= 16:9 or V=D3; Zhang realized that the value for diameter in this formula was inaccurate, noting the discrepancy as the value taken for the ratio. Zhang then attempted to remedy this by amending the formula with an additional D3, hence V=D3 + D3 = D3. With the ratio of the volume of the cube to the inscribed sphere at 8:5, the implied ratio of the area of the square to the circle is √8:√5. From this formula, Zhang calculated pi as the square root of 10 (or approximately 3.162). Zhang also calculated pi as = 3.1466 in his book Ling Xian (). In the 3rd century, Liu Hui made the calculation more accurate with his π algorithm, which allowed him to obtain the value 3.14159. Later, Zu Chongzhi (429–500) approximated pi as or 3.141592, the most accurate calculation for pi the ancient Chinese would achieve. Astronomy In his publication of AD 120 called The Spiritual Constitution of the Universe (靈憲, Ling Xian, lit. "Sublime Model"), Zhang Heng theorized that the universe was like an egg "as round as a crossbow pellet" with the stars on the shell and the Earth as the central yolk. This universe theory is congruent with the geocentric model as opposed to the heliocentric model. Although the ancient Warring States (403–221 BC) Chinese astronomers Shi Shen and Gan De had compiled China's first star catalogue in the 4th century BC, Zhang nonetheless catalogued 2,500 stars which he placed in a "brightly shining" category (the Chinese estimated the total to be 14,000), and he recognized 124 constellations. In comparison, this star catalogue featured many more stars than the 850 documented by the Greek astronomer Hipparchus (c. 190–c.120 BC) in his catalogue, and more than Ptolemy (AD 83–161), who catalogued over 1,000. Zhang supported the "radiating influence" theory to explain solar and lunar eclipses, a theory which was opposed by Wang Chong (AD 27–97). In the Ling Xian, Zhang wrote: Zhang Heng viewed these astronomical phenomena in supernatural terms as well. The signs of comets, eclipses, and movements of heavenly bodies could all be interpreted by him as heavenly guides on how to conduct affairs of state. Contemporary writers also wrote about eclipses and the sphericity of heavenly bodies. The music theorist and mathematician Jing Fang (78–37 BC) wrote about the spherical shape of the Sun and Moon while discussing eclipses: The Moon and the planets are