Mathematics of Medieval Asia. Julie Belock Salem State Mathematics Department October 15, 2007. Decline of Mathematics in Europe. “Dark Ages” – 5 th to the 11 th centuries Decline of the Roman Empire.
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Salem State Mathematics Department
October 15, 2007
Bureaucracy was established, including standards for weights and measures
Education became necessary
Civil service exams were instituted (these were in use through the 19th century!)
Civil servants were required to be competent in various areas of mathematics (among other subject areas)
These were designed to be teaching texts. The original authors are unknown. Most of what we know about them comes from later commentaries.
gou: base, gu: height, xian: hypotenuse
The red rectangles have equal areas.
Now for [the purpose of] looking at a sea island, erect two poles of the same height, 5 bu [on the ground], the distance between the front and rear [pole] being a thousand bu. Assume that the rear pole is aligned with the front pole. Move away 123 bu from the front pole and observe the peak of the island from ground level; it is seen that the tip of the front pole coincides with the peak. Move backward 127 bu from the rear pole and observe the peak of the island from ground level again; the tip of the back pole also coincides with the peak.
What is the height of the island and how far is it from the pole?
1247: Shushu jiuzhang (Mathematical Treatise in Nine Sections) of Qin Jiushao
Qin used a method involving binomial coefficients (Pascal’s triangle) and synthetic division on the counting board.
Whilst making love a necklace broke.
A row of pearls mislaid.
One sixth fell to the floor.
One fifth upon the bed.
The young woman saved one third of them.
One tenth were caught by her lover.
If six pearls remained upon the string
How many pearls were there altogether?
-From Ganita Sara Samgraha of Mahavira, ~850
(possible reason: if R=3438, the circumference = 21601.6, close to 21600 = 360×60. Then each minute of arc corresponds to approx. one unit of length on the circumference.)
= ith sine difference
= ith arc
h = 3 ¾ °
Brahmagupta gave no justification for the formula.
Hisab al-jabr w’al muqabalah (“The science of reunion and reduction”)
*“al-jabr” is the source of the word “algebra”*
One square, and ten roots of the same, are equal to thirty-nine dirhems. That is to say, what must be the square which, when increased by ten of its own roots, amounts to thirty-nine?
Al-Kwarhizmi gave a written explanation of how to solve this; he then justified with geometry, literally completing the square.
This rectangle has a total area of 39.
The area of the new, large square is 39+25 = 64.
Thus, its side must have length 8, and so x = 3.
Berlinghoff, William and Gouvea, Fernando. Math Through the Ages, Oxton House Publishing, Farmington, ME, 2002.
Katz, Victor. A History of Mathematics, Brief Edition, Pearson Addison Wesley, Boston, 2004.
Katz, V. (editor), The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton University Press, Princeton, New Jersey, 2007.
MAA PREP Program, “Mathematics of Asia,” June 10 – 15, 2007 (course notes).
Swetz, F.J., The Sea Island Mathematical Manual: Surveying and Mathematics in Ancient China, The Pennsylvania State University Press, University Park, Pennsylvania, 1992.