Our chief sources of information concerning ancient Egyptian geometry are the Moscow (1850 B.C.) and Rhind (1650 B. C.) papyri. A Scottish scholar and antiquary, A. M. Rhind discovered in 1858 in Egypt and bought an ancient Egyptian papyrus found in some ruins in Thebets. The Rhind papyrus is a collection of arithmetical, geometrical and miscellaneous problems, including some area and volume applications. The papyrus is a copy of 1650 B. C. of much earlier writings of the latter part of the 1900 B. C. The entire work emphasizes the two concepts that particularly characterize the mathematics of the early Egyptians: 1) the consistent use of additive procedures and 2) computations with fractions. Most problems are of practical nature. Some problem may present a challenge even to the modern student; e. g., "Find the volume of a cylindrical granary of diameter 9 and height 10 cubets."

The Moscow papyrus also referred to as the Golenischev papyrus for the man who owned it before its acquisition by the Moscow Museum of Fine Arts was probably written about 1850 B. C. Although it contains only 25 problems, it is similar to the Rhind Papyrus. This work shows that the Egyptians were familiar with the formula for the area of a hemisphere and the correct formula for the volume of a truncated square pyramid'

V =a²+ab+b²h

3

The solution is expressed only in terms of the necessary computational steps for the given numerical values: height of 6 and the bases of sides 4 and 2. There are various conjectures about how the Egyptians could develop this procedure, but the papyrus offers no help. This formula is often referred to as the Egyptians' "greatest pyramid". A challenging and exciting discovery of more than a century ago of these two mathematical texts gives fascinating exercise to the student of mathematics, both modern and ancient. Of the 110 problems in the papiri 26 concern the computation of land areas and volumes. The ancient Egyptians recorded their work on stone and papyrus resisting the ages because of Egypt's dry climate. There is no documentary evidence that the ancient Egyptians were aware of the Pythagorean theorem. Nevertheless early Egyptian surveyors realized that a triangle with sides of lengths 3, 4 and 5 units is a right triangle. Egyptian geometry arose from necessity. The annual inundation of the Nile Valley forced the Egyptians to develop some systems for redetermining land markings; in fact, the word "surveying" means "measurement of the earth". The Babylonians likewise faced an urgent need for mathematics in the construction of the great engineering structures (marsh drainage, irrigation and flood control) for which they were famous. Similar undertakings and geometrical accomplishments occurred in India and China. The ancient Indians and Chinese, however, used very perishable writing materials (bark bast and bamboo) and due to the lack of primary sources we know next to nothing about mathematics in ancient India and China.

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3.1.12. The Pythagorean Theorem

"The Pythagorean" theorem is one of the most important propositions in the entire realm of geometry. Despite the strong Greek tradition that associates the name of Pythagoras with the statement that "the square on the hypotenuse of the right-angled triangle is equal to the square on the sides containing the right angle", there is no doubt that this result was known prior to the time of Pythagoras.

It is possible that Pythagoras gave the proof of the theorem based on the proportionality of similar figures. With the later realization that all lines are not necessarily commensurable, this proof became invalid. Thus, at the time of Euclid's "Elements" there was no need for a more adequate proof. Euclid's Proposition 1,47 is the Pythagorean theorem with a proof universally credited to Euclid himself. Proclus's speculation was simply that Euclid rewrote the proof in order that he might put the proposition in his first book to complete it. There is also considerable evidence that the first book was written to lead to the climax of this theorem and its converse.

In 1907 L. S. Loomis published his book “The Pythagorean Proposition”, a work that contained 370 proofsof this theorem. Probably no other theorem in mathematics can be demonstrated by such a wide variety of algebraic and geometric proofs. The Pythagorean theorem and the proof are so important in mathematics that Loomis writes in his book: "I noticed two or three American texts on Geometry in which Euclid's proof of the Pythagorean theorem does not appear. I suppose the author wishes to show his originality or independence — possibly up-to-datedness. He shows something else. The leaving out of Euclid's proof is like the play of "Hamlet" with Hamlet left out".

Choose the proper alternative or give your own answers.

Is Pythagorean theorem general or special?

1. General. 2. Special. 3. Fundamental.

What role do fundamental theorems play in mathematical reasoning, in proofs, justification and in science?

1) every fundamental theorem is a landmark in the history of maths. 2) they are precise and concise arguments. 3) they are convenient shortcuts to proofs. 4) their application is the best justification. 5) they serve as points of reference in maths. 6) they are the main guiding threads in scientific theories.

Why do mathematicians re-prove fundamental theorems? (e. g., Pythagorean theorem)?

1) they enjoy doing it. 2) it's their hobby. 3) re-proving theorems is mental gymnastics. 4) it is the "food" for the mind. 5) Euclid gave the proof of a special case of the Pythagorean property. 6) it's simple to give a proof of this theorem. 7) the first proof is, as a rule, not rigorous and elegant. 8) they want to become more famous. 9) to display their ingenuity. 10) to broaden the range and scope of the theory, where it was originally proved.

Every high school leaver remembers the Pythagorean theorem for the rest of his life. Why is it unforgettable?

1) because of the legendary fame of its creator — Pythagoras 2) because, according to the legend, Pythagoras sacrificed 100 oxen to the Gods for its proof. 3) due to the mastery of high school teachers' presentation of the theorem. 4) thanks to the simplicity of its proof. 5) because there exist too many proofs. 6) Geometry begins with this theorem, 7) the theorem runs "like golden thread" throughout mathematical history. 8) because of the beauty and elegance of its proof. 9) the theorem is an obvious consequent of lots of other theorems. 10) it holds for all right triangles and for all Pythagorean triples (= a set of three positive whole numbers x, y and z such that x^{2} + y^{2} = z^{2}, e. g., 3, 4, 5 and 5, 12, 13). 11) the theorem leads directly to the famous Fermat's theorem x^{n} + y^{n} = z^{n}.

Suppose you are to prove the Pythagorean theorem. What proof (geometric, algebraic, etc.) do you prefer? Study the models of proofs in books devoted to the Pythagorean theorem available in the library, choose one up to your liking and demonstrate it in class, expressing all the formalized statements of the proof in words.

3.1.13. Èç êíèãè An Introduction to the Abstract Algebra. F. M. Hall (1964)