Transcript Mathematics of Medieval Asia Julie Belock Salem State Mathematics Department October 15, 2007

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
rd
(3
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.
From the Sea Island Mathematical Manual
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?
Find y and h = x + b
x
h
b
y
a1
a2
d
x
h
b
y
a2
a1
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 placevalue numerals
• Place value numerals (including zero) first
appear in written works ~800 CE
• However: references to a base 10 placebased 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 (11141185) 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 string
How 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
sin α
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.
(Modern notation)
i
= ith sine difference
i
= ith arc
h = 3¾°
sin  i     sin  i  

2h
 i   i 1  
2
2h
2
 i   i 1 
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
x  10 x  39
2
Al-Kwarhizmi gave a written
explanation of how to solve this; he
then justified with geometry, literally
completing the square.
x
x2
10x
x
10
This rectangle has a total area of 39.
x
x2
x
5
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.