A final year project discussing the history and significance of the Pythagorean theorem in the ancient world.
The presentation provides information on the Babylonians, the Egyptians, the Indians and Chinese before moving on to Ancient Greece and Pythagoras himself.
2. Pythagorean Theorem
A simple proof:
Combine 4 triangles to form a square of length (a+b) such that the centre
square has side c. Therefore we obtain:
(a+b)2=2ab + c2
Simplifying we reach the identity:
a2+b2=c2
There are over 400 different proofs
of the Theorem.
4. Babylon
★ There are several clay tablets that show
knowledge of Pythagoras’s theorem.
★ The ones we will look at are: Plimpton 322,
YBC 7289, the Susa tablet and the Tell
Dhibayi tablet.
5. Plimpton 322
★ This tablet dates to around 1800-1650
BCE.
★ The tablet itself is damaged, a large
portion missing.
★ But it is believed to be a list of
Pythagorean triples with a method for
their calculation.
★ Though this is mostly guess work and
some mathematicians believe this
tablet has nothing to do with
Pythagoras at all.
6. YBC 7289
★ This tablet dates to
around 2000-1600 BCE.
★ The contents of the
problem are relatively
simple, but show a clear
use of Pythagoras’s
theorem and a very
accurate estimate of √2.
7. The Susa Tablet
★ Discovered in Susa in 1936 CE, this tablet
dates to the Old Babylonian Period (a similar
time to the previous tablets).
★ The tablet contains a problem about an
isosceles triangle and a circle through the
three vertices.
★ Given the side lengths of the triangle (in this
case 50, 50 and 60), the tablet describes a
method for calculating the radius of the circle
and makes use of Pythagoras’s theorem.
8. The Tell Dhibayi Tablet
★ In 1962 around 500 clay tablets were found, believed
to have originated from around 1750 BCE.
★ One of these tablets contains a problem of finding the
lengths of sides of a rectangle given the area and the
length of the diagonal.
★ For modern day mathematicians, this is a simple
algebraic exercise, but the Babylonians approached
the problem in quite an interesting way, using
Pythagoras’s theorem along the way.
9. Egypt
The ancient Egyptians are well known for their
architecture, but how did they manage to
build so perfectly?
10. Egyptian Rope Constructions
★ It is believed that the ancient
Egyptians used rope to help them
create perfect right angles for their
buildings.
★ The rope was knotted at 12 equally
spaced places and then arranged as
shown on the right to create a
perfect 3-4-5 right-angled triangle.
11. Were 3, 4 and 5 special?
The relationship between the numbers 3, 4 and 5 were
very important to the Egyptians, so much so that it is
believed the ancient Pyramid of Giza is actually a
representation of these numbers.
“The Great Pyramid, ONE Structure, representing the
number THREE by its triangular faces, the number
FOUR by its square base, and the number FIVE by its
apex and four corners.”
12. Were 3, 4 and 5 special?
Another example of the important of these numbers in
ancient Egyptian culture is the Egyptian 3-4-5 triangle, in
which each side of the triangle is given a name:
“The upright, therefore, may be likened to the
male, the base to the female, and the
hypotenuse to the child of both, and so Ausar
(Osiris) may be regarded as the origin, Auset
(Isis) as the recipient, and Heru (Horus) as
perfected result.”
14. China
Sources
The Arithmetical Classic of the Gnomon and
Circular Paths of Heaven (or Zhoubi
Suanjing), circa 1st millennium BC
Nine Chapters on the Mathematical Art (or
Jiuzhang Suanshu), circa 2nd century BC,
and its later commentaries
Sea Island Mathematical Manual (or Haidao
Suanjing), circa 3rd century AD
15. Earliest source: Zhoubi Suanjing
“Therefore fold a trysquare so that the base is three in
breadth, the altitude is four in extension, and the
diameter is five aslant. Having squared its outside, halve
it [to obtain] one trysquare. Placing them round together
in a ring, one can form three, four and five. The two
trysquares have a combined length of twenty-five. This is
called the accumulation of trysquares.”
16. Hsuan-thu
Also contained in the Zhoubi Suanjing is the
famous Hsuan-thu diagram, which displays the
statement in pictorial form.
17. Zhao Shujing’s abstracted proof
A modern translation:
Let the gou be a, the gu be b, and the xian be c.
The area of the triangle is 0.5(ab), therefore the area
of four triangles is 2ab
Then by the second diagram:
c²=2ab+(a-b)²
=2ab+a²-2ab+b²
=a²+b²
Alternatively, by the third diagram:
c²+2ab=(b+a)²
c²+2ab=a²+2ab+b²
c²=a²+b²
a
b c
18. Nine Chapters of the Mathematical Art
Had an influence on Chinese mathematics
similar to Euclid’s “Elements” did in Greece
The final chapter is entitled “gou-gu” and
contains numerous problems requiring use of
the gou-gu theorem to find unknowns using
algebraic methods.
19. Liu Hui
Influential Chinese scholar who
produced a number of works, mainly
extrapolating on knowledge found in
older texts such as the Nine Chapters.
Produced a unique proof of the Gou-gu
based on axiomatic assumptions,
though parts of his proof are now lost.
20. Liu Hui’s applications to surveying
In his “Sea Island
Mathematical Manual”,
written 263AD, Hui provides
applications of the Gou-gu
to problems in land
surveying.
21. Uniqueness of Chinese achievement
China provided both explicit statements and
unique proofs of the theorem.
As Chinese scientific development was
relatively self-contained compared to other
civilisations, it is likely that the Chinese Gou-
gu originated independently of other cultures.
22. India
Sources
The Sulba sutras: Baudhayana sutra (ca. 800BC),
Apastamba sutra (ca. 600AD) and Katyayana sutra
(ca.170BC)
The Brahmanas (ca. 900BC)
23. Sulba Sutras
Roughly translates as “rules of the chords”, a
series of 9 texts dated throughout the first
millennia BC.
Their contents are an amalgam of ritual altar
construction, spiritual knowledge and
mathematics (including approx. of root 2,
circling the square).
The more prominent authors are Baudhayana,
Apastamba, Katyayana and Manava.
24. Baudhayana (ca. 800BC)
Wrote the earliest of the sutras, and in the
first chapter gives the following statement:
1.9: The diagonal of a square produces double the area
[of the square] …
1.12: The areas [of the squares] produced separately by
the lengths of the breadth of a rectangle together equal
the area [of the square] produced by the diagonal
1.13: This is observed in rectangles having sides 3 and 4,
12 and 5, 15 and 8, 7 and 24, 12 and 35, 15 and 36.
25. Apastamba and Katyayana
Similar verses are also found in the
later sutras of Apastamba (600-
540BC), in chapter 1 verse 2, and
Katyayana (200-140BC), in chapter 2
verse 7.
26. No Proof?
In contrast to the Chinese mathematicians,
Indian scholars did not actually prove the
theorem until Bhaksara in the 16th century
AD.
However, statements contained in the sutras
were viewed as established rules, i.e.
theorems.
27. Mahavedi
Translating as “great altar” or “entire altar”, the
Mahavedi was used in Vedic ritual practices for centuries.
Procedures for construction of these altars are found in a
number of the sulba sutras.
28. Mahavedi
Analysis on the geometry of the altar reveals
that 18 unique pythagorean triples (plus their
mirrors) are found inherent in its design.
29. The Brahmanas
Dated circa 900BC, the Brahmanas are a commentary on
the Vedas, the underlying texts of the Vedic religion.
The Brahmanas contain methods to construct the
Mahavedi and other altars, although the construction
provided differs from the one found in the later sutras.
Does this mean that the creators of the Mahavedi knew of
our theorem even longer ago? We don’t know.
30. Significance
Despite not producing a proof, the Ancient
Indian scholars most certainly understood the
Pythagorean theorem and were able to
construct numerous pythagorean triples.
They applied this understanding to indigenous
religious practice and more advanced
mathematical constructs.
31. Mutual History: Approximations of pi
The accolade of “best approximation of pi”
passed hands from India to China for
centuries.
Their method of approximation revolved
around repeated uses of the Pythagorean
theorem applied to polygons inscribed within
circles.
33. Pythagoras The Explorer
★ Pythagoras travelled
extensively through Egypt,
Mesopotamia and India.
★ Acquired religious
philosophies as well as
mathematical knowledge,
including geometry and
astronomy.
India
34. Pythagoras The Cultist
★ Pythagoras created a secret society
of mathematicians known as the
Pythagoreans.
★ As the leader, Pythagoras
influenced the followers of his own
beliefs.
★ Any advances made by the
Pythagoreans were attributed to
Pythagoras.
ALL IS NUMBER
35. Pythagoras The Mathematician
★ Pythagoras believed that
numbers were more than just
values and attributed
characteristics to them.
★ He believed that integers could
be used to explain operations in
nature.
★ But he struggled with the notion
of irrational numbers.
36. The First Proof
★ The first known formal proof
appears in Euclid’s Elements.
★ The birth of this theorem resulted
in a new wave of thinking in many
areas of mathematics.
37. Geometry & Arithmetic Link
Arithmetic:
Deals with
the discrete
Geometric:
Deals with the
continuous
Pythagorean
Theorem:
a2+b2=c2
38. Concept of Infinity
★ The Pythagoreans attempted to determine a rational
solution to √2.
★ They produced recurrence relations to generate
solutions to:
x2-2y2=1
★ This implies the notion of limits:
Xn √2 as n ∞
yn
39. Plato & Aristotle
★ Plato founded an academy and
became known as the “maker of
mathematicians”.
★ Made mathematics an essential
part of the curriculum.
★ Aristotle, who joined the
academy, proved the irrationality
of √2.
ALL IS NUMBER
40. Pappus of Alexandria
★ Produced 8 books
known collectively as
the Mathematical
Collection.
★ Discovered a
generalisation of
Pythagoras’ Theorem
known as Pappus’ Area
Theorem.
B
C
A
P
41. Pappus of Alexandria (2)
★ Pappus followed up on
Pythagoras’ belief of mathematics
in nature.
★ He noticed how bees construct
their hives with hexagonal cells,
which have a greater surface area
than that of a square or triangle.
42. “The Pythagorean theorem is mathematically
universal, likely to arise in any sufficiently
advanced civilization. Other such cultural
universals are the concept of π—the ratio of
diameter to circumference in the circle—and the
Euclidean algorithm.”
John Stillwell, Mathematics and Its History, Third Edition (Springer, New York, 2010), p. 70
Editor's Notes
Parallels can be struck up between the rope-geometricians of Egypt and their spiritual assignment of mathematical beauty
As you can see, verse 1.12 gives explicit statement of the theorem, and 1.13 gives examples that result in integer solutions for the hypotenuse (aka Pythagorean triples)
Katyayana was also an influential grammarian, providing groundbreaking work on semantics
