![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.”](https://image.slidesharecdn.com/pythagorasfinal-150517114838-lva1-app6892/85/Pythagoras-final-15-320.jpg)
![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.](https://image.slidesharecdn.com/pythagorasfinal-150517114838-lva1-app6892/85/Pythagoras-final-24-320.jpg)
Pythagoras(final)
- 1. The Significance of Pythagoras’ Theorem in the Ancient World By David O’Connell, Rachel Glover & Alastair Titchmarsh
- 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.”
- 13. Section 2: Use in China and India
- 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.
- 32. Section 3: Pythagoras and the Greeks
- 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
- #24: Parallels can be struck up between the rope-geometricians of Egypt and their spiritual assignment of mathematical beauty
- #25: 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)
- #26: Katyayana was also an influential grammarian, providing groundbreaking work on semantics