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| BADR HUSSAIN |
ABUT TOURMAN........ name : Badr Hussain n.name : bibo gender : male age : 30 job : travel agent s.status: single d.birth :08.02.1968 location:Egypt-Cairo address :5,Adly st.down town Cairo.Egypt tel.# :002-02-2666542 | ||||||||||||||||
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1.Cheop's pyramid in Giza was considered as one of the seven wonders of ancient world, and it is the only one still existing. Of course there are some more pyramids not only in Giza but also in Dahshour , Abussir, and other places. Pyramid, as burial monument of Pharaohs, took its final shape after a long evolution. The faith of ancient Egyptians in life after death, led them to construct special type of tombs(1). The deceased was put in the burial chamber with all the necessary things for his afterlife, as food, utensils etc, and all his relatives had to give him offerings during special ceremonies which were taking place in a funeral chapel, close to the burial chamber. Thus the tomb was divided in two rooms: the burial chamber and funeral chapel, and since earliest era it had always north-south orientation. These two rooms were underground, and a flat stone on the ground served to indicate the position of the tomb. In the next phase of evolution a hill of sand or mud-bricks was erected above the tomb where the stone with the name of the deceased was erected, and the entire construction was surrounded by a wall. The shape of the hill was oblong and the main facade was at its east side, where the stela was put. In order to let the soul of the deceased visit the body, they gave to this stela the form of a false-door always at the east side and with the name of the deceased on it. In a special hollow inside the hill they put statues of the deceased for receiving the offerings. The side wall was made of mud-bricks and in order to reduce the side pressure caused by the material inside it, they made its sides inclined, so that all sides of the construction were inclined. The workers of excavations gave to this type of tomb the name �mastaba� after the wooden benches by the doors of today's villages of Egypt, which are made of the same material. This form of mastaba took its final form by the beginning of the 4th dynasty (~2950 B.C.). It was formed of an oblong mastaba orientated from North to South and a yard. Under the mastaba there was the burial chamber as well as the statue room, called �serdab� . In front of the mastaba, beside the false door at the East side there was a place where they used to worship the deceased by offerings. This worship place was at first in an open air while later was covered and divided into a great number of rooms. The royal tombs followed the same model. But all innovations and evolutions started by the kings, who wanted always their tombs to be distinguished. They made their tombs bigger than the usual, to show their grandeur, and they surrounded them by a wall to cover their entrance. But how big can be a tomb? Next evolution was the reduction of the height of the wall to such a level that the wall could cover only the entrance of the burial chamber, but not the entire height of the construction. The result of this innovation was the two steps mastaba. The remains of these mastabas are destroyed, especially in their upper part. Next step to differentiation of royal tombs was the construction of more than two walls of increasing height from outside to inside, and the result was the step mastaba. A characteristic example of this form is the step mastaba of Djeser (~2950 B.C.) in Saqqara, which is called also step pyramid, but its base is oblong instead of square. The closest antecedent of pyramid was built in Meidoum, near Fayoum. It had not only square base, but the steps of the side faces started to be covered by inclined surfaces. This pyramid is attributed by some Egyptologists to Choni, the last Pharaoh of the 3rd dynasty, and by some others to Senefru, the first Pharaoh of 4th dynasty. Senefru built two pyramids in Dahshour. What is characteristic for the southern pyramid is that the angle of slope of its lower part is greater than this of the upper part, and it is called Bent Pyramid. The northern one, the so called Red Pyramid is the oldest real pyramid. Pyramid had already replaced mastaba as royal tomb, and the worship place of the deceased was converted into a temple; both pyramid and temple formed the royal burial monument. The shape of the pyramid is connected with the sun worshipping. In the Pyramid Texts it is mentioned that the king used the sunbeams as a ramp to ascend to the sky and the sun made the sunbeams stronger to support the king. Thus the Egyptologists conclude that the step pyramid and the pyramid were the means for the Pharaoh to cross the distance between the earth and the sky. The Pharaohs of 4th dynasty Cheops, Chephren and Mycerinus built their pyramids in Giza.The next pyramids were built in Abussir -not far from Giza- by the Pharaohs of 5th dynasty (~2700 B.C.) 2.Particularly remarkable and characteristic feature of the construction of pyramids in Giza is its accuracy. In the right angles of the base of Cheops pyramid appear very small errors:(2) in NW corner -0' 2�� in NE � +3' 2�� in SE � -3' 33�� in SW � +0' 33�� This accuracy in construction of right angles is an obvious indication of advanced geometrical knowledge around 2900 B.C. The orientation of Cheops pyramid is astonishing not only for its accuracy but also for the method used to construct it. The NS direction differs only 2' 30�� at its west side, 5' 30�� at its east side from the real North and the EW direction differs 1' 57�� at its south side and 2' 28�� at its north side from the real West(3). How were they able to reach such accurate results? One way is the observation of sunrise or sunset during the equinox: the shade of the plumb line on the horizontal level at the moment of sunrise or sunset gives the EW direction. But according to Borchardt's remarks the ancient Egyptians were not able to use the above way because of the following: 1)They could make this observation only twice a year. 2) According to our knowledge the Egyptians at the era of construction of pyramids were not able to estimate the exact days of equinox. 3) The above method would give an approximate error of 15', which is greater than the real, because of the quick transposition of the sun on the horizon at the equinoxes, so that the position of the sun is rarely at spring or fall point at the moment of sunrise or sunset. Similarly if they had estimated the directions of both points of sunrise and sunset at the equinox and then bisected the angle of these directions, so that the bisector could give the SN direction, then the error would have been much greater. Another way is the determination of the SN direction using the stars. At that time in the place of our Polar star, there was � Draconis, a star of magnitude 3.8 , whose distance of real North was 1� 15'. If � Draconis had been used, then the error would have been around 1� 15'. The position of the North could also have been found as middle of the segment which is determined by the two positions of a star with difference of 12 hours, which was possible only at the New Kingdom period. It might have been found also as middle of the segment determined by the position of upper and lower culmination of a star, but prerequisite for this is a special instrument for the measuring the height of a star, not known at that time too. Then the above methods are out of question. The way that can give the accurate NS direction is the determination of the points of rising and setting of a star that remains for a short time under the horizon. The bisector of the angle ROT gives the NS direction, (R: rising point , O: observer , T: setting point ). Borchardt claims that this was the way used by the ancient Egyptians for the determination of the North and he mentions about harpedonaptae and the rite of stretching the cord. N T R O S We cannot answer definitely how they were able to construct right angles or to find correctly the orientation. But these are problems which cannot be solved without advanced geometrical and astronomical knowledge. About the competence of ancient Egyptians in Geometry, the Greek Mathematician Democritus (~460-370 B.C.) refers that �in construction of lines, undoubtedly, no one is better than me, even the so called harpedonaptae in Egypt�. The comparison with Egyptians was done to show his expertness. 3.Associated Mathematics with Pyramids construction Pyramids were constructed at the era of Old Kingdom, during the 4th and 5th dynasty (3000-2500 B.C.). Unfortunately there are no written calculations found about Pyramids construction. The written evidences that exist at our disposal, are only a distant echo, since they originate from papyri about thousand years later. These are Rhind Mathematical Papyrus [RMP] and Moscow Mathematical Papyrus [MMP]. Rhind Mathematical Papyrus was a teaching book, its aim being the transmission of mathematical knowledge to the future generations. What is preserved to us is a copy dated from Hyksos period of an older original Papyrus that goes back to the Middle Kingdom, as it is mentioned in the title of the Papyrus: �Accurate reckoning.The entrance into the knowledge of all existing things and all obscure secrets. This book was copied in the year 33, in the 4th month of the inundation season, under the Majesty of the king of Upper and Lower Egypt, cA-user-Rec, endowed with life, in likeness to writing of old made in the time of the king of Upper and Lower Egypt, Ne-ma-cet-Rec (Amenemhet I). It is the scribe Ach-mos� who copies this writing.�(4) Moscow Mathematical Papyrus is also dated from the Middle Kingdom. 3a.The �seked� of a pyramid. In Rhind Mathematical Papyrus [RMP] (5) (~1600 B.C.) there are some problems concerned with the �seked� of pyramids (RMP 56-59). But what exactly is the seked of a pyramid? The word skd (6) is related to the causative form of the verb kd (7) that means build. Then skd is a concept that deals with building. The study of problems 56-59 of RMP shows that the seked denotes the slope of the inclined faces of a pyramid, in a way that reminds the cotangent (8) of an angle. If half of the pyramid side is b palms and its height is a cubits the seked is the quotient b palms/ a (cubits). [ Cubit and palm are units of length: 1 cubit mh is 7 palms. 1 palm ssp is 4 fingers db ](9). Then the seked denotes that the slope is b/a palms horizontally for every rise of one cubit in height. The existence of the notion of seked was essential to build pyramids, in order to keep the same slope in each inclined face. Every craftsman was able to attain the given slope of the pyramid, raising 1 cubit the height for every �seked� of length horizontally. Since the ancient Egyptians could only use unit fractions, the use of palms to measure the length, instead of cubits, was providing more accurate measuring. For example the seked 5 1/25 mentioned in RMP 56, if it was calculated in cubits/cubits it would have given 1/2+1/5+1/50, which would have been difficult to be realised. The angle of slope of the inclined faces of some pyramids are: (10) Cheops 51� 52' Chephren 52� 20' 53� 10'+ 4' (11) Mycerinus 50� 45' Dahshur (southern stone pyr.) 54� (lower part) 43� 21' (upper part) The slope of the pyramids studied in RMP are (11): seked cotx x(angle of slope) the slope is close to the pyramid of Problem56 5 1/25 palms 18/25 54� 14' 46�� Dahshur Problems57-59 5 1/4 palms 3/4 53� 7' 48�� Chephren Here is the hieratic text (12) with my translation and comments: | UNBELIEVABLE HISTORY | ||||||||||||||||
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