All the ancient canons, which proceeded from mathematical regularity, natural forms of the human body structure, recreated on this basis the system of proportions, which was the result of artistic and figurative generalization. The choice of measures in the work of masters is a secondary process. The Egyptians, being empiricists, found perfect, ideal proportions, which were the basis for canonical types of images. Numbers, becoming sacred, began to be thought of as the embodiment of the divine principle.

When we talk about measurement, we assume two forms of it - speculative, based on representation and forming the sphere of philosophical concepts, and concrete-practical, based on the bodies and objects of the visible world. In both cases, we are dealing with the categories of space, time, and motion.

Movement is a process that causes changes in time; however, there is also a certain long path that can be measured by linear measures. If the path runs in three-dimensional space, then you will need volume measures to fix it. All measures are interconnected; the mass of bodies and the direction of their movement in space and time can be grasped with the help of thought and a dimensional tool, using for clarity such geometric shapes as triangles, squares, cubes, circles.

Mechanical motion has a quantitative characteristic and obeys a visible order. According to its laws, the building plans of temples and tombs are developed, plastic compositions that acquire an external immobility in a complete form with potential movement that takes hidden, symbolic forms.

The symbolic connection between the divine and human principles can be traced in the iconography of the ancient Egyptian gods, many of whom in their hypostases take anthropomorphic images; according to the ancient Egyptian versions of the origin of the world, gods and people, the gods gave people their image. Statues were represented as" bodies " of gods, receptacles of spiritual life. When a person after the resurrection became the incarnate Osiris, his spiritual substance found itself in a material shell, i.e. in a body. Hence the spiritual and the physical were a single whole among the Egyptians. The Memphis legend of the origin of the world says: "He (Ptah - NP ) begot the gods, he created the cities, he founded the nomes, he placed the gods in their sanctuaries, he established their sacrifices, he founded their temples, he created their bodies according to the desire of their hearts. And the gods entered into their bodies of every wood, of every stone, of every clay, of every stone.-

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Table 1

Oko Hora Udmsat, containing basic divisions into fractions

things that grew on it and in which they took their images. And all the gods and their Ka (counterparts) gathered around him. - N. P.)" 1 .

The Egyptians called the image the word twt (tut); since the time of the Ancient Kingdom (III millennium BC), this word has also been applied to a statue; it could be images of a god, pharaoh, or any person, regardless of the iconographic type, material, or size. The same word denoted a figure executed not only in a round sculpture, but also on a plane. The noun twtw can be translated as similarity 2 . So here, in the ancient Egyptian text, we first encounter the understanding of the image as a likeness. The ancient Egyptian phrase twt'nh (tut-anh) literally means "living statue". The Egyptians used the same word twt for the reflection of a figure in the eye, which corresponded to the Greek word ("image", "figure").

By means of mythopoetic figurative symbolism, the Egyptians did not contrast concrete and abstract vision. The eye of Ujat (the life-giving eye) carried the ability not only to resurrect the dead, but also served as a dimensional unit. According to the ancient Egyptian myth about the duel between Horus and Set, as a result of which the evil Set pulls out the eye of Horus, the eye returned to Horus gains magical power, its image becomes an amulet.

In the measurement system, fractions were indicated using the eye (Table 1). According to one version of the myth. Seth took the eye and cut it into 64 pieces. Thoth, the God of counting and wisdom, fused all these parts together, but 1/64 could not be found. The phases of the moon symbolically represented the constantly recurring struggle of the Chorus and Set. So, the image of parts of the dissected eye became the designation of fractions: the pupil corresponded to 1/4, the eyebrow-1/8, the corners of the eyes along the eyelid-1/2 and 1/6. The measure hnw (henu) formed the following row: 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, which coincided with the duration of the musical and rhythmic structure. As a result, the sum of this series was 63/64 with a loss of 1/64 of the whole.

Not only the Egyptians, but also other peoples of antiquity and the Middle Ages, independently of each other, created their own measuring system based on the proportions of the structure of the human body and its parts. But if the Egyptians were hiding behind this philosophical and religious views, then other peoples used similar relations purely practically, finding them simple and convenient.

The works of ancient masters-architects, sculptors, painters-impress with the constructive logic of thinking, clarity of form, proportionality of all elements of the composition, harmony of proportions, which is the basis for the synthesis of all types of art, starting with monumental architecture and ending with monuments of small forms-from-

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the division of art craft. The constructive clarity of the thinking of ancient artists is organically combined with the depth of imaginative content, with their inner spirituality. Works of art serve as the main source that allows us to approach the study of the creative method of ancient masters. Even in later epochs, up to the Middle Ages, working methods were considered secret. As noted by Academician B. A. Rybakov, "the mystery of calculations and recipes was characteristic of all medieval masters: even passing on the legacy of teachers and their experience to students, they tried to encode their advice" 3 .

Based on studies of art monuments, a clear pattern has been found in the techniques and methods of work of masters, which allows us to deduce the canon system for Ancient Egypt .4

The Greek word canon in the original sense meant a measure, a rule. The root of [Greek. it is found in several variants of a number of Semitic languages .5 Words with this root in these languages meant the noble reed (orundo donax), from which the rocker arms for scales and measuring rods were made. The latter served as a real standard of length measure. Semites in remote times associated the noble reed with a certain measure of length (Ezekiel 40: 5) or balance in the sense of the yoke of the balance (Isaiah 46 : 6). The transfer of this meaning from the sphere of specific application to the field of speculative meaning was already made by the Greeks. Hence, in the original understanding, the canon was reduced to a purely practical aspect of measurement. In the future, the fixed measure was represented as an ordered quantity and then began to replace the canon in its meaning of order, norm, and law.

Measures among the peoples of all countries went back to the proportions of the human figure, and therefore are based on anthropomorphic units. The great philosophers and artists of antiquity - Herodotus, Plato, Vitruvius-knew and wrote about this; of the Renaissance-Pierro della Francesca, Leonardo da Vinci and our contemporaries-A. F. Losev, S. S. Kobuladze, B. A. Rybakov, I. P. Shmelev, the French architect A. Fournier de Cora, the Norwegian artist E. K. Kieland and many others others. Plato wrote in the Timaeus that it is impossible to establish a connection between two quantities without a third-and this last one unites them, because there is no better connection than that which alone makes several things one .6

Among the various proportional relations, there is one that has the most perfect balance of parts for the eyes - this is the golden ratio. The golden Ratio is not a field of mathematical abstraction, but a natural principle for expressing the laws of equilibrium. The Greek sophist Protagoras formulated his thesis as follows: "Man is the measure of all things"; the Egyptians expressed the proportionality of the proportions of architecture through intermediary gods in their humanoid hypostasis-Thoth, Seshat, Nut. The Egyptians early came to believe that the universe was based on order, which they attributed to the divine structure. The regularity observed in the surrounding world, in the structure of the cosmos, the cyclical nature of terrestrial processes, the calculation of the calendar and the time of Nile floods-all obey numerical symbolism. Thus, geometry, being sacralized, remained in line with the urgent tasks.

In extant mathematical papyri, texts containing economic regulations, and simply in simple exercises in the field of arithmetic and geometry, the measure of cubits is constantly found. Elbow measurements were applied to both large and small items. There were several types of cubits, which were a kind of measurement standards that carried large and small units.

The ancient Egyptian cubit as a measure has been mentioned since the IV Dynasty (Ancient Kingdom). In Egypt, there were several types of cubits - among them the "Pharaohs" elbow mh

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Table 2 Comparative measures of small and large (Pharaoh's) cubits

nsw-t (mech-bear) and the so-called small elbow - mh srr (mekh-scherer), which is 6/7 Pharaoh's (Table 2). The word mh denoted the numerical concept of the elbow, the numerical value followed this word 7 . The length of the cubits ranged from 52.5 to 54 cm, corresponding to the dimensions of the cubits of measuring models found in tombs. There are several similar exhibits in the Museum of Egyptian Art in Cairo, which at one time probably were standards of measures.

For the classical type of elbow, which combined symbolic and constructive functions, one can take the elbow that was in the Museum of Egyptian Art in Turin. Its research was carried out by the largest Egyptologist R. Lepsius 8, who set its length at 52.5 cm. The Turin elbow was made of wood (Table 3). On the opposite side of it there are various divisions: fingers, palm, fist, small and large spans, as well as elbows: dsr, rmn elbow, taken from the shoulder, and the "Pharaonic" elbow. Correspondences between these elbows are established by dividing the standard elbow into 28 parts, where each corresponds to a finger. In a parallel row, there is a list of 28 names of the gods, which is read from right to left. The first of them is the name of Ra, who opens the Great Nine. Then follow the eight gods of the Ennead, followed again by Ra, followed by Horus with his sons, and Thoth in the 15th finger, which corresponds to the XV nome of Upper Egypt - Hermopolis, the place of dwelling and worship of this god. In general, the Turin locus, in addition to 28 gods, also contains a list of nomes: 22 of Upper Egypt and 6 of Lower Egypt. The cubit under consideration thus establishes a comparative comparison of dimensions expressed in different units in the names of gods and nomes symbolically related to each other.

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For the first time, the connection between the Egyptian system of linear measures and the units of the canon of proportions was pointed out by R. Lepsius, 9 and this is confirmed by texts, mostly mathematical and partly concerning metrology, as evidenced by Herodotus.10 According to Herodotus, in Egypt, since ancient times, it is possible to establish the presence of two interconnected measures of length - the small cubit and the "Pharaoh's" cubit. The first one was probably the initial measure, a kind of standard of measurement, which was used for everyday purposes almost from the pre-dynastic period and up to the Sais XXVI dynasty (VII century BC). After this time, the small cubit practically fell out of use and was replaced by the "Pharaoh's" one, which was directly dependent on the small one. The small elbow corresponded to the length of the forearm from the elbow to the end of the thumb, which was approximately 45 cm. It also included smaller divisions - six palms or, more precisely, hands in a width of about 7.5 cm. This measure, in turn, was divided into four fingers, and the fingers, in accordance with the ancient Egyptian system of fractions, were divided into halves, thirds, quarters, eighths, etc. The smallest fraction of the division-1/16-was the smallest unit of length of the metric system and was approximately 1 mm. Naturally, the name "metric" system is used in our understanding. Among the Egyptians, there is also an image of a hand with a measuring instrument; reading this sign causes different interpretations.

As smaller divisions, the size of the fist, the width of the hand and fingers were used. Two-thirds of the small elbow is the length of the arm from the forearm to the wrist. The four small cubits are four times contained in the full height of the figure. This value was denoted by the Egyptian word hpt (xenem) 11 and corresponded to a measure of one fathom of water - about 1.82 m. The word hpt in the ancient Egyptian language meant "to embrace", "to embrace" and had as a determinative the image of hands in girth By means of this value, the growth of the figure was recorded.

In ancient Egypt, elbows as measurement standards were used not only in the form of special objects, but also fixed directly on monuments. On the pedestal of the statue of the seated Pharaoh Ramesses II of Luxor, there is a black horizontal stripe applied on the side of the pedestal, where the deities of the Nile are depicted, linking the sma heraldic emblem, symbolizing the unity of the South and North of the country. The horizontal bar embedded en creux is at the level of the colossus ' knee. Its length of 54 cm corresponds to an elbow of 28 fingers. This is one of the most ancient measures, the use of which passed to the Copts and Arabs; it is also used in measuring the rise of the Nile water level during floods. Such a measure, which is mostly found on stone statues and in compositions where the Nile deities or their emblems are present, has become conventionally called the "black elbow".

A similar measure - a cubit of 54 cm-is also found in the system of Old Russian measures, but, as noted by B. A. Rybakov, "in Ancient Russia from the XI to the XVII century. there were seven types of fathoms and cubits that existed simultaneously" 12 . This means that the ancient Russian master architects and sculptors used the same measures as the Egyptians, and, of course, independently of each other. The ratios of proportions were based on the simplest constructions of triangles, squares, rectangles and their diagonals, which gave the main values used in construction practice and in sculpture.

By depicting a full-length human figure, the Egyptians specified a measure of one fathom of water, or one breaststroke (about 1.82 m). In addition to purely linear measures, the Egyptian fathom could transmit both linear and angular values. This measure is comparable to the arcs of a circle of 360 degrees divided by 60 degrees and 45 degrees . As an arc measure, the bras correlated with the time cycles of the Egyptian calendar-so in the following way:-

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Table 4

Fragment of the composition from the Queen's mastaba. Meresankh III (IV din. The ancient kingdom. Saqqara.)

360 contains 36 braces, which corresponds to 36 decans of the time cycle. The length of an arc of 60 relates to the chord in the same way that the width of the reach of the outstretched arms relates to the height of the human figure, which can be expressed in terms of Thus, a figure inscribed in a semicircle of radius equal to its height has an arc of 60 in its coverage.

The composition with the image of a male figure with arms spread out to the sides has been known since the Ancient Kingdom. In addition to the one-brace measure, this figure is the unit that balances most of the elements of the composition in proportions. A piece of cloth is draped over her outstretched arms, the ends of which hang from her wrists. This measure is also contained in the length of rope stretched from the last figure to the first of the group of men pulling the snare (Table 4).

The measure of one breast-breadth, corresponding to four small cubits, belongs to the Egyptians. Later, researchers and creators of various canon systems, starting from antiquity and ending with the Renaissance, proceeded from such relations. Their commonality is based on the structure of the human body in the golden ratio, and therefore the methods of expressing proportions are largely similar. The figure's height of 4 cubits (6 palms across four fingers) is accepted by both Vitruvius and Leonardo da Vinci. In their works, the figure with outstretched arms fits into a square and a circle. The center of the circle falls on the navel, which is also the point of division of the figure in the golden section. In the future, the measure of outstretched arms in 4 elbows will be accepted in the ancient and Renaissance canon for an ideal figure (Vitruvius, Leonardo da Vinci).

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Table 5

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The construction logic of the masters of Ancient Egypt proceeded from the same principles as the architects of subsequent epochs - the Arabs, the masters of Ancient Russia. The book of B. A. Rybakov repeatedly mentions the work of Abdul-Waf (940-998) and the contemporary Tmutarakan graph 13 . The simplest relations derived in it were known to the Egyptians and used by them: the construction of double squares, an equilateral triangle, an equilateral square, a square inscribed in a circle, as well as a system of so - called "dynamic rectangles" based on the ratio of the square and the diagonals of subsequent rectangles, giving-values:, etc .14.

In the work of B. A. Rybakov, there is an indication of the correlation of linear measures of length: cubits - with a circle derived by the Egyptians arithmetically, which implies that a small elbow of 46 cm is expressed through the side of a square inscribed in a circle with a diameter of "Pharaohs" cubit (52 cm) - the ratio or 0.886 15 .

A similar kind of geometric conjugacy of quantities can be traced in Old Russian measures. This regularity in the geometric conjugation of the Old Russian "straight" and "oblique" fathoms was deduced by B. A. Rybakov [16], noting that a straight fathom is a side of a square, and an oblique one is its diagonal -

This principle of the ratio of the sides of a square with a diagonal is the basis for the method of "growing" a composition constructed by sequentially inscribing squares in circles originating from one center, and the radius of each subsequent circle will be equal to the diagonal of the square inscribed in the previous one. Compositions of art monuments of Ancient Russia are based on this principle; similar relationships can be found in Egypt and Mesopotamia.

In the construction practice of the Egyptians, a small elbow of 0.45 m was more often used as a measure. This is confirmed directly by the monuments themselves, which indicate the calculations made by the ancient Egyptian master. An exceptionally rare group of monuments consists of plans of architectural structures. Among them, you can note the plan of the tomb, inscribed on papyrus, stored in the Turin Museum. The publication of this document belongs to Lepsius 17 . In 1889, the Egyptologist G. Derssi, during excavations at Biban el Molouk, found a large number of ostracons covered with drawings and inscriptions; among them was a plan (Table 5) similar to that depicted on the Turin papyrus 18. In contrast, the hand-drawn outline of the tomb on a 0.835 m long stone fragment was a cursory sketch, probably for auxiliary purposes. Although the plan is drawn on the rough surface of the stone very roughly (and the stone was broken into four pieces), this sketch allows you to establish the units of measurement used in measuring the royal tombs.

A comparative comparison of several plans makes it possible to consider them canonical, because they have an identical sequence of rooms descending into the depths of the rock, starting with a sloping staircase and ending with a sanctuary carved into the rock. Thus, the plan, being a kind of model of the royal tomb, could also contain canonical proportions, which correlated the dimensions of individual religious premises and the passages connecting them (Table 6).

On the plan of the ostracon, the designations of quantitative measures are faintly distinguishable, because the paint is almost worn off, only a yellowish outline remains, by which you can make out some dimensions: the passage leading to the sacred chamber was 30 cubits long, 6 cubits wide, 7 cubits high. A small cubit (0.45 m) was taken as a unit, which is also confirmed by comparison with the plan of the Turin papyrus. According to the inscriptions and marks in the plan, you can determine the purpose of religious premises associated with the litany of the sun.

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Table 6

Comparative data of dimensions in the plan, presumably related to tomb No. 6

Purpose of premises

Size according to plan

Measurements of tomb No. 6 by actual dimensions

1 corridor

30 cubits = 13.50 m

door 1.00 m = 18.00 m corridor 17.00 m

2 corridor

30 cubits = 13.50 m

door 1.00 m = 11.00 m corridor 10.00 m

3 corridor

30 cubits = 13.50 m

door 1.00 m = 12.45 m corridor 11.45 m

1 camera

10 cubits = 4.50 m

door 1.00 m = 4.45 m room 3.85 m

2 camera

20 cubits = 9.00 m

door 4.64 m = 10.54 m room 6.90 m

3 camera

25 cubits = 11.25 m

door 1.40 m = 12.30 m room 10.90 m

To compare ancient Egyptian measures and metric system measurements, G. Derssi provides a comparative table of measurements based on the drawing of the plan in cubits (taking into account that the plan was made by hand) and meters, respectively, with real measurements of the interior of the tomb. The ostracon plan found in the tomb of Ramesses VI bears a strong resemblance to the dimensional plan. However, the author calls into question whether the plan belongs to this particular tomb, since it is a kind of canonical model; since the differences in the dimensions of tombs with this plan are very insignificant, it could also belong to the tomb of Seti I, which lies near the location of the ostracon.

It is interesting that B. A. Rybakov found drawings-"babylons" of the IX-XII centuries 19 on the Taman hillfort and the Old Ryazan hillfort , which show a connection with the building principles of the ancient Egyptians and their measures of length. Cut on a slab of pink sandstone squares - "babylons" contained measures of 19 cm (the side of the largest square, equal to the Russian "small span" - 1/8 straight fathom). The same measure is 1/8 of an ancient Egyptian fathom, or half a small cubit. This convergence is due to the fact that the source of measures was the human body.

Interesting information is contained in the chapter of the book by B. A. Rybakov, which deals with the design work of the architect. The author draws this information from the Slavonic version of the "Tale of Solomon and Kitovras" (XII c.). "outline" is mentioned here, which is equivalent to the concept of a plan. Wooden measures of 4 cubits each were used as standard measuring instruments. "Obviously," writes B. A. Rybakov, "a magical 'architect' Kitovras was endowed by the author of the legend with real accessories of the Russian architect in the form of fathoms made of wood, divided into 4 cubits " 20 .

The Egyptians specified the fathom (the so-called sea fathom - 1.82 m) by means of the full height of the figure containing 6 foot lengths (6 feet). In Russia, the" sea fathom " was defined by B. A. Rybakov as approximately 183 cm, i.e. four cubits of 45.7 cm. Like the image of the Russian centaur with wands on the walls of Ge-

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Orga Cathedral in Yuriev-Polsky (1230), an ancient Egyptian master around the XXVIII century BC depicted the real architect Hesir in relief on the wooden paneling of his mastaba tomb in Saqqara.

In the New Kingdom era, the proportions of the canonical figure were metrically expressed through the" black " elbow of 54 cm. The structure of a person is comparable to the Egyptians with an elbow of 24 fingers, which corresponded to a quarter of the coverage, or 1/4 of the brace-a unit of 96 fingers. Therefore, a cubit of 54 cm is a sacred measure, different from the natural proportions; it is obtained from the calculation of the size of one breaststroke of 45, which is about 1.85 m. Then one finger is approximately equal to

The" black " elbow consists of 28 fingers, hence its size is 54 cm. Thus, by means of the "black" elbow, sacred proportions were established, in accordance with which the masters made statues and built temples, as well as natural ones emanating from the structure of the body. From this we can deduce the canonical measures of proportions found in ancient Egyptian monuments, which constitute a constant correspondence of parts and the whole in the adopted measures of length:

1 palm = 4 fingers

1 fist = 6 fingers

2 palms = 8 fingers

3 palms = 12 fingers = 1 small span

3.5 palms = 14 fingers = 1 large span

4 palms = 16 fingers = 1 elbow dsr

5 palms = 20 fingers = 1 elbow rmn

6 palms = 24 fingers = 1 small elbow

7 palms = 28 fingers = 1 "pharaoh's" elbow

1 breaststroke = 96 fingers

1/4 breaststroke = 24 fingers = 46.2 cm = 1 small elbow

1/6 breaststroke = 16 fingers = 30.87 cm = 1 elbow

So, the measures of elbows discussed above contained linear and volumetric values. Evidence from ancient Egyptian sources can serve as confirmation. The texts contain considerable information, but do not provide any visual material or practical disclosure of its application, which is especially valuable for the study of monuments.

By means of cubits, it was possible to find the expression of volumetric measures - this is confirmed by the models of cubic cubits; in addition, the Egyptians also used special measures of volume.

The already mentioned measure of hnw, known as the fluid measure, was about 0.45 liters; the measure (hekat) was equal to 1/10 of hnw. The relationship between one cubic cubit and the volume measures in question can be traced in Table 7. The measure (container for grain, mass in the amount of 20 known since the Middle Kingdom) was equal to two-thirds of a cubic cubit and was used to measure spherical space. At the same time , a measure came into use - a grain measure of about 4.785 liters, 22 also used for myrrh, frankincense, and even gold. It contained divisions into 64 parts, which was associated with one of the versions of the myth about the duel of Horus with Set.

The derived relations are confirmed by problem N 64 (1), contained in the mathematical papyrus Rind 23 . It clearly illustrates the mathematical calculation methods of the ancient Egyptians, who used the decomposition of numbers into series to replace multiplication by addition and division by subtraction (Table 8). The essence is to divide 10 measures of barley (10 for ten people. The word prw probably meant difference. And, indeed, the difference is 1/8 . Subtracting 1 from

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Table 7

Comparative table of volume measures in their ratio to cubic cubit

10, we get the remainder 9. If one had a difference of 1/8, then half of the difference will be 1/16. Multiplying 1/16 by 9 can be replaced by adding 1/2 + 1/16 part . According to this condition of the problem: 10 measures (10 ) fall on 10 people-the Egyptian scribe develops the solution method, representing the measure in 1 as 1/2 + 1/2, splitting the unit in half. Thus, it has a row (the lower one is the 6th digital line of the papyrus).

Here, there are a number of odd numbers whose numerators make up an arithmetic progression. The condition of the problem may seem primitive at first glance, connected with a narrowly practical goal, but the course of reasoning indicates that the Egyptians were able to derive a beautiful generalized solution from a specific simple example with finding the arithmetic mean term, for which a special term is used - "middle part".

Speaking about the meaning of ancient Egyptian mathematics, one should not primitivize it, as the well-known German specialist in the history of ancient mathematics O. Neugebauer is inclined to do 24 . Series of numbers of progression, harmonic proportions-this is the material that the Egyptians perfectly mastered in practice and in theory. The choice of fixed measures is not accidental, and it is based on a deep understanding of the relationship of phenomena.

The study of ancient Egyptian papyri strongly proves that Egyptian mathematicians were well versed in the field of harmony of numbers and proportions. The numerical relationships that express the structure of the compositional structure of monuments of fine art and music coincide. The metrorhythmic units of a piece of music are integers, halves, quarters, eighths, sixteenths, thirty-twoths, and sixty-fourths. Acoustic laws of sot-

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Table 8

Problem N 64(I) of the Rind papyrus

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They correspond to the same harmonic proportions that are laid down in the basis of measures that meet the canonical rules for constructing the human figure. Schematically, this canvas can be represented as follows: the segment taken as an integer is divided in half into two units (1 + 1), then into 3 parts (2 + 1), then, respectively, into 5, 9, 17 parts - (4 + 1), (8 + 1), (16 + 1). We get a series of numbers that are multiples of two: 1, 2, 4, 8, 16. In the Rind papyrus, the expression of numbers through n + 1 has already been found. In this case, a series of numbers n forms a geometric progression. The top row of the table shows the relationship between binary and ternary systems. Similarly, the scale is built on the numbers 2, 3, 5, 9 or the ratio-9:8,2:3,1:1and a symmetrical mirror series 1 : 1, 3 : 2, 8 : 9. The same ratios will correspond to the scale of pure intervals of the musical scale 1 : 1 (unison), 1 : 2 (octave), 2 : 3 (fifth) and in circulation-3: 2 (quart), 4 : 3 (fifth). The number 5 provides the following consonant intervals: 5: 4 (large third), 8: 5 (small sexta), 5: 3 (large sexta). Dissonances are arranged in such ratios-9: 5 (small septima), 9: 8 (whole tone), etc.

The scale of musical overtones gives a natural scale. The natural scale has a parallel with the spectrum, which is divided into a range of colors. In monuments of fine art, measures take on visible forms of expression in proportions, contributing to the coverage of large compositions with a single glance. A piece of music evolves over time. Since the sound itself is large, it covers more parameters and belongs to several spheres at once, having a temporal and spatial extent. But the sound is thus limited in its being. The system of measures of sound, the vibrations of the string remain a mystery; the visible form of expression of these measures are numbers. The number leads to the foundations of musical harmony, which traditionally goes back to Pythagoras in his designation of the order that reigns in the world. However, it is possible that the Egyptians anticipated Pythagoras.

The idea of ancient Egyptian music can be made only indirectly, on the basis of extant musical instruments and their images. Thus, attempts were made to establish the structure of the Egyptian musical scale. The first of them belonged to the Russian scientist A. Machinsky, who chose harps as the most convenient tool for research. If we assume that the proportions of instruments are sufficiently accurately maintained in reliefs and paintings, then the structure of harps can be calculated from the ratio of string lengths to the longest and by comparing the average ratios of similar instruments .25 In ancient Egypt, arc and angle harps were used, the type of which was determined by the shape of the resonator. Images of the angular harp in the hands of the god Bes from the relief of the temple of Isis on the island of Philae are used as a sample by the researcher in the field of canon proportions Schwaller de Lubitsch 26, performing painstaking calculations to detect patterns of numerical relations (Table ). The points that lie at the intersection of the lines corresponding to the divisions: 1, 2, 4, 8 and 1, 3, 9, 27 determine the ratios 1: 2 (octave), 2 : 3 (fifth), 3: 4 (quart) - the result is a harmonious chord C-C-G-C.

The doctrine of harmony is based on the philosophy of numbers. Musical harmony is a combination of sounds in a chord; the merging of parts into a whole is expressed acoustically in sound-pitch relations laid down in the proportions of simple, pure intervals. Harmonic proportions can be represented by a scale of musical intervals that form a strict order based on the law of symmetry. The presence of a measure makes it possible to subordinate elements to each other, forming a whole.

The phenomenon of musical intervals is an expression of order in direct and derived relations; the phenomenon of vibration has a physical nature and a quantitative expression, depending on the scale of the structure of musical intervals. However, the scant information we can glean about ancient Egyptian music is still available to us.-

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Table 9

Natural scale that makes up the harmonic scale of musical intervals

through the medium of visual art or the logical method of calculation and comparison, in no way can they fill up the idea of how it sounded 27 .

The Egyptians understood numerical relations not abstractly, but organically, building compositions of monuments based on them. In the painting of the tomb of Pharaoh Ramesses IX (Table 10), the composition is built according to the ratios of the sacred Egyptian triangle - thus the ritual symbolism echoes the numerical one. The legs of the triangle formed by the body of the snake make up segments, of which the first is equal to half the height of the figure with raised hands, the second-a larger segment of dividing its height in the golden section. The distance from the ends of the fingers of the raised hands to the crown determines the following dimensions : the center of the circle in which the figure is inscribed (without hands), from-

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Table 10

Fragment of a composition from the tomb of Pharaoh Ramesses IX

it is located three such units away from the upper limit, and its radius is two. In the circle we write a pentagon, the sides of which are equal to half the height of the figure with raised hands (1/2 N). Then the figure without hands from the top of the head to the foot (h) will correspond to the height of the pentagon. The following relations are observed in geometric constructions: the diameter of a circle relates to the diagonal of a pentagon in the same way as this diagonal relates to its height.

The measuring system of Ancient Egypt is connected with the mechanism of a grid of squares - a mechanical device that expresses the proportions of figures depicted in canonical poses. This mechanism turned out to be a fairly simple and flexible device that clearly demonstrates the proportionality between the whole and its parts. Proportions determined the choice of measures based on the ratios of numbers considered sacred in ancient Egypt. As already shown, the proportions were based on the height of the figure: the head, mostly equal to the size of the foot, was 1/6 of the height, the arm with the bent hand from the elbow to the fist (the size close to the small elbow) was placed five times in the figure. It should be borne in mind that the size of the elbow is not taken as a fixed abstract measure of length, but in accordance with the proportions of this figure, because it is its elbow, its fist, its foot that act as units of measurement.

Subsequent researchers and creators of various canon systems, from antiquity to the Renaissance, also proceeded from such relations. Their commonality is based on the structure of the human body in the golden ratio, and therefore the methods of expressing proportions are largely similar.

In conclusion, we can say that all the ancient canons, which proceeded from mathematical regularity, natural forms of the structure of the human body, recreated on this basis a system of proportions, which was the result of artistic and figurative generalization. The choice of measure in artistic practice is a secondary process, revealing and fixing the established norms. The Egyptians, fixing the solution found in the course of practice, obtained perfect ideal proportions, which were then used as the basis for canonical types of human activity.

page 20

figures and iconography of the gods; numbers, becoming sacred, began to be thought of as the embodiment of the divine principle.

notes

Mamye M. E. 1 Drevneegipetskie mifi [Ancient Egyptian Myths], Moscow, 1956, p. 84.

A. Erman, H. Grapow. 2 Worterbuch der Aegyptischen Sprache. В., 1955. Bd. V. S. 257 (19).

Rybakov B. A. 3 Iz istorii kul'tury Drevnoi Rus ' [From the History of Culture of Ancient Russia]. Moscow, 1984, p. 82.

Pomerantseva N. A. 4 Esteticheskie osnovy iskusstva Drevnego Egypti [Aesthetic foundations of the Art of Ancient Egypt]. Moscow, 1985, pp. 100-115.

Oppel H. 5 Kanon. Zur Bedeutungsgeschichte des Wortes und seiner lateinischen Entspechungen // Regula-Norma. Philologus, Supplementband Leipzig, 1937. Bd. XXX. Het 4. S. 1.

6 Plato. Timaios, 31. .BC.

Erman, H. Grapow. A. 7 Op. cit. Bd. II. S. 120 (3, 4).

Lepsius R. 8 Altaegyptische Elle und Ihre Eintheilung // Abhandlung der Koniglichen Akademie der Wissenschaften zu Berlin. В., 1865. S. 53.

Lepsius R. 9 Die Langenmasse der Alten. В., 1884.

10 Herodotus II, 149.

A. Erman, H. Grapow. 11 Op. cit. Bd. III. S. 71.

Rybakov B. A. 12 Decree. soch. P. 8.

13 Ibid., p. 91.

Vladimirov V. N. 14 Proportions in Egyptian architecture] / / Universal History of Architecture, Moscow, 1944, vol. 1, p. 7.

Rybakov B. A. 15 Edict. soch. P. 91.

16 Ibid., p. 85.

Lepsius R. 17 Grundplan des Grabers Konig Ramses IV // Abhandlung der Koniglichen Akademie der Wissenschaften zu Berlin. В., 1867.

Daressy G. 18 Un plan egyptien d'une tombe royale // Revue archeologique. Serie 3. P., 1898. V. 32. P. 235-240.

Rybakov B. A. 19 Decree. op. P. 87.

20 Ibid., pp. 83, 84.

Erman, H. Grapow. A. 21 Op. cit. Bd. III. S. 363 (1, 2).

22 Ibid. S. 174(11-15).

Peet E. 23 The Rhind Mathematical Papyrus. L., 1923. P. 64.

Neugebauer O. 24 Exact Sciences in ancient Times, Moscow, 1968, pp. 83, 84.

Matchinsky A. 25 A propos de la gamme musicale Egyptienne / / Kruzhok po izucheniyu Drevnego Vostoka pri Gos. Hermitage Museum, Moscow, 1935. 2. p. 24, 25.

Schwaller de Lubicz R.A. 26 Le Temple de l'Homme. Apet du Sud a Louqsor. P., 1957. V. 1. P. 369.

Pomerantseva N. A. 27 Otzvuk drevneegipetskoy muzyki [The echo of ancient Egyptian music]. Iskusstvo (Appendix to the newspaper "First of September"). 2001. N 23 (239).

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