Again, we find that the hypotenuse of this triangle is precisely 51.84º . . . the same, exact, degree angle as the sides of the Great Pyramid of Giza.
5040 x 3960 miles is 14:11 ratio.
To make it really easy and simple for the layman to grasp this:
If we now apply the 14 x 11 rectangle shown in figure 3, we note that the rectangle is neatly divided into 11 x 11 for the Earth and 3 x 3 for the Moon – together creating a 14 x 11 ratio rectangle.
These numbers relate to the earlier significance of Earth radius 3,960 divided by 11 = 360, and Moon radius 1,080 divided by 3 = 360.
Gary Osborn
Copyright © Gary Osborn 2012. Updated 2014. All Rights Reserved.
It has been established that each of the four inclined sides of the Great Pyramid of Giza are at the angle of 51º 51' (in arc minutes), and in angle degrees they are 51.84º.
What is the real significance of the Great Pyramid side angle of 51.84º?
How was the angle of 51.84º arrived at?
From what source did this angle originate?
In January 2010, my attention was drawn to a geometrical diagram of the combined proportions of the Earth and Moon.
The diagram was from an article by Howard Arrington based on author and Chartered Surveyor, John Mitchell’s discovery published in his book The View Over Atlantis (1969).
In his book, author John Michell presents a diagram in which he places the Moon on top of the Earth and draws a triangle combining the radius of both – the side angles of which happens to be the same as the side angles as the Great Pyramid of Giza – 51.84º. (Note also the green triangles flanking the Moon. These are Pythagorean Triangles of 3–4–5 proportions, which we will be studying in some depth later).
Figure 1: Diagram from John Michell’s book The View Over Atlantis (1969).
I then followed what John Michell did . . . .
First, we draw up two circles. The diameter of the first circle relates to the diameter of the Earth, and the diameter of the other relates to the diameter of the Moon, as shown in Figure 2.
With both the circles of the Earth and Moon to scale, we then place the Moon on top of the Earth as-per John Michell’s diagram (figure 1).
Figure 6: The angle of 51.84º derived from the combined radius proportions of the Earth and Moon.
Figure 5: The 5040 x 3960 (11 x 14 – ratio 11:3) Triangle connecting the radius of the Earth and the radius of the Moon.
Gary Osborn
Copyright © Gary Osborn 2013. All Rights Reserved.
Figure 2: The correct sizes and proportions of the Earth and Moon measured in miles.
Immediately upon seeing it, I thought that there was more to this diagram, and again in terms of ‘degree angles.’ The way in which the Earth and Moon were positioned with the Moon placed on top of the Earth, again reminded me of the ‘Ancient Geometric Formula’ mentioned by the 32nd Degree Freemason Frank C. Higgins on page 12 of his book Ancient Freemasonry, who believed the geometry, angles and numbers that could be derived from it, encapsulated and preserved knowledge of “cyclic time periods” and the “structural proportions of the universe,” and all since remote antiquity.
“The crowning secret of the ancients, as well as of our own time, is that the study of the structural proportions of our universe as evinced in lines of force and direction, cyclic time periods, celestial areas and visible parts, reveals the fact that it is of definite form, perfectly balanced proportion and just such a synthesis of the principles of pure geometry as to show that the features exhibited are from the same causative source as that which geometrizes the snowflake, the crystal and the blossom. Hence the expressions used in the Masonry of the Grand Architect and the Great Geometrician of the Universe. “The ancient Egyptians adopted as the key to this great cosmic philosophy the famous Pythagorean triangle of 3x4x5 proportions, which is the basis of the celebrated forty-seventh proposition of Euclid.”
Ancient Freemasonry: An Introduction to Masonic Archeology by Frank C. Higgins (1919), p. 12.
The base of Michell’s triangle is 7920 – being the diameter of the Earth in miles.
The vertical axis of the triangle is 5040 miles – being 3960 + 1080 – the combined radius of both the Earth and Moon. Using trigonometry, the tan of 3960 + 1080 is 1.27272727273 – otherwise 14 x 11 pi.
The Moon is 1/27th the size of the Earth. Measuring the angle of a 14 x 11 rectangle from corner to corner, results in a value somewhere between 51.842º and 51.843º so Geoff Simmons tells me, which is nitpickingly precise but goes to show how accurate the result of this angle is. We really don’t need to worry about the last decimal digit, because working out the trigonometry of the combined radiuses of the Earth and Moon gives us the precise side angle value of the Great Pyramid of Giza . . . 51.84º.
(Geometry means “Earth Measure”)
“The Great Pyramid is not only an engineering structure conforming to clear geometrical rules but also is a sacred temple of initiation into the mysteries of rebirth, reincarnation, and cosmic existence. To be sure, the ancient architect-priests used geometrical and mathematical canon to design the Great Pyramid, but this canon was an intrinsic part, not of an engineering manual, but of sacred books used for the highest system of occult initiation. Geometry was sacred. Art was sacred. Architecture was sacred. Hieroglyphs were sacred. They were all part and parcel of a sacred knowledge, a sacred science if you will, taught to high initiates in the temples. The mind of man, which they deemed the most sacred thing of all creation, was to be made adept to this sacred science or knowledge in order to find the creator and experience eternity. Those who planned and designed the Great Pyramid were not architects in our modern definition of the word. In fact there is no need for architectural prowess to design a regular pyramid. Any schoolchild can easily design a pyramid. These ancient men however, were high initiates and magicians, brought into the occult sacred science of symbols and cosmic ambience. The Great Pyramid in not only a masterpiece of architecture and engineering, it also is a masterpiece of sacred science and esoteric knowledge.”
Robert Bauval,
Secret Chamber Revisited: The Quest for the Lost Knowledge of Ancient Egypt (2014).
Keeping the above quote in mind, and noting Higgins’ comments about the Pythagorean 3–4–5 triangles and Euclid’s ‘47th Proposition’ – also known as the ‘Pythagorean Theorem’ in relation to this formula – I thought I would check this combined Earth-Moon diagram for the angles I know so well. And so to make further discoveries to which my intuition was guiding me, I replicated Mitchell’s drawing using the latest ‘mile radius’ data we have of both the Earth and Moon.
The Earth’s Diameter at the Equator is 7,926.28 miles (12,756.1 km).
The Earth’s Diameter at the Poles is 7,899.80 miles (12,713.5 km).
The Earth’s Equatorial Radius is 3963.14 miles.
The Earth’s Polar Radius is 3949.9 miles.
The diameter of the Moon is 2159.14062 miles.
The radius of the Moon is 1,079.57031 miles.
Earth Radius using rounded off figures: 3,960 miles.
Moon Radius using rounded off figures: 1,080 miles.
John Mitchell used these same rounded-off values. I also learned from researcher Geoff Simmons – a long time correspondent of mine – that Michell placed a stylized (“canonical”) value on the Earth’s mean diameter/radius – 3960, and another on the Moon’s diameter/radius – 1080. These same figures produce the following:
Earth Radius 3,960 divided by 11 = 360.
Moon Radius 1,080 divided by 3 = 360.
As the values of both 3960 and 1080 were significant to the author John Michell (both of which can also be reduced to the number 9 – i.e., 3+9+6 = 18 . . . 1+8 = 9 . . . 1+0+8+0 = 9 – a method used by the ancients to easily encode numbers relating to the 25,920 year ‘Cycle of Precession’), I thought it best to remain with these rounded-off values. So . . . .
Radius of Earth: 3,960 miles.
Radius of Moon: 1,080 miles.
I made sure that my drawings of the Earth and Moon were both in proportion and exactly to scale. The process is simple using the latest artwork software. In drawing each circle one will just ‘key-in’ the mile-radius figures of each in millimetres and then reduce both together so that one has a comfortable-sized image to work with.
Artwork by Gary Osborn. Copyright © 2012 All Right Reserved.
Figure 3: An 11 x 14 rectangle produces the precise angle of 51.84º from diagonal corner to corner.
Artwork by Gary Osborn. Copyright © 2012 All Right Reserved.
Figure 4: The Moon placed on top of the Earth.
Artwork by Gary Osborn. Copyright © 2012 All Right Reserved.
Now we draw a triangle on the image.
Again, the base of the triangle is 3960 – being the radius of the Earth – and the upright is 3960 + 1080 = 5040 – being the combined radiuses of both the Earth and Moon.
Artwork by Gary Osborn. Copyright © 2012 All Right Reserved.
Artwork by Gary Osborn. Copyright © 2012 All Right Reserved.
Figure 7: Applying the 11 x 14 rectangle, resulting in a 3 x 11 rectangle for the Moon radius and 11 x 11 square for the Earth radius.
Artwork by Gary Osborn. Copyright © 2012 All Right Reserved.
This is really not surprising as the diameter of the Earth and Moon relate to one another according to the ratio – 11:3.
And now knowing this, we can use a much simpler method to draw up the relative proportions of the Earth and Moon and I present this simpler method so that anyone can do it.
First we draw up two squares – one at 11 x 11 and one at 3 x 3.
For example, using computer art software we simply draw the first square at 110 x 110 millimetres and the second square at 30 x 30 millimetres.
We then draw two rectangles, both at 3 x 4. Again, draw two rectangles with the width at 40 millimetres and the height at 30 millimetres.
We then arrange these squares and rectangles as shown below on the left.
If we now circle both the large 11 x 11 square and the small 3 x 3 square as shown in the center diagram, we have the relative sizes and proportions of the Earth and Moon to scale at ratio 11:3 . . . simple.
It would appear that this is exactly how John Michell arrived at the Great Pyramid side angle of 51.84º. (See figure 1).
Note again the Pythagorean Triangle/Triples of 3–4–5 proportions, which is also expressed in the ‘47th Proposition of Euclid’ – otherwise known as the ‘Pythagorean Theorem’ – a subject we will touch on throughout the following three presentations.
Figure 8: Drawing up the Earth Moon to the ratio of 11:3 to find the Great Pyramid angle of 51.84 degrees.
Artwork by Gary Osborn. Copyright © 2012 All Right Reserved.
We can safely establish then that this is how the Great Pyramid of Giza angle of 51.84º was arrived at . . . by first knowing the sacred and combined proportions of the Earth and Moon. And it is a good possibility that this is why the inclined sides of the Great Pyramid of Giza were built to this same angle, as this angle incorporates the combined radius proportions of both the Earth and Moon . . . as well as the Sun as we shall soon discover.
It would perhaps surprise many to know, that the geometry of the final image on the right – which presents us with the combined proportions of the Earth and Moon and the 51.84º side angles of the Great Pyramid – was encoded by the French Baroque-period artist, Nicolas Poussin in his second ‘Self Portrait’ dated 1650. In fact, it was this painting by Poussin that had first led me to discover the 23.5º Earth’s axis angle within the angle geometry of the Great Pyramid of Giza, as shown in cross-section.
As we can see, both Poussin’s pyramid-cut stone ring, which he wears on his little finger, and the ‘All-Seeing Eye’ in the diadem of the goddess in the painting behind him, are connected by the precise angle of 23.5º. This is exactly as this angle appears within the Great Pyramid of Giza, beginning from the south-corner vertex and running through the center of the so-called King’s Chamber. There are also precise Pythagorean Triangles of 3–4–5 proportions in the painting, which are not revealed here, but will be revealed in another presentation.
Figure 8: The combined Earth-Moon proportions and Great Pyramid side angle of 51.84º and Earth’s axis line of 23.5º, in Nicolas Poussin. Self Portrait (Version II, 1650. 78 cm x 94 cm). Musée du Louvre, Paris, France.
Artwork by Gary Osborn. Copyright © 2012 All Right Reserved.
What Poussin encoded in his paintings lends credibility to this revelation and more that we will discover using this same geometry. If nothing else, it shows that people throughout history knew, understood, and believed in this sacred geometry and the connections made with it, and this is something I will also reveal more of later.
Figure 9: The 51.84º triangle from which the angle geometry of the Great Pyramid was derived.
Artwork by Gary Osborn. Copyright © 2012 All Right Reserved.
Looking at the images above, surely this would imply that knowledge of the size (circumference/diameter/radius) of the Earth and Moon was known long before the Great Pyramid of Giza was built.
It is a fact that the ancient Greeks already knew the diameter of the Earth is around 13,000 km (8,000 miles). By observing the Moon carefully during a Solar Eclipse, and seeing how the Earth’s shadow fell on it, the Greeks found that the diameter of the Earth’s conical shadow at the distance of the Moon was about two-and-a-half times the Moon’s own diameter. As the Moon is 110 Moon diameters away, then the Moon’s distance from Earth must be more than 400,000 km away. From this they then calculated the Moon’s diameter of 3,700 km (2,160 miles).
Also, as those who are proficient in mathematics would know, the combined Earth radius and Moon radius (square root of phi – √ Φ), resulting in the hypotenuse angle of 51.84º, is also the ‘Golden Triangle,’ phi 1.618 (Φ) – the mathematical expression being A²+B²=C².
Figure 10: ‘Squaring the Circle’ and Phi 1.618 Φ.
Artwork by Gary Osborn. Copyright © 2012 All Right Reserved.
I have also “squared the circle” as John Michell had done and as so many have done since and this is important as we will discover in a subsequent chapter. Although there have always been initiates among us who already understood these things from what was preserved and passed down to them through tradition, only now, and through the most intelligent among us, has our civilization reached a point where we are able to grasp, read and comprehend the complex and advanced, geometrical and mathematical knowledge that has been built into the Great Pyramid . . . and does this not tell us something?
As one source expresses it:
“The architects of the Great Pyramid of Giza were extremely wise beings, with an advanced knowledge of math and astronomy far beyond the standard of their time, and I would say even far beyond our own current knowledge.”
Egyptian Numerology: the Pythagorean Triangle and Its Esoteric Meaning. Some Historic Notes and Brief Comments on Sacred Geometry. Antonietta Francini, S.R.C., M.D., with Benefactor Taciturnus,
F.R.C. Rosicrucian Digest No. 1. 2009.
American researcher David Sereda has come to the conclusion that the “builder of the Great Pyramid knew the square of the circle and the cube of the sphere.”
Sereda says that “the pyramid is actually a hidden sphere.” (See Square of the Circle & Cube of the Sphere by David Sereda).
In other words, a sphere at the same height of the Great Pyramid would be equal in ‘cubic volume’ to the cube of the pyramid.
Sereda adds that this would only work at the Great Pyramid angle of 51º.51’. (51 arc degrees and 51 arc minutes).
It will not work at any other angle.
Π x 5151 = Φ
Π (pi) x 5151 (slope angle of Great Pyramid 51º.51’) = Φ (phi).
In view of what we have learned already, it is quite absurd to think that the Egyptologists still cling to the belief (merely a theory) that the Great Pyramid was only a ‘tomb’ for a vainglorious Pharaoh of the Old Kingdom, 4th Dynasty, named Khufu (or Cheops – his Greek name), and nothing more.
I will now reveal how the angle geometry of all three Giza Pyramids are based on the Pythagorean Triangle of 3–4–5 proportions and the ‘47th Proposition of Euclid’ (or ‘Pythagorean Theorem’) and how these sacred geometries also encapsulate the combined Earth-Moon diameter, 11:3 ratio proportions.
It is now evident that the side angle value of the Great Pyramid of Giza (seked of 5½ palms (11 x 14) – yielding the angle of 51.84° or 51°51’ in arc minutes), was derived from the combined 3/11 ratio proportions of the Earth and Moon.
The combined Earth radius and Moon radius (square root of phi – √ Φ), resulting in the hypotenuse angle of 51.84º, is also the ‘Golden Triangle,’ phi 1.618 (Φ) – the mathematical expression being A²+B²=C².
The architects of the Great Pyramid also knew the square of the circle and the cube of the sphere – Π (pi) x 5151 (slope angle of Great Pyramid 51º.51’) = Φ (phi).
Pages in this Series of Presentations