The needs of life and everyday practice determined the emergence of mathematical knowledge among the ancient Egyptians. In the third millennium BC, a state emerged in the Nile Valley. The work of his tax office would have been unthinkable without the ability of an extensive staff of scribes to solve at least the simplest arithmetic, and sometimes even more complex, algebraic and geometric problems. It was impossible to count livestock, determine the size of field plots, calculate taxes, and construct majestic buildings without certain mathematical skills. Originating in ancient times, Egyptian mathematics had a great influence on the development of mathematical science in neighboring countries, especially in Greece. The" father of history " Herodotus wrote: "I believe that geometry was invented there (in Egypt), and from there it came to Greece." 1 Archimedes spent many years of his life in Egypt, getting acquainted with the mathematical achievements of this country. In general, ancient Egyptian mathematics, although it was inferior to Babylonian mathematics in its development, nevertheless made a very significant contribution to the treasury of universal culture.

Egyptian numerals were invented in ancient times, apparently simultaneously with the signs of hieroglyphic writing. These figures are quite simple. So, small vertical dashes were used to write numbers from one to nine. A sign resembling a shackle or horseshoe (shackles for cattle) was used to indicate ten. The image of a twisted rope served to record the concept of a hundred. The lotus stalk represented a thousand. A raised human finger corresponded to ten thousand. The image of a tadpole was a symbol of the hundred thousand. The figure of a squatting deity with his hands raised represented one million. The Egyptians used the decimal system of calculation, in which ten characters of the lowest row could be replaced by one sign of the next step. So, ten dashes corresponded to one ten sign. Ten tens could be replaced with one hundred. Ten hundred characters were equivalent to one thousand, and so on. The Egyptians wrote most often in a horizontal line from right to left, first depicting larger digital signs, and then smaller ones. But they did not come up with the positional principle of using numbers, in which the dignity of the latter depends on their location. Ancient Egyptian numerical records for this reason seem rather cumbersome.

Addition and subtraction operations were carried out, however, even with such a digital signature.

1 Herodotus, II, 109.

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the system with ease. For this purpose, digits of the same denomination were added or subtracted, followed by replacing, if necessary, ten characters of the lowest category with one character of a larger value. But multiplication and division were performed with great difficulty. Here, the doubling method was used, which consisted in doubling the multiplier until the desired product was found, or the divisor was doubled until the number was found, multiplying by which the specified divisor could get the specified divisible. Let's explain what was said: suppose a scribe was given a task to multiply 6 by 8. To establish the product, he first learned that 8 = 2x2x2, and then successively multiplied 6 by 2, then 12 by 2, then 24 by 2. If it was necessary to divide 48 by 6, then the action was performed in reverse order: scribe I chose a number that, if multiplied by 6, would give 48, and I reasoned this way: if you double 6, you get 12; if you double it again, you get 2 - 1; if you double 24, you get 48. This means that the number you are looking for is equal to doubling three times, i.e. 8.

The Egyptians also used fractions, but mostly those where the numerator was one. To express, for example, 5/8, they repeated 1/8 five times - In hieroglyphic writing, the fraction was represented by the sign of the human mouth (in hieratic writing - in the form of a dot). Under this sign, a number was written out, indicating which part of the integer corresponds to this fraction. Some fractions were expressed in hieroglyphs specifically. So, 1/2 was indicated by a stylized image of an edge; 1/4-by an oblique cross. There were special signs for fractions 2/3 and 3/4. Specific were also fractions that conveyed the concept of a part of the measure of bulk solids "hekata", equal to 4.785 liters. Here the scribes specially processed the legend of the struggle of the evil god Set with the god Horus, the son of Osiris. According to an ancient myth, Set tore the eye of Horus apart, but the wise god Thoth restored it. Images of certain parts of the torn eye of Horus began to designate fractions in 1/2, 1/4, 1/8, 1/16, 1/32 and 1/64 "hekata".

The measure of length for the Egyptians was the elbow, equal to 52.3 centimeters. The elbow, in turn, consisted of 6 palms, and each palm was divided into 4 fingers. The main measure of area was considered to be a cross-section of 100 square meters. to your elbows. A special "river measure" was equal to 20 thousand cubits (10.5 km). The main measure of weight "deben" corresponded to about 91 grams.

The ancient Egyptians established mathematical patterns and found ways to solve specific problems experimentally. As a rule, the logical reasoning used to justify the correctness of solving mathematical problems is not known. It is therefore difficult to judge whether they themselves had a sufficiently deep understanding of the mathematical regularities they had discovered. Scientific treatises of the Egyptians of that time with mathematical analysis have not reached us. It is quite possible that they did not exist at all. Of course, even in the time of the Ancient Kingdom, that is, in the third millennium BC, pyramid builders had to be able to solve complex geometric problems. Apparently, long before Pythagoras, they were practically aware of the features of a right triangle with sides related to each other as 3: 4:5.However, it is difficult to say how deeply they delved into the geometric patterns that they used when they discovered them gradually, purely experimentally. So, the classic, now well-known form of the pyramid of Khufu (Cheops) was found by the Egyptians after a long search, at the source of which was the more ancient, stepped pyramid of Pharaoh Djoser, built by the famous architect and doctor Imhotep (the prototype of the Greek physician Aesculapius).

The level of mathematical knowledge of the ancient Egyptians is currently judged mainly by the preserved mathematical papyri. The largest of these are the London mathematical papyrus "Rind" 2 and the Moscow mathematical papyrus from the collection of the Pushkin Museum of Fine Arts 3 . Stored in the British Museum, The Rhind was first published in 1877. This text, which contains solutions to 80 problems, was rewritten in the 18th century BC, that is, during the Hyksos period of domination in Egypt, from some older mathematical work of the Middle Kingdom era. The opening lines of the papyrus say: "Recorded

2 See T. E. Peet. The Rhind Mathematical Papyrus. L. 1903.

3 Cm. W. W. S truve. Mathematischer Papyrus des staatiichen Museums der scho-nen Kunste in Moskau. B. 1930.

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This book is in the 33rd year of the reign, the month of the 4th flood, of King Aa-Userr of Lower Egypt, gifted with life, according to an ancient manuscript entry made during the time of King Ni-maat-Ra of Upper Egypt. Yahmog the scribe wrote down this copy." As noted by papyrus publisher T. Peet, the document "is not a mathematical treatise in the modern sense, that is, it does not contain a series of rules for solving problems of various kinds, but consists of a number of examples." 4 This concrete character is characteristic of all the Egyptian mathematical texts known to us. Vygodsky noted that all of them "contain either the scheme of the solution or its verbal recipe, but they do not contain either an analysis of the problem or a justification for the given recipe" 5 .

The problems presented in Rind are very diverse and mostly have purely practical significance: calculating the area of a field or the capacity of a basket and barn, dividing property between heirs, and so on. Some papyrus problems (NN 24-38) can be called algebraic, since in them,as in problems NN 1, 19 and 25 of the Moscow papyrus, we are talking, in essence, about solving equations in which the concept of the unknown ("x") corresponded to the word "heap". Egyptian mathematicians also knew how to raise a number to a power and extract the square root. The Rind contains problems of theoretical interest, although their presentation is given in the dogmatic form of a ready-made scheme without any analysis or proof. So, in problem N 64, it is proposed to divide ten measures of grain between ten persons in such a way that the difference in the amount of grain received by them forms an arithmetic progression. The task text reads: "You are told to divide 10 'hekat' of barley between 10 people so that the difference between each person and his neighbor is Uz 'hekat' of barley. The average share is 1 "hekat". Take 1 out of 10; the remainder is 9. Make up half of the difference: this is 1/16 "hekat". Repeat it 9 times. Here is the result: 1/2 and 1/16 "hekat". Attach it to the middle lobe. Now you must subtract 1/3 "hekat" for each face until you reach the end of"6 . At the end of the problem, all ten fractions are given, and addition checks the solution, showing that the sum is really equal to 10.

In problem N 79, we are talking about a geometric progression. The condition refers to 7 houses, 7 cats in each house, 7 mice eaten by each cat, 7 ears of corn eaten by each mouse, and 7 measures of grain produced by each ear. You need to determine the total number of all houses, cats, mice, ears of corn, and measures of grain. The correct answer is 19507. Of exceptional scientific interest is problem No. 50 for determining the area of a circle. The method of solving it is extremely simple: it is proposed to square 8/9 of the diameter of the circle. The essence of the method is as follows. The Egyptians noticed that the area of a square with sides of 8/9 of the diameter almost completely corresponds to the area of a circle. In fact, if you fit a square into a circle, the area of the square will be smaller than the area of the circle. On the other hand, if you fit a circle into a square, the area of the square will be larger than the area of the circle. But if you build an intermediate square, its area will almost coincide with the desired area of the circle. The sides of this square should be 8/9 of the circle's diameter. The resulting slight discrepancy was of no practical significance to them. As modern researchers have found, a similar inaccuracy would have been obtained if the ancient Egyptians had determined the area of a circle by the formula S =nd 2 / 4, taking the value for π not in 3.14, but in 3.16. The question, however, is whether the Egyptians had an idea of the number π as the ratio of the circumference of a circle to its diameter. Apparently, they still didn't. Therefore, it is hardly fair to declare the discovery of a number close to the true value of π as a merit of ancient Egyptian mathematicians.

The Moscow mathematical papyrus, which contains 25 problems, testifies to the success of the ancient Egyptians in mathematics. It was rewritten in the Middle Kingdom era (during the XII dynasty) from an older text, studied first by the leading Russian Egyptologist Academician B. A. Turaev, and published in 1930 by Academician V. V. Struve. The papyrus provides solutions to quite complex problems, such as determining the volume of a truncated pyramid, determining the surface area of the hemisphere. Really

4 T. E. Peet. Op. cit., рl. A, p. 33.

5 M. Y. Vygodsky. Arithmetic and Algebra in the Ancient World, Moscow, 1967, pp. 57-58.

6 T. E . Peet. Op. cit., p. 122.

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problem # 14 looks like: "If you are called a truncated pyramid 6 cubits high, 4 cubits in the lower side, 2-in the upper side, calculate the area from four, squaring it; it turns out 16; double 4, it turns out 8. Calculate from two, squaring it; it turns out 4. Add up those 16s with those 8s and those 4s; you get 28. Calculate 7z from 6; it turns out 2. Calculate 28 twice; it turns out 56. See: 56. You found it right! " 7 . Even more difficult is the task N 10-to determine the surface area of the hemisphere. The essence of the solution was to multiply the diameter of the hemisphere by the length of the semicircle forming this hemisphere 8 .

Ancient Egyptian mathematics has gone through a centuries - long path of development from simple accumulation of facts and observations to gradual attempts to comprehend them and generalize them. Even if we assume that the theoretical depth of analysis of the Egyptian mathematicians was insufficient from the modern point of view, we must not forget that these were some of the first steps that humanity took on the path of understanding the secrets of nature. In this sense, it is difficult to overestimate the significance of the discoveries of obscure experts in mathematics of ancient Egypt.

7 W. W. Struve, Mathematischer Papyrus..., S. 135.

8 See ibid., pp. 157-169.

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