Mathematics of Medieval Asia

Julie Belock

Salem State Mathematics Department

October 15, 2007

Decline of Mathematics in Europe

• “Dark Ages” – 5th to the 11th centuries

• Decline of the Roman Empire.

• The Christian Church was the one stable institution; although Latin texts were copied and used to learn Latin, many ideas of classical Greece were suspect, as coming from ancient pagans.

Medieval Era: ~ 5th – 13th centuries

• China

• India

• Arabic world

Ancient / Medieval China

• History is organized into dynasties

• 221BCE: China was united into a single empire

Han Dynasty (~200BCE – 400CE)

• 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)

Mathematical Texts during Han

• Zhoubi suanjing (Arithmetical Classic of the Gnomon and Circular Paths of Heaven) – contained an argument (not quite a proof) for the Pythagorean Theorem.

• Jiushang suanshu (Nine Chapters of the Mathematical Arts) – contained problems and solutions, many were geometric.

These were designed to be teaching texts. The original authors are unknown. Most of what we know about them comes from later commentaries.

Mathematics from the Arithmetical Classic and the Nine Chapters

• Computations, including square roots, using counting boards

• Gou-gu theorem (i.e. Pythagorean Theorem)gou: base, gu: height, xian: hypotenuse

• Surveying problems• Areas and volumes, including approximation of • Systems of linear equations: method nearly

identical to Gaussian elimination

Counting Boards

•Rods were set up in columns; units in the rightmost place, and higher powers of ten as you moved left.•Blank column represented zero•Vertical and horizontal arrangements alternated•Red rods used for positive numbers, black rods for negative.

“Proof” of the Gou-gu Theorem

• From the Arithmetical Classic

• Only used a 3-4-5 right triangle, but hints at generalization

Li Hui (3rd century CE)

• “The Chinese Euclid”• Wrote a commentary on the Nine Chapters; his

edition is the surviving one, and the most important Ancient Chinese mathematical text.

• Expanded the section on surveying problems and named it separately: Haidou suanjing (Sea Island Mathematical Manual)

• Used the “out-in” principle of rectangles to solve surveying problems

The “out-in” method

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?

From the Sea Island Mathematical Manual

Find y and h = x + b

x

b

y a1 a2

h

d

x

b

y a1a2

h

d

Mathematics of Medieval China

1247: Shushu jiuzhang (Mathematical Treatise in Nine Sections) of Qin Jiushao– Solving polynomial equations

Qin used a method involving binomial coefficients (Pascal’s triangle) and synthetic division on the counting board.– Chinese Remainder Theorem for solving

simultaneous congruences

Transmission to and from China

• Not much is known prior to 16th century, when Jesuit missionaries entered China and translated Euclid’s Elements

• Chinese were using a base-10 number system on the counting boards; there is evidence that traders brought the counting boards to India in 5th-6th centuries

Medieval India

• Earliest written mathematical references (~300 CE) were in religious texts (Vedic hymns)– Geometry of ritual altar building

– Decimal numbers

– Extremely large numbers; concept of the infinite

• Mathematics was written in Sanskrit, the language of priests and scholars

Development of decimal place-value numerals

• Place value numerals (including zero) first appear in written works ~800 CE

• However: references to a base 10 place-based numbers system appear earlier

• Chinese traders brought counting boards to India – this may have influenced the development of the number system.

Concept of Zero

• First written evidence: 876 CE, but the concept existed earlier.

• Brahmagupta (598 – 670) gave the first written rules for computing with zero and negative numbers.

• Mahavira (800 – 870) and Bhaksara II (1114-1185) also refined the ideas later on – but they still struggled with the idea of division by zero.

Mathematical Highlights of Medieval India

• Decimal numerals, zero and algebra rules• Geometry of rectilinear figures, circles, solids• Trigonometry of sines and cosines• Solutions of 1st and 2nd degree indeterminate

equations (“Diophantine equations”)• Iterative approximations• Combinatorial algorithms• Finite/infinite series, “infinitesimals”, power

series (precursors of calculus – 13th century)

Indian problems were often posed in verse…

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 stringHow many pearls were there altogether?-From Ganita Sara Samgraha of Mahavira, ~850

Trigonometry

• First developed by the Greeks to aid in astronomy– Hipparchus of Rhodes– Claudius Ptolemy

Chord β β

• Hipparchus used a circle of radius 3438;

(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.)

• Ptolemy used a circle of radius 60.

• The sine of an angle was the length of the associated chord, not a ratio as we use today. Both computed tables of values of the chord for different angles.

Indian trigonometry• Computed “half-chords” instead of chords• Used a circle of radius 3438, so they may have

known of Hipparchus’ work

α

sin α

α

Brahmagupta (598 – 670)

• Indian sine tables contained values for angles that were multiples of 3 ¾°; they began with sin 90° = R = 3438, sin30°=R/2 and used Pythagorean theorem and half angle formulas for the rest.

• Brahmagupta developed an interpolation procedure to find sines of other angles.

12

2

1 22sinsin iiiiii hh

i

i

= ith sine difference

= ith arc

h = 3 ¾ °

(Modern notation)

Brahmagupta gave no justification for the formula.

Why is it called the sine?

• The Sanskrit word for half-chord is jya.• When Arabic mathematicians translated the Indian

sine results, they created a new word for it: jiba• When Europeans eventually discovered and

translated Arab trigonometry, they mistook jiba for jaib, meaning “cove” or “bay”.

• They used the Latin sinus for this. Sinus had come to mean any hollow or cove-shaped area.

The Kerala School (1300 – 1600)

• Infinite series that were equivalent to Maclaurin series expansions for the sine, cosine and tangent.

• Semi-rigerous proofs (“demonstrations”) were provided that often used induction.

• Most of the series were attributed to Madhava (1349 – 1425), but none of his works survive; written evidence is found in later commentaries

Though there are striking similarities between the results of the Keralese school and those of 17th century Europe, there is no evidence that these ideas were known outside of Kerala before the 19th century.

Medieval Arabia

• 7th Century: the beginning of Islam • 766: Baghdad was established by Caliph al-

Mansir as the capital of the caliphate.• Libraries were established; the Ancient Greek

mathematical and scientific works began to be translated into Arabic.

• Islamic culture encouraged learning; Islamic mathematicians were supported by the rulers and religious authorities.

Mathematics of Medieval Arabia

• Improvement of the Indian decimal number system, which had spread to at least Syria by the mid-7th century

• Development of algebra, including linking it to Greek geometry.

• Another influence of the Greeks: they understood the importance of proof.

• Induction, sums of powers, Pascal’s Triangle

• Solution of cubic equations

• Combinations

• Computations of areas and volumes – extending the work of Archimedes

• Trigonometry – extended the works of the Greeks and Indians to establish the other five trig functions

• Spherical trigonometry

Al-Khwarizmi (~780 – 850)

• Wrote a treatise on computation with Indian numerals.

• Wrote a major work on algebraic rules and problems (but note that symbols still were not used):

Hisab al-jabr w’al muqabalah (“The science of reunion and reduction”)

*“al-jabr” is the source of the word “algebra”*

Example:

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?

In modern notation, solve

39102 xx

Al-Kwarhizmi gave a written explanation of how to solve this; he then justified with geometry, literally completing the square.

x

x 10

x2 10x

This rectangle has a total area of 39.

x

x 5

x2

5

x x2

5

The area of the new, large square is 39+25 = 64.

Thus, its side must have length 8, and so x = 3.

Transmission to Europe

• During the Crusades, Europeans brought back many Arabic texts; these were translated into Latin.

• Leonardo de Pisa (Fibonacci) traveled throughout the Middle East, brought back texts and promoted the use of Hindu-Arabic numerals throughout Italy.

Bibliography

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.