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THE KEYS OF THE TEMPLE
David Furlong 1997
Page 93
SACRED GEOMETRY AND THE GIZA PYRAMIDS
"The alignment of the pyramids Grand Gallery pointed to the centre of one of the circles
Above the entrance to Plato's Academy at Athens was the legend 'Let none ignorant of geometry enter here'. To the ancient Greeks, pure geometry lay at the heart of all things. It was the way of reconciling the world of the divine with the form of the world we see. The golden mean proportion, for example, can be depicted in terms of geometry but not number. It can be drawn, but the number that represents it cannot be written down as it runs to an infinite number of decimal places. Geometry can be seen as a way of defining what is otherwise indefinable.
Our knowledge of the use of pure geometry in ancient Egypt is more tenuous. We do not have any papyri which give the geometrical equivalent of the equations of Plato, Thales and Euclid, epitomising ancient Greek thought. However, Plato considered that Egypt possessed a profound canon of knowledge based on harmony and proportion. We can infer that the ancient Egyptians were as adept with the compass and the rule as their Greek counterparts. This knowledge would have influenced their art and architecture. Unravelling how the Egyptians might have selected the proportions they used is a way of reaching back into the roots of their civilisation. / Page 94 / The patterns on the Marlborough Downs also appeared to be based on pure geometry. Our next step, therefore, must be to draw together these disparate parts of the puzzle and discover the underlying geometry that unites them.
" The term 'sacred geometry' can be misleading as the fundamentals of geo-m~tric proportion are found extensively throughout the natural world as well as in art and architecture. Why are some elements sacred and not others? There is no easy answer to this question. Nevertheless, a tradition has grown up which has placed a special emphasis on certain geometric relationships and proportions found most commonly in the design of buildings used for religious purposes. To the general observer, these pro- portions are simply pleasing. Artistically, this is analogous to music. Using different groupings of notes and internals, harmonious or discordant sounds can be created. Some music, such as Gregorian chants, can put us more in touch with spiritual feelings. Other music puts us directly in touch with our emotions. Indeed, one of the great philosophers, Pythagoras, demonstrated the links between music, sound, number and form.
Three basic geometric shapes are central to religious tradition: the circle, the triangle and the square (Fig. 29). (Diagram omitted) These were taken to represent three levels of our being: spirit, mind and body. Like systems of counting, no one knows who first used a compass. It was probably just a piece of /
Fig. 29. The three basic shapes in geometry
Page 95 / string and two pegs,"6ut this development paved the way for a symbolic exploration into the realm of ideas and forms. All regular geometric shapes can be derived using a compass. God, who has sometimes been called the 'Great Geometer', is often depicted wielding a pair.
'Linked to geometry was the study of numbers. Whole numbers were considered the ideal. A completeness was perceived in them; while frac-tions represented numbers in the stage of becoming. In this sense, they were sometimes considered as the dynamic power of the divine moving through creation. Whole numbers were knowable, but ratios such as pi..." "... could only be approximations and therefore unknowable. This was the unfathomable hand of God that permeates all things.
But while individual numbers are either rational (whole numbers) or irrational (fractional numbers), geometry can bridge this divide. The circle can both represent a rational whole number principle in its diame-ter as well as an irrational function along its circumference. A square and its diagonal also produce a similar phenomenon..."
Page 98
Fig. 34. Pentagon derived from vesica (Diagram omitted)
"...The triangle was seen as a transitional form between the square and the circle. Eventually it came to represent a triad of gods and goddesses, usu- ally, as in Egypt, a father, mother and son. This concept formed a central pivot of many religious belief systems and manifests itself in Christianity as the Father, Son and Holy Ghost. The triangle's most perfect form was considered to be equilateral, where the sides and the angles are all equal. Another triangle that was widely used was the one attributed to Pythagoras, although it certainly predated him by a very long time. Its / Page 99
Fig. 35. Construction of golden mean from 2 x 1 rectangle (Diagram omitted)
Fig 36. Construction of golden mean from a square and a circle (Diagram omitted)
/ sides are in. the whole number ratio 3:4:5. This triangle produces the simplest version of a right-angled (90°) triangle with sides that can be expressed as whole numbers. Because of the simple numeric ratios employed, it was used in surveying as well as in art and sculpture. The pyramid of Khafre is based on it.
The circle, triangle, square and rectangle form the basis of all sacred architecture. They were, by tradition, related to each other by specific pro- portions. These proportions attempted to portray the inherent harmony of the cosmos. One such proportion was the 'gnomon' which was defined /
Page 100
Fig. 37. The construction of a pentagon (Diagram omitted)
Fig. 38. The golden mean proportion within a pentagram. (Diagram omitted)
Page 101 / by Aristotle as any figure which, when added to an original figure, leaves the resultant figure similar to the original.' In other words, the ratios between each additional step are always maintained. An example of this is the 'golden mean' proportion which can be expressed through the num-bers 1, 1,2,3, 5, 8, 13,23, and so on, where the ratios between any two adjoining numbers rapidly converge as you move up the series. The Fibonacci series is the most well known example of a gnomic ratio, but others exist.
Robert Lawlor, in his book Sacred Geometry, gives examples of 'gno- monic' spirals such as the one based on the Fibonacci series, which is derived from the ratio of 1 :2. These expanding patterns are sometimes called 'whirling squares', giving rise to spirals which are frequently found in the natural world (Fig. 39). (Figure 39 omitted)
Examining 'gnomons' of different ratios, I made a significant discovery. One of the 'gnomons' based on the ratio 1:3 is directly relevant to the Giza / Page 102 / pyramids. It so happens that from this one ratio the basic proportions of Khufu's, Khafre's and Menkaure's pyramids can all be derived (Fig. 40). Figure 40 omitted ) The development starts by drawing three abutting squares in line, to cre-ate a 3 by 1 rectangle. A square is then drawn on the longer side at each stage of its development.
The first square creates a rectangle with a ratio of 3:4. Doubling it reveals the ratio of Khafre's pyramid, 6:4. Adding two more squares, in turn, to the 3:4 rectangle gives the ratio of Khufu's pyramid, 7:11. /
Page 103 / Another square"gives the proportions of the pyramid of Menkaure, 11: 18. This device of adding squares, starting with a 3 by 1 rectangle, dramati-cally reveals that the pyramids reflect a natural mathematical progression in their height to base ratios. Whether through chance or choice they are linked by a harmonious geometric series.
What could have been so significant about a 3: 1 ratio? Perhaps it reflected the symbolism of the Egyptian trinity of Osiris, Isis and Horus. We will probably never know for sure, but this pattern gives a valuable insight into Egyptian methods.
This discovery is also in accordance with what is known about Egyptian design methods which always seem to have been derived from squared grid patterns. There are numerous examples existing in Egyptian art, showing painters and sculptors first drawing a grid on the wall that is to be painted or carved so that fixed proportions could be maintained. The simple numeric ratios of these grids lie at the heart of all the great artistic achievements of the Egyptians.
This method was also used by many of the great Renaissance artists, such as Leonardo da Vinci. In ancient Egypt, this is exemplified in the Great Pyramid and establishes a further link with the pattern on the Marlborough Downs.
THE
PARANORMAL
Stuart Gordon 1992
Page388
"LEIBNITZ, Gottfreid Wilhelm van (1646-1716) This 'greatest intellectual genius since Aristotle' (Macneile Dixon) claimed that matter is but another name for energy, and that space and time are inseparable. He said, 'The world is not a machine. Everything in it is force, life, thought, desire. The world, in brief. . . is a living society'.
Page389 Believing that all living individuals always have existed and always will exist, he regarded death as a longer, profounder version of sleep, in which the continuity of the individual life is not wholly broken. He held the universe to be composed of monads; living and active beings who reflect the universe each from its own angle, each in its own degree.
From his Monadology: 'There is nothing waste, nothing sterile, nothing dead in the universe; no chaos, no confusions, save in appearance.'
LEONARDO DA VINCI (1452-1519) Leonardo, whose genius led him in so many prophetic directions, was clearly a believer in reincarnation. In his Notebooks he wrote: 'Behold now the hope and desire to go back to our own country, and to return to our former state, how like it is to the moth with the light! . . . this longing is the quintessence and spirit of the elements, which, finding itself imprisoned within the life of the human body, desires continually to return to its source.' And again: 'Read me, 0 Reader, if you find delight in me, because very seldom shall I come back into this world.'
THE KEYS OF THE TEMPLE
David Furlong 1997
Page 88
"The angle of slope of Khafre's pyramid is the same as that of the pyra- mid of Pepi II, one of the kings of the 6th Dynasty who reigned from 2278 to 2184BC..."
"The angle of slope of Khafre's pyramid is the same as that of the pyra- mid of Pepi II, one of the kings of the 6th Dynasty who reigned from 2278 to 2184BC. This pyramid is now in ruins, but the angle of slope has been deduced from what remains of the outer casings. The construction of the later Egyptian pyramids was inferior to the Giza group and they have suffered greatly through the passage of time. Many have now collapsed into heaps of rubble. However, in Khafre's case, the angle of his pyramid, (based on the 3:4:5 triangle) is highlighted in three of the problems (num-bers 57, 58 and 59) posed on pyramids in the Rhind Mathematical Papyrus. So it was a ratio that was well known to the ancient Egyptians.
In fairness to those Egyptologists who maintain that the ancient Egyptians did not know about the 3:4:5 triangle, the length of the hypotenuse - 5 - is never given. But mathematical problems, and those involving the pyramids, are always calculated in terms of the seked of the angle, the ratio of height to base. In the case of the 3:4:5 triangle, the seked is the ratio of 3 to 4. But as no mention was ever made of the length of the hypotenuse, it was inferred that the Egyptians had never worked out the length of the third side.
Are we really supposed to believe that a people who could design and build a monument to the precision shown in the Great Pyramid, or the pyramid of Khafre, inv,olving exact measurements over considerable dis- tances, would never have measured or worked out the lengths of the hypotenuses of the triangles they were using? Surely, any people seeking precision of measurement would check all lengths as a matter of course as part of their general exploration of number, form and geometry. It would be an essential part of their working methods. So, I would maintain that simply by using the ratio 3:4 in their building plans, they implicitly knew the length of the third side. "
"...However, in Khafre's case, the angle of his pyramid, (based on the 3:4:5 triangle) is highlighted in three of the problems (num-bers 57, 58 and 59) posed on pyramids in the Rhind Mathematical Papyrus. So it was a ratio that was well known to the ancient Egyptians..."
"In the case of the 3:4:5 triangle, the seked is the ratio of 3 to 4.
A.P. Rossiter 1939
Page 15 "The Egyptians,…" "…made good observations on the stars and were able to say when the sun or moon would become dark in an eclipse (a most surprising event even in our times), and when the land would be covered by the waters of the Nile: they were expert at building and made some discoveries about the relations of lines and angles - among them one very old rule for getting a right-angle by stretching out knotted cords with 5, 4, and 3 units between the knots.
Reight wah scribe, if there iz a point to labour, set up the labourer, and labour that point.
The scribe quick to noblesse oblige, counted the number of single letters in each of three words.
5 . . . 4 . . . 3
EARTHMOONSUN
3
THREESUN
4
FOURMOON
5
FIVEEARTH
Page 98
Father, Son and Holy Ghost.
The triangle's most perfect form was considered to be equilateral, where the sides and the angles are all equal. Another triangle that was widely used was the one attributed to Pythagoras, although it certainly predated him by a very long time.Its /
Page
99
sides are in. the whole number ratio
3:4:5.
This triangle produces the simplest version of a right-angled (90°) triangle with sides that can be expressed as whole numbers.
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THE KEYS OF THE TEMPLE
David Furlong 1997
Page 94
"...The triangle was seen as a transitional form between the square and the circle. Eventually it came to represent a triad of gods and goddesses, usu-ally, as in Egypt, a father, mother and son. This concept formed a central pivot of many religious belief systems and manifests itself in Christianity as the Father, Son and Holy Ghost. The triangle's most perfect form was considered to be equilateral, where the sides and the angles are all equal. Another triangle that was widely used was the one attributed to Pythagoras, although it certainly predated him by a very long time. Its / Page 99 / sides are in. the whole number ratio 3:4:5. This triangle produces the simplest version of a right-angled (90°) triangle with sides that can be expressed as whole numbers.
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Scofield References
Jeremiah B.C. 590
Page 809 8 x 9 + 72 7 + 2 = 9
Chapter 33 Verse 3 x 33 = 99
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