#### Pythagorean Theorem

For those of us who still remember our schooling days, the subject of Mathematics was, for lack of a better word, special. Some students really struggled with it, while for others, manipulating numbers was as easy as breathing. They probably all became waiters, after all: “On a waiter’s bill pad numbers dance. Reality and unreality collide on such a fundamental level that each becomes the other and anything is possible, within certain parameters.”*

What we all remember with varying degrees of fondness is the Pythagorean Theorem, named after the Greek Mathematician Pythagoras of Samos:

a² + b² = c²

a = side of right triangle

b = side of right triangle

c = hypotenuse

Little is really known about Pythagoras himself, and likely he left no writings. However, it is believed he founded a philosophical and religious brotherhood where he taught astronomy, metaphysics, music, number theory and various other subjects such as philosophy and intuitive truths.

It is only from the commentary of his students and later writings that we know about the man himself and his teachings.

Pythagoras himself did not invent the theorem. Theories suggests that he travelled to Egypt and Babylon where he picked up the concept and later taught it to his students. Evidence for this comes in the form of The Rhind Mathematical Papyrus.

#### The Rhind Mathematical Papyrus

The Rhind Mathematical Papyrus, named after Alexander Henry Rhind, who purchased the papyrus in 1858, is one of the oldest known mathematical documents to have survived to modern times. The Rhind Papyrus is also known as the Ahmes Papyrus after the scribe who copied it.

The manuscript’s writings covers algebra, fractions, geometry and trigonometry, dating back to 3,550 years ago (approx. 1550 BC-1650 BCE) and predating Pythagoras by about a 1000 years.

The papyrus is approximately 33 cm wide and consists of various parts making it over 5 m. long. It is written in hieratic script, a cursive form of hieroglyphics, and is divided into 87 problems or exercises. These exercises cover a wide range of mathematical concepts, including arithmetic, geometry, and algebra.

It shows that the Egyptians used first-order equations, geometric series and a second-order algebraic equations, related to the Pythagorean theorem a² + b² = c². This is not at all surprising, given the Egyptians fondness for triangles, which where, in a sense, sacred. This is what is now referred to in modern times as the Sacred Triangle of Egypt, and forms the basis of construction of the Egyptian pyramids.

These equations involve determining an unknown quantity by setting up proportional relationships. It also includes examples of geometric series, which are sequences of numbers where each term is found by multiplying the previous term by a constant factor.

It also describes how to obtain an approximation of π, and it gives its value to be 3.16, accurate to within less than 1% than its current known value.

Additionally, the Rhind Mathematical Papyrus contains one of the earliest known attempts at squaring the circle, a famous problem in mathematics. The problem involves constructing a square with the same area as a given circle using only a compass and a straightedge. The papyrus describes an algorithm for approximating the area of a circle, but it falls short of providing a solution to the squaring the circle problem.

#### Mathematical Achievements and Contributions

The Rhind Mathematical Papyrus demonstrates several key achievements and contributions of ancient Egyptian mathematics:

1. Unit Fractions: The use of unit fractions as a means of representing non-unit fractions is a distinctive feature of ancient Egyptian mathematics. This method made calculations more manageable and efficient.
2. Geometric Approximations: The papyrus provides one of the earliest approximations of π (pi) as 3.125. While not very accurate by modern standards, it reflects the ancient Egyptians’ understanding of the relationship between the circumference and diameter of a circle.
3. Practical Mathematics: Many of the problems in the papyrus are practical in nature, reflecting the real-world applications of mathematics in ancient Egypt. These applications included land surveying, taxation, and trade.
4. Algebraic Techniques: The use of algebraic techniques to solve linear equations is evident in several problems. The method of “false position” employed in the papyrus is a precursor to algebraic methods used in later mathematics.

#### Conclusion

The Rhind Mathematical Papyrus is a remarkable testament to the mathematical achievements of ancient Egypt. It sheds light on their practical and theoretical mathematical knowledge, including their use of unit fractions, geometric approximations, and algebraic techniques.

The Manuscript is currently housed in the British Museum and is considered a remarkable treasure for historians, mathematicians, and anyone interested in the history of mathematics.

* This is a joke from the novel “Life, the Universe, and Everything”, by Douglas Adams.

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