu Chongzhi began his calculations. Together with his thirteen-year-old son, they worked day and night for over ten days, progressing from a regular hexagon to a regular 96-sided polygon. The result they obtained for the regular 96-sided polygon was 0.000002 zhang less than Liu Hui's result.
Zu Chongzhi's son said to him, "We have calculated very carefully; it's impossible for us to be wrong. It's likely that Liu Hui made a mistake." Zu Chongzhi disagreed with his son's view. He said, "To overturn Liu Hui's result, we must have evidence." So, the father and son tirelessly recalculated everything from scratch. This time, the result showed that Liu Hui's calculation was correct. Although it took over ten extra days, Zu Chongzhi learned a lesson: errors are inevitable during calculations, and to reach a correct conclusion, each step must be double-checked. Thus, Zu Chongzhi worked tirelessly, calculating all the way up to a 12,288-sided polygon and then a 24,567-sided polygon. He discovered that the difference between these two was only 0.0000001. Zu Chongzhi understood that, in theory, he could continue further, but practical limitations prevented him from doing so, even though he did not want to stop. Through repeated calculations, he determined that pi fell between 3.1415926 and 3.1415927. In the history of mathematics, this was the first time anyone had calculated pi to seven decimal places, a record that stood for a full thousand years.
After that, Zu Chongzhi also calculated approximate values of pi as fractions, with the approximate ratio being 22/7 and the precise ratio being 355/113. This result was over a thousand years earlier than in Europe. Because he was the first person in the world to propose this precise ratio, some foreign scientists advocated naming it "Zu's Ratio."
In addition to his achievements in astronomy and mathematics, Zu Chongzhi also made very outstanding contributions to mechanical manufacturing, inventing machines such as a "timer," "water-powered mill," and "thousand-li boat," which greatly improved production efficiency.