# Numeric record system of early history: Numbers among Magura cave paintings. Recognition and types. Numeric expressions. Numeric operations. Similarities to Egyptian numeric system.

**Abstract:**

The process of decoding the ancient paintings of Magura cave repeatedly leads to the recognition of numeric records among the paintings. Therefore it has become necessary to define what a number must look like in order to recognize it when on encounter. This has led to the discovery of 234 numeric expressions among the total number of 712 paintings which is 33% of the total volume of the ancient record stored in Magura cave, which determines the paintings as a sophisticated and merely scientific record as long as mathematics is a sign of well-developed cultures.

Indeed, there are thirteen different cases of numeric records in Magura cave, every one of them suitable for a specific field of use. This makes Magura numeral system probably the most sophisticated known to exist even more that one can find at least three mathematical operations there – summation, multiplication and division as well as marking of specific positions and representation of parallel processes and sub processes.

Meanwhile, 27,7% of the paintings deal with the number 3 and its products thus completely describing the various manifestations of the triune Goddess-Mother who was believed to had created the manifested world, meaning that Magura mathematics is mostly oriented to the description of cosmogony processes and events.

After all of this it is of no wonder that Magura numeral system is very similar to the Egyptian one and not only. Roman numeric system is also similar to it and we can even recognize traces of Magura mathematic approaches in our modern mathematics. A culture so developed and bright that it has served as the root of modern, enlightened world.

Full title:

**Numbers among Magura cave paintings. Numeric record system of early history. Recognition of paintings as numbers. Types of numbers and cases of their usage. Numeric operations and expressions.**** Similarities to Egyptian numeric system.**

Short title:

**Numeric record system of early history: Numbers among Magura cave paintings. Recognition and types. Numeric expressions. Numeric operations. Similarities to Egyptian numeric system.**

Author list

Author:

eng. Kiril Lyubchov Kirilov

Affiliation:

Board member of NGO “Archaeological society of Belogradchik”, Bulgaria, Vidin district, Belogradchik, 3900, 8 Petko Kovacha str.

Corresponding author:

eng. Kiril Lyubchov Kirilov

kirilov_kl@abv.bg

**Introduction:**

I am starting with the reservation and the acknowledgement that not everything is clear and known and a lot more needs to be studied in order to finish the theme of numeric records among Magura cave paintings. It is even possible that a new view point or an evidence in the future can totally change the present understanding on the topic and therefore a new research may be published to present different statements. But so far a lot of data and analyses have been gathered so that the present publication on numeric records among Magura cave paintings can be written.

In the first place, it has to be stated that numbers are abstract. Yes, they can describe natural qualities of real objects – weight, length, number of elements. But in the same time numbers don’t really need any object to be bound to in order to exist. Numbers go beyond the necessity of material manifestation.

Indeed, a culture, in turn, does not necessarily need numbers to exist. There may be and probably are cultures which do not deal with measurable features of nature and its objects or phenomena. But a well-developed culture is very probably and even necessarily to describe the world and its phenomena using numbers to state specific dimensions and qualities of what goes on around it. Numbers are necessary to pass on information on important processes, durations, quantities thus allowing for the creation and development of decent and even of sophisticated technology. Of course, numbers serve in no way to mankind if they are not written down in a standard way which can be understood and reproduced by all involved individuals.

It is an important part of my research on Magura cave paintings and on what knowledge has been stored in the paintings to try and find out whether there are numeric records among the paintings and even read the written numbers if such exist. Because, I can suggest various concepts of the meaning of the paintings but if my suggestions do not lead to recognition of specific numbers which describe natural and important processes, this means that my suggestions do not have the vitality to thoroughly read the paintings and are most probably wrong.

I admit that I have former information on the topic as I have managed to read a few numbers along the scenes but now I have to prove that those are indeed numbers and to lay out the proof in a systematic way.

Let’s start with a very simple understanding of what a number should be. A number has to represent some quantity and different numbers have to represent different quantities. Therefore we have to look for similar paintings but with varying numbers of elements as these elements are likely to mean quantities. Figure 1 is a good illustration.

Fig. 1. An example of what can be numeric records among Magura cave paintings.

Here we see that the paintings are similar in construction (in shape) which means that they are referred to similar ideas or even they represent one and the same idea – the idea is an object or some phenomenon. But the paintings vary in the numbers of their elements (in the case – the number of the horizontal lines) thus representing the varying numeric properties of the various specimens of that type of object, idea or phenomenon.

If the number of lines did not matter then all of the paintings should have the same number of horizontal lines. But as they differ in the number of lines, we know that the exact number matters. Therefore the different number of the lines is on purpose and serves to transfer information about a property which can be counted. This is how we know that these paintings are numeric records of a certain type.

Now let’s go forward and test other types of paintings. It is easy because the logic is clear and all the paintings have been indexed and separated in groups of similarity in the Complete catalog of all Magura cave paintings (1).

Fig. 2. An example of what can be numeric records among Magura cave paintings.

Fig. 3. An example of what can be numeric records among Magura cave paintings.

Fig. 4. An example of what can be numeric records among Magura cave paintings.

Fig. 5. An example of what can be numeric records among Magura cave paintings.

The logic is the same: we encounter groups of similar paintings but with differing numbers of repeating elements. Therefore the number of the elements matter and the purpose of this is to transfer information about measurable and countable properties of some type of objects or phenomena. This determines the paintings on Figures 2 to 5 as numbers.

Then another example of paintings is shown on Figure 6.

Fig. 6. Groups of similar paintings.

The task is to check whether these are also numeric records. We see repeating types of paintings which contain repetitions of the same elements but the number of the elements in the various groups is always equal so it does not matter. Therefore, these paintings do not represent numeric records.

Then we can check the symbols T shown on Figure 7.

Fig. 7. Symbols T among Magura cave paintings.

Are these symbols numbers? They are not because the number of their elements is always equal and their counting brings no additional knowledge.

Figure 8 represents a similar situation:

Fig. 8. Two groups of similar paintings.

Nor the symbols Y, neither the symbols V nor Л are numbers as they always consist of the same number of elements. But the symbols V and Л are part of numeric records in the case of paintings shown on Figures 9, 10 and 11:

Fig. 9. A group of horizontal zig-zag lines.

Fig. 10. A group of horizontal zig-zag lines.

Fig. 11. A group of vertical zig-zag lines.

All of these zig-zag lines are repetitions of the same symbols – V (VVVVV) or Л (ЛЛЛЛ) and we can count the number of the elements as the number is different in every paintings. This determines the zig-zag lines as numeric records as well. Then it is logical to ask whether the elements themselves are numbers indeed – we don’t know this so far but it is possible to find groups of paintings where they act as numbers. This will be done later in this publication when the presentation goes to various cases and examples.

Then we have another example of numeric records in the form of stars – Figure 12.

Fig. 12. Numeric records in the form of stars.

All the stars have similar construction – a circle with rays, but all the stars differ in the number of ray therefore they express numbers as well.

There are some other types of numeric records shown of Figure 13.

Fig. 13. More examples of numeric records in Magura cave paintings.

Now we can sum up the different types of recognized numeric records as it is shown of Figure 14.

Fig. 14. Types of numeric records among Magura cave paintings.

There are twelve different types of numeric records used among Magura cave paintings and I want to present one more, unexpected type:

Fig. 15. Mathematical proportions of the elements of the paintings.

The symbols shown on Figure 15 are part of a larger group of similar paintings that I have recognized as the symbols of energy / spirit during my research. Here is their complete presentation – Figures 16 to 19:

Fig. 16. Various examples of the group of spirit symbols.

Fig. 17. Various examples of the group of spirit symbols.

Fig. 18. Various examples of the group of spirit symbols.

Fig. 19. Various examples of the group of spirit symbols.

It is visible that all of these symbols contain a vertical column and a circle or an arc over it. I measured the lengths of the columns and the arcs of some of these paintings and compared them searching for any ratio. Thus I found that many of the symbols represent exact proportions column / arc, for example 1/1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/8 and 1/16. These proportions may “turned over” thus creating their reciprocal which are 1, 2, 3, 4, 5, 6, 8 and 16 – all of them integers. It is obvious that so many integers can not have been integrated into the paintings of this class by chance but this is done on purpose, therefore the paintings are not the product of a spontaneous painting. Even the opposite – the paintings are the result of intentional design and consecutive drawing (but not paintings indeed) which to ensure the creation of the desired proportions.

As I still have not studied the whole group of these paintings which are 108 in total, I do not know exactly how many of them represent such proportions, but the potential is that all of them do so. If so, the total number of paintings which represent numeric records in Magura cave may be 234 out of 712 paintings in total. This means that up to 32,87% of all paintings in Magura cave deal with numbers which is a very significant part. This research must change the attitude to Magura cave paintings which for a very long time has been as to primitive paintings of prehistoric daily routine where chequer paintings were decided to be crop fields, strips of segments – rivers and zig-zag paintings – snakes. It is now obvious that all of these are indeed numbers and 20% to 30% of the paintings are completely abstract as they represent numbers.

Another feature of the paintings is that there are 13 types of numeric expressions among them – why is it necessary to have such a diversity of numeric expressions?

What I think is the following: Magura cave paintings show an intention to represent information with the smallest amount of symbols which allow for the information to be read and complete. There is no better option than combining the phenomenon you want to talk about with its numeric properties in one single painting. Thus you not only reduce the volume of the record but you also make it clear that the number refers to this phenomenon.

On the other hand, and probably this is the most important reason, different types of numbers describe different types of phenomena and objects. Here are the examples.

Figure 20 shows the simplest of all numbers where a single vertical line represents 1 specimen of a certain type.

Fig. 20. One specimen of a certain type.

It has to be kept in mind that this is not the digit 1 as there are no other digits as it should be: 2, 3, 4, 5, etc. This means 1 specimen. Here we have an example of the number two meaning that there are two elements of the same type – Figure 21.

Fig. 21. An example of the number 2.

As the vertical lines are identical, they represent similar objects and even the same type of objects and count their quantity.

Then we have multiple examples of the number 3 – Figure 22.

Fig. 22. Multiple examples of the number 3.

Here we have six examples of the number 3 represented by series of vertical lines and we can find many more expressions of the same number among the paintings – Figure 23.

Fig. 23. Various examples of the number 3 among Magura cave paintings.

Among which examples we also find the triune depiction of the Goddess-Mother or the Materializing Spirit – Figure 24.

Fig. 24. The triune depiction of the Goddess-Mother.

The total amount of all paintings which represent the number 3 is 42 out of 712 paintings in total thus forming 5,9% of the total volume of the records. This determines the number 3 as the most commented number among Magura cave paintings. It is not only this, but many of the numeric records represent the numbers 6, 9, 18, 27 and 81 – all of them divisible by 3. In addition, all symbols of spirit are 108 of which 54 are male, 27 are female and 27 are others – again numbers divisible by 3. Meanwhile, all symbols which contain the element T, which in Magura stands for God-Speech, are 108 – again a product of the number 3.

Thus we have 197 paintings which deal with the number 3 or with its development and manifestation which is 27,7% of the total volume of Magura cave record. Then we may conclude that Magura cave paintings mostly represent the number 3 itself as well as its manifestations. What is the reason for this?

I can only guess as there are no live priests and adepts of Magura school to tell me what is the idea but the very presence of a triune god who creates the universe is a powerful clue. As you saw, similar paintings can be put together into groups of similarity and we can arrange them from the most complex to the simplest thus finding the parent of every group – the simplest painting in the respective group. This can for all the paintings and the analysis reveals three initial symbols which combine and exchange elements thus creating every painting which can be found in Magura cave. This means that Magura cave paintings reveal the creation of the whole world on the basis of three initial elements.

Without having the absolute knowledge on the topic I think that these two reasons explain the large commitment of the paintings to the number 3 and its manifestations.

Then bigger numbers can be expressed in this way – Figures 25 and 26.

Fig. 25. The number 4.

Fig. 26. The number 6.

The principle is clarified through observation of the supplied examples – the total number is obtained through summation of the elements. To find the number in this types of paintings we have to sum all lines, i.e. to count them.

Note that in shown examples all the lines are merely equal in length but there are cases when some of the lines differ in length – Figure 27.

Fig. 27. An example of a more complex type of numeric record with vertical lines where some of the lines differ in length.

Or we can stylize the example to make it more clear and obvious – Figure 28.

Fig. 28. Stylized representation of the above numeric record.

We read these numeric records in the following way: (the total number of the long lines)x1 + (the total number of the shorter lines)x1/2 as the shorter element here has half the size of the longer ones. Thus we get 14×1 + 1×1/2 = 14,5. It is obvious in this case that the half-line should be only one as it means ½ and there is no sense to have two of them as it would mean 1. There should be no sense to have even three of them as it is 1,5 and we can write it as iI but not as iii.

Following the same logic, if we want to express the number 8,5 we should write the following (Figure 29):

Fig. 29. A demonstration on the number 8,5.

On the basis of these examples it seems that Magura culture did not use digits but used to represent the numeric information through the respective numbers of similar elements.

This type of numeric expressions had an advantage when we want to talk about the order and the serial number of an element or about its specific position in a process (Figure 30):

Figure 30. Positions, order, serial number of an element.

Marking any of the elements with a dot above or under it, we state that the very position is special in some way. Thus we can not only talk about 14 or 5 or 10 elements as parts of something but we can also talk about the position of the elements. Let’s take this example with 14 elements – they may the fourteen days of a process. By writing all the 14 lines we say the number 14, but by marking a special element we can say “the fifth”, “the seventh”, etc. This is the way to express the order of elements when there is no speech. What other way can we suggest to say “The process lasts for fourteen days and on the fifth day do something”? The way is to use the mentioned type of numeric record because it allows for such statements.

Without being completely sure of it because I have not studied everything about numbers among Magura cave paintings, I think that not only dots are used to mark specific elements but also horizontal lines can be used as it is shown on Figure 31.

Fig. 31. An example of marking a number of consecutive elements as special.

When the horizontal line is beneath a set of vertical lines, it acts like dots beneath all of them thus saying that all of the underlined elements are special in the process.

There is another example in which the horizontal line is joined with the elements – Figure 32.

Fig. 32. Marking a series of elements as a sub process of the main process.

In this case we state that the four elements form a sub process which is a relatively separate and distinguishable part of the whole process.

Then we have an example of marking a specific position shown on Figure 33.

Fig. 33. An example of marking a specific position in the scene of rebirth.

Here we have the total number of 9 vertical lines keeping in mind that the female figure represents the fourth element. This figure means a pregnant woman and this fragment of the scene states that pregnancy lasts for 9 months. This is also an example of how the process and numbers that concern it are combined to create a clear and short record. By placing the figure of the pregnant woman among the 9 elements they have stated that these 9 elements concern the pregnancy. By placing the figure of the pregnant woman on the place of the fourth element (month) they have stated that the fourth month of the pregnancy is special for something. Indeed, gender id determined during the fourth month and the development of the embryo reaches such a stage when it is already decided to be a human.

And then there is an example of a similar idea expressed with the use of a tape as a numeric record – Figure 34.

Fig. 34. Using the tape type of numeric records.

In this example square number six (from the right hand side to the left hand side as this is the direction of writing and reading in Magura cave) is separated into four sub squares which means something but I still have no knowledge about it. Then elements 10 and 11 are divided into 3 sub rectangles. And element number 8 is broken I two parts by a solid object. This tale is a part of the scene of mankind’s history and every square of the strip is one era thus allowing us to calculate specific dates. Then the division of some of the squares states that there was something special in those eras. And the breaking of square number eight makes it clear that some catastrophe happened in this period.

Examples of marking specific elements may be very complex as it is shown on Figure 35.

Fig. 35. A complex example of marking specific positions in a numeric record.

I don’t know the meaning of this record as I can’t find the exact process or object that it concerns.

The examples with this type of numeric records can grow up to a despairing complexity – Figure 36.

Fig. 36. The record of months’ and weeks’ duration.

Here we have the number 14,5 where 3 elements have been united by a single horizontal line thus determining a sub process with its duration and position and these are elements 4, 5 and 6. Then, the vertical lines of the symbol beneath point to elements 3, 4, 5, 6, 7, 8, 9 and 10. In the same time, there is a symbol of turning over in the upper right part. All of this happens on the background of the represented crescent which states that the whole process starts at new moon. At the end of the left side there is a symbol of end, depletion. Is this scene related to menstruation and ovulation? Indeed, there are two parallel processes in this scene and the lower one corresponds to certain days of the process above.

This is a process of 14,5 days starting at the beginning of a lunar month. Elements 4, 5 and 6 are special and for a sub process. And there is a parallel process of eight days lasting from day 3 to day 10 and its fifth day is more special. What is this? Do you know? As it starts on new moon and lasts for 14,5 days, it must finish at full moon.

It has been demonstrated that this type of numeric records are capable of expressing very complex ideas. Now we come to the last example of these numbers – Figure 37.

Fig. 37. An example of a decimal number.

When it comes to writing down big numbers it is understandable that using series of vertical lines is inconvenient as we have to write down a hundred lines to represent the number 100. This is too much lines and the probability for a mistake is too big. Magura culture has solved this problem by creating and using articulated numbers in which groups of lines are arranged into ranks. Ranks are distinguishable from each other by specific signs. In the example shown here the five left most lines are joined by lines, the middle six lines are separate and in the right side we have three repetitions of the combination IH. Thus we clearly recognize three groups of elements corresponding to three ranks.

To read such a number, one has to start from the left most ranks, count the numbers of elements and multiply it by 1, then count the number of elements in the following rank to the right and multiply it by 10, then count the number of elements to following ranks of the right side and multiply it by 100 and so on till all the ranks have been processed. In the end, one has to sum all this numbers to find the total sum of the record. This is indeed the same way we write and read numbers. For example, the number 365 with the difference that we read and write from the left side to the right. The number 365 reads in the following way: 3×100 + 6×10 + 5×1 and this is the number of elements it contains.

Now I want to explain how this Magura number of 365 was read.

This is how I started by writing the number in a more familiar way:

IIIIIIIIIIIIHIHIH

Then I paid attention to the fact that the five lines to the left are joined by thinner lines forming one block which has been obviously meant to be relatively separate from the rest of the elements. I decided to use an additional symbol in order to represent that fact:

iiiiiIIIIIIIHIHIH

Initially, there was a supposition that the number may be binary. It was clear that a single vertical line stands for 1 element therefore the only possibility was that an H symbols meant 0 (zero). Then the number had to look this way:

11111111111101010

And I converted it into decimal number using a convertor. The result was:

130150

Which number is found in no process of importance. Therefore the chosen way to read it had to be wrong as it did not lead to the reading of a significant number.

After this I tried another approach by separating the number into groups:

11111 111111101010

In the decimal system the result of this are two numbers as it follows:

31 4074

Which are again numbers with no recognizable significance. Therefore this approach had to be wrong as well. A final division of the number was done:

11111 111111 101010

Its corresponding decimal result is as it follows:

31 63 42 – again nothing significant.

I had a doubt whether the division was correct and thought about the following:

Not 11111 111111 101010 but 11111 1111111 01010.

But the last group to the right side 01010 in mathematically no different from 1010 because the zero in the beginning ads no meaning to the rest. Therefore it could not be 01010 because there is no sense. Apparently the wright division was 11111 111111 1010101 with the only correction needed to note that the numeric expression was not binary.

Anyway, the efforts were not in vain as an understanding was built that the number record had to be separated into three parts like this:

iiiii IIIIII IHIHIH

Then it was very easy to just count and write numbers down:

iiiii IIIIII IHIHIH

5 6 3

Now, this is a significant number but we only have to read it in the opposite direction – from the right side to the left as all paintings and scenes in Magura cave are read. I had to remind of this earlier but did not. The days in a year are 365.

With this it became clear that a group IH stands for 1×100 thus determining the symbol H to mean the ranks of hundreds. And, maybe, it is not just a coincidence that the Greek word for 100 is Heka, the English word is Hundred, the German word is Hundert and so on.

A great help in the process of reading this number was the spotting that there was another similar composition at another place among the paintings (Figure 38):

Fig. 38. Another repetition of the pattern HIHIH among the cave paintings of Magura.

With this I knew that the pattern HIHIH is normal to Magura numeric expressions and that helped me go on with the correct division of the number 365.

Some principals of numeric records of Magura cave paintings were discovered. First of all, numbers are written and read from the right side to the left side. Then, Magura culture used decimal numeric system with the only difference that they did not use digits to express the magnitude of the respective ranks but used repetitions of similar elements. The elements in the respective ranks are grouped in a distinguishable way which allows for a recognition off all the elements belonging to the respective rank. The magnitude of the respective rank is obtained through counting of its elements. The total number is obtained by summing the magnitudes of all ranks multiplied by respective decimal divisions. Magura culture used decimal numbers and the symbol H stood for 100. It has to be admitted that it is unclear whether an H found alone among the paintings still means 100 or it does so only in a combination with other symbols referred to numeric expressions.

Probably, if I want to write down the number 200, I have to do it this way:

IIH

As the H states its hundreds and II states they are two. But if I want to write down the number 201 it will not be done this way:

IIIH

As this should mean 300. I would rather do it this way:

IIHIH

As it says 1 + 100 + 100.

I am not totally familiar with the whole logics of Magura cave numbers but this is the beginning of the process. Maybe the number 200 has to be written down as

IHIH?

I presented the very large capabilities of the series of vertical lines to express numbers and positions. Now is the time to move to some numeric operations that I have recognized among Magura cave paintings over time.

Let’s go back to an example of the number 3 – Figure 39.

Fig. 39. An example of the number 3.

This is an integer and what do we have to do to create a fraction from it? There is an example among Magura cave paintings – Figure 40.

Fig. 40. An example of a fraction.

What is the way to show that a thing is opposite to something? The way is to rotate it. It depends on the shape of the painting – in the case of vertical lines the only recognizable rotation is 90°. Therefore what is shown on Figure 40 may mean two things – 1/3 or x3. Whether it is used as 1/3 or as x3 is determined by the position of the symbol. Here is an example of 1/3 – Figure 41.

Fig. 41. An example of division by 3 or of multiplying by 1/3.

One of the horizontal lines is missed on purpose as it is replaced by the basic number we operate with – in this case this is the zig-zag line above. Now this zig-zag line is understood as the basic number and the replacement of one of the horizontal lines with it means that the operation concerns the zig-zag number record. Then we have two horizontal lines beneath which mean division by 3 keeping in mind that the third horizontal line is replaced, substituted.

In mathematics division by 3 is the same operation as multiplying by 1/3. Therefore 3 horizontal lines below mean multiplying by 1/3 or division by 3.

Then we have another example of the opposite operation – multiplication by 3 – Figure 42.

Fig. 42. An example of multiplication by 3.

Here we have a block of 27 elements which means the number 27 and it is the basic number we operate with. The two lines above mean multiplication keeping in mind that one of the lines is substituted with the block of 27 elements thus showing that the block is the basic numeric record to operate with. This may be read as “do two more times of that” which is in total three times the initial thus we know it is indeed multiplication by 3.

This numeric record is a part of the solar calendar in Magura cave where the record is located between two out of four positions of Earth around the Sun. The gap between an equinox and a solstice is one season. The number 27 represents one orbital lunar month or one month. This numeric record says that there are three months of 27 days each in a season. Therefore it is known for sure that this process is multiplication by 3.

Note this: chequer-like numbers are used when we are interested in the dimensions of a complete process without being interested in specific elements and their properties. Therefore elements are presented as a solid block with no differences among them. The total number of the days in this record is 81 but this is not interesting in the case. What is interesting is that there are 3 periods of 27 days each and this is the reason to use exactly this type of numeric record as it best fits the intention to transfer specific information.

Here is the place to say that only the rotated integer is not enough to clarify whether it is used for division of for multiplication. Only when it is related to a numeric record we can say which of the operations takes place. If the rotated integer is under – this is division. If the rotated integer is above – this is multiplication.

Now there is a more complex example which combines multiplication and division – Figure 43.

Fig. 43. An example of multiplication and division operations at the same time.

We have a block of 9 elements. Above it, there are two horizontal lines which mean multiply by 3 thus creating the number 27. Then, above all of this, there is a symbols for a turn over which means “then divide” and now the division is by 2 as only the two lines are sent beneath the basic numeric record. As a result we get 27/2 which is 13,5.

The reason to have this number again in the solar calendar is the duration of the Earth’s year which is 365 days which in turn is not multiple of 27 – the duration of an orbital lunar month. Therefore there are 13,5 orbital lunar months in a year. There is a star painting in the calendar with 13 rays and 13 segments between them, respectively. This star states that something happens 13 times and then repeats. These are the 13 orbital lunar months within a year but ½ of a month remains to be filled. For this reason there is the expression of Figure 43 to represent the necessary half of a month to complete the year.

Star-shaped numeric records were already mentioned. Here I want to say it that the stars are used to express repeating processes, cycles. They contain a number of periods within which, when completed once, start again. Therefore the shapes are round to symbolize rotation and repetition.

Then we come to two situation where elements do not mean numbers on their own but mean numbers in a combination with other symbols – Figure 44.

Fig. 44. Two examples of creation of numbers using elements which are nor numbers on their own.

Here we see two V elements mentioned in Figure 8 where they were determined as non-numeric on their own. But we see that they can represent numbers in combination with other elements which are numbers on their own.

I think that a V element means 5 or 10 and it is for a reason that Roman numeric system uses the same symbol with a similar or equal meaning – because later cultures are based on previous cultures thus the Romans inherited some numeric knowledge from the much earlier Magura culture.

It has to be additionally checked but I haven’t done it so far whether a V element is 5 or 10 – more real examples of the paintings must be examined in their context to find out the exact meaning.

In the same time, this study is a bit harder as there Л elements as well. A zig-zag has to be written and read from the right side to the left side and this shown us what is the first element – V or Л. This is illustrated on Figure 45.

Fig. 45. An illustration of opposite elements in two equally long zig-zag lines.

As you can see, both zig-zag lines contain 3 repetitions of elements but the left one starts with an Л element which determines all of its element as Л-elements, and the one on the right side starts with s V element which determines all of its elements as V-elements.

It is obvious that Л and V elements are opposite in their direction and this must alarm us to understand them as opposite in some way. Maybe they are reciprocal and one of them is 10 and the other is 1/10? Or maybe one of them means 5 and the other means 10? This has to be found out in the future if possible. Possible means if there are enough examples among the paintings and if the context of the examples is clear enough to allow for precise interpretations.

Then we have an example of a more complex zig-zag numeric record – Figure 46.

Fig. 46. An example of a more complex zig-zag numeric record.

It starts with a V element therefore all of its elements are V. We can count three V elements and half of an element on the left side. What does it mean? I still do not know. Maybe, if a V element is 10, the half stands for 5?

Zig-zag numeric records can also be used to represent positions of elements – Figure 47.

Fig. 47. Positions on zig-zag numeric records.

Probably, the rules of reading are similar to the case of series of vertical lines. I don’t know.

This is so far on the topic of numeric records among Magura cave paintings. Only one thing more remains to be done – a brief comparison to Egyptian numeric system.

Figure 48 shown what hieroglyphs Egyptians used to write down the different decimal ranks within a number. They used distinguished hieroglyphs to show where the respective ranks start and stop. A similar approach was used in Magura numbers by clearly dividing large numbers into sub groups of distinguishable elements to make it clear where a decimal ranks starts and stops. This means that the logic is completely the same, only the symbols used differ.

Fig. 48. Hieroglyphs used in Egypt to write down the different decinal ranks of numbers.

Then we have an example of an Egyptian number – Figure 49.

Fig. 49. An example of an Egyptian number.

The number is 4622. It is written from up to down and from the left side to the right side – just as we do which determines this record to be of later periods compared to Magura cave records as enough time has passed to change the direction of the writing and reading. Here we see the same construction of the number – a certain number of equal elements in every ranks shows the magnitude of the rank and we have to multiply this magnitude by the respective decimal rank – just as in Magura. The only two differences are in the symbols used (except for the symbol for 1 which is the same) and the direction of writing and reading.

Here we see the element Л stood for 10 and maybe it is the same in Magura as the logic of both numeral systems is very similar. Then, maybe the element V means 5 or 1/5 but it is more likely to mean 5 as it has probably transferred to Roman numeral system at a later period.

Then we come to division and fractions in the Egyptian numeral system – Figure 50.

Fig. 50. An example of a fraction (division by 3) in the Egyptian numeral system.

The logic is very similar to the division records among Magura cave paintings.

At this stage of the study I think that Egyptian numeral system is a later development of Magura numeral system. It has been simplified and the various cases of use of different numeric records have been skipped to reduce complexity and to make it more suitable for everyday use.