Mencius: The Second Sage of Confucian Thought

Mencius, also known as Mengzi (c. 372–289 BCE), stands as one of the most influential figures in Chinese philosophy. Revered as the “Second Sage” after Confucius, Mencius played a crucial role in shaping and expanding the teachings of Confucianism. Born as Meng Ke in the state of Zou—modern-day Zoucheng, Shandong—he lived during the Warring States period, a time of political turmoil and moral decline. Like Confucius, he traveled widely, offering counsel to rulers and advocating for benevolent governance rooted in ethical principles.

At the heart of Mencius’s philosophy is his belief that human nature is inherently good. He taught that every person is born with the innate capacity for virtue, a concept he described through the “four beginnings”: compassion, shame, respect, and moral discernment. These, he argued, form the seeds of benevolence, righteousness, propriety, and wisdom—the foundations of moral life. For Mencius, the task of education and self-cultivation was to nurture these innate virtues rather than impose morality from outside.

Mencius’s political thought was equally profound. He believed that rulers bore a moral responsibility to care for their people, insisting that good governance should be based on virtue, not fear or coercion. He argued that a ruler who neglects the welfare of his subjects forfeits the “Mandate of Heaven” and can rightfully be replaced—a bold view that earned him the reputation of being a champion of the common people and an early advocate of proto-democratic ideals.

Although Mencius briefly served as an official in the state of Qi, he spent most of his life as an itinerant scholar, promoting his vision of moral government. His teachings were later compiled in The Mencius, a text that became one of the Four Books of Confucianism and a cornerstone of East Asian moral and political philosophy. Through his enduring emphasis on compassion, justice, and human potential, Mencius helped shape not only Confucian thought but also the ethical foundations of Chinese civilization itself.
Mencius: The Second Sage of Confucian Thought

Jia Xian (1010-1070 AD) - Chinese mathematician

Jia Xian was a Chinese mathematician during the Song dynasty. He described Pascal’s triangle, more than six centuries before Pascal, and used it to calculate square and cube roots.

Jia Xian is also known as Chia Hsien. Jia Xian was a government official and studied with the mathematician and astronomer Chu Yan and was well versed in mathematics, writing many books on the subject. He had a relatively low military office during the reign of Emperor Renzong of the Song Dynasty.

Jia Xian is known to have written two mathematics books: Huangdi Jiuzhang Suanjing Xicao (The Yellow Emperor's detailed solutions to the Nine Chapters on the Mathematical Art), and Suanfa Xuegu Ji (A collection of ancient mathematical rules).

Jia Xian improved the theoretical level of ‘The Nine Chapters of the Mathematical Art’ and created Jia Xian Triangle and Zengcheng root-extraction method.
His diagram is similar to Blaise Pascal’s triangle, which was discovered independently later in the West.
Jia Xian (1010-1070 AD) - Chinese mathematician

Liu Hui, Chinese mathematician

Liu Hui, a Chinese mathematician who lived in the Kingdom of Wei. Kingdom of Wei was a north-central kingdom and one of the three separate kingdoms after the fall of the Han dynasty in the early 3rd century.

Liu Hui not only improves the algorithm of extraction, but also gives an explanation for extracting the square root.

Based on a method he developed, he obtained the most accurate approximation yet of the value of π (=3.1416). About 300 years later, another Chinese mathematician, Zu Chongzi, worked out π to 7 digits (3.1415926).

Liu Hui’s most influential work was his Island Mathematical Manual.  In this book, Liu Hui shows how to use pairs of similar right triangles to measure inaccessible distance.

He completed his commentary on The Nine Chapters on the Mathematical Art in 263. The Nine Chapters on the Mathematical Art was mathematical cannon of the 1st century BC that played a similar role in the East to Euclid’s Elements in the West.
Liu Hui, Chinese mathematician 

Chou Kung (12th century BC)

Chou Kung or the duke of Chou was the brother of Wu Wang the founder of the Chou Dynasty. He was a political figure who solidified the power of the Zhou dynasty. On the death of his brother, he resisted the temptation to size the throne and chose instead to serve as counselor to his nephew.

Chou Kung put down the final Shang rebellion and established a governmental organization that lasted for nearly 800 years.

He is remembered for his name in the Chou li (Rites of Chou) one of the oldest of the ancient Chinese scientific treatises, in which he supposedly takes part in a dialog with someone called Shang Kao.

Their conversation concerns the properties of the right-angled triangle, the gnomon, the circle and the square, as well as measuring heights and distances.

The work is important in providing hard evidence for the state of early Chinese mathematics. The most significant feature of the work is a demonstration of the proof of Pythagoras’s theorem for triangles with sides of 3, 4 and 5 units.

The proof consisted of diagrammatically stacking triangles one on top of the other.
Chou Kung (12th century BC)

Zu Chongzhi – Chinese mathematician and astronomer

Zu Chongzhi’s (429-500) ancestral home was Ji in Fanyang commandery. He came from a family of astronomers who specialized in calendar making.  In his youth he already demonstrated a broad knowledge of ancient learning.

Emperor Xiaowu of the Song dynasty assigned him to the Hualin academy. He also gave him a residence a chariot and clothing. Zu Chongzhi is best known for revising the calendar. His revision was based on a new estimate of the length of the year, an estimate that was accurate to 50 seconds. A court of rival accused him of ‘distorting the truth about heaven and violating the teaching of the classics.’

Zu Chongzhi was a very capable geometer and astronomer. He was one of the greatest mathematicians of ancient China. He computed π to be 355/113 a value not bettered until 1000 years later by al-Kashi (1430).

In astronomy he measured the length of the year to be 365.2429 days, by extensive observations of the length of shadows around the winter solstice, and improvement over contemporary values.

Around 494, Zu Chongzhi was appointed commandant of Changshui which put him in charge of the border area.

He wrote ‘An bian Lun’ (Disquisition on securing the frontier). He also expanded the military farms in the border region, which resulted in increased agricultural production.
Zu Chongzhi – Chinese mathematician and astronomer