Tiny Wiki : Fast loading, text only version of Wikipedia.

Liu Hui

Liu Hui (, fl. 3rd century) was a Chinese mathematician who lived in the Wei Kingdom. In 263 he edited and published a book with solutions to mathematical problems presented in the famous Chinese book of mathematics known as ''The Nine Chapters on the Mathematical Art''.

He was a descendant of Marquis of Zixiang of Han dynasty, corresponding to now Zixiang township of Shandong province. He completed his commentary to the Nine Chapters in year 263.
He probably visited Luoyang, and participated in measurements of sun shadow.

Mathematical work

Liu was one of the first mathematicians known to leave roots unevaluated, giving more exact results instead of approximations. Along with Zu Chongzhi, they were among the greatest mathematicians of the ancient world.Needham, Volume 3, 85-86. Liu Hui expressed all of his mathematical results in the form of decimal fractions (using metrological units), yet the later Yang Hui (c. 1238-1298 AD) expressed his mathematical results in full decimal expressions.Needham, Volume 3, 46.Needham, Volume 3, 85. He also provided commentary on the mathematical proof that is identical to the Pythagorean theorem of the Greek Pythagoras (c. 580 BC-500 BC).Needham, Volume 3, 22. Liu Hui called the figure of the drawn diagram for the theorem the "diagram giving the relations between the hypotenuse and the sum and difference of the other two sides whereby one can find the unknown from the known".Needham, Volume 3, 95-96. In terms of the treatment of plane areas and solid figures, Liu Hui was one of the greatest contributors to 'empirical' solid geometry. For example, he figured out that a wedge with rectangular base and both sides sloping could be broken down into a pyramid and a tetrahedral wedge.Needham, Volume 3, 98-99. He also figured out that a wedge with trapezoid base and both sides sloping could be made to give two tetrahedral wedges separated by a pyramid.

In his commentaries on the Jiuzhang suanshu, he presented (among other things):
*an algorithm for calculation of π in the comments to chapter 1.Needham, Volume 3, 66. He calculated pi to

3.141024 < \pi < 3.142074

with a 192 (= 2⁵ � 6) sided polygon . Archimedes used circumbscribed 96 polygon to obtain inequality

\pi <\tfrac{22}{7}

, and then used an inscribed 96-gon to obtain inequality

\tfrac{223}{71} < \pi

. Liu Hiu used only one inscribed 96-gon to obtain his π inequalily, and his results were a bit more accurate than Archimedes'.Needham, Volume 3, 100-101. But he commented that 3.142074 was too large, and picked the first three digits of π=3.141024 ~3.14 and put it in fraction form

\pi = \tfrac{157}{50}

. He later invented a quick method and obtained

\pi =3.1416

, which he doubled checked with 3072-gon(= 2⁹ � 6), he was quite happy about this result.

Nine Chapters had used the value 3 for the π formula, but Zhang Heng (78-139 AD) had previously estimated it to the square root of 10;
*Gaussian elimination;
*Cavalieri's principle to find the volume of a cylinder.
Source: Wikipedia