When I put out my bedroom lights and rested my head on a pillow last night the phrase – ‘As Above, So Below’, from the ancient text, the Emerald Tablet, which is a cornerstone of Hermetic philosophy and alchemy, suddenly popped into my head.
Then, I had what I can only describe as being an Epiphany.
I saw in my mind an image of a Pyramid placed on top of an inverted mirror image of itself with a circle around the circumference of the ‘holistic’ diamond shaped image, and also around the transverse radius.
Then it struck me like a thunderbolt – this image represents a globe.
You can draw it on a piece of paper.
I keep a pad of paper next to my bed with a pen in case I wake up in the middle of the night with a new idea, and did exactly that.
I then had a peaceful night’s sleep and when I was awoken by the Sun’s rays this morning another idea occurred to me.
I knew that the Great Pyramid of Giza was originally covered by smooth limestone facing blocks, i.e. the steps underneath were not visible. But of course in designing the Pyramid they must have been drawn on a Plan.
Then it occurred to me what if each course i.e. layer of these steps when drawn horizontally on a spherical map of the earth i.e. a globe, could be designated as a line of latitude?
Then as I lay in bed I imagined the Great Pyramid of Giza visually turned on one side.
Then it occurred to me, if when the Pyramid was turned on its side, could each of the course steps i.e. each layer of the steps, vertically be designated as a line of longitude?
Then I realized, if they could, this would create a grid.
My next thought was, if e.g. at the intersection of any grid lines a fixed point in space i.e. a bright star, observable with the naked eye in the night sky could be marked as a fixed point on the grid e.g. by creating a channel inside the masonry of the Pyramid that pointed directly to the star on a particular date, then somehow using math to factor in precession, could a map of the sphere created by two pyramids be used as a navigational tool at sea, i.e. to chart a position by reference to lines of latitude and longitude?
So, in the office this morning, I asked AI – ‘Could a grid on a map of a sphere be used by sailors in Antiquity to navigate by the stars?’
AI replied – ‘Yes, ancient mariners could have used a grid on a spherical map to navigate by the stars, as the concept of spherical coordinates and astronomical positioning was understood in antiquity, notably by figures like Hipparchus, who compiled stellar catalogs and used spherical trigonometry. While ancient star charts were less precise than modern ones and lacked the detailed tables and accurate chronometers of later eras, sailors used instruments like the astrolabe and quadrant, combined with knowledge of star movements and constellations, to determine their latitude and general position at sea.
Understanding the Spherical Nature of Navigation
- Spherical Coordinates:
The Earth is a sphere, and ancient astronomers like Hipparchus understood that a spherical coordinate system, which uses two angles to define a point on a surface, was the natural way to describe positions on a sphere.
Hipparchus developed models using spherical trigonometry to understand celestial movements.
Key Tools for Ancient Celestial Navigation
This device could measure the altitude of celestial bodies and, with the help of tables or charts, help determine latitude.
Originally an astronomical instrument, the quadrant was adapted for navigation to measure the height of stars above the horizon.
While rudimentary compared to modern charts, ancient mariners used star maps to identify constellations and celestial bodies.
How They Used the Grid (Conceptually)
- Latitude:
In the Northern Hemisphere, sailors could determine their latitude by measuring the angle of the North Star (Polaris) above the horizon, as this angle is roughly equivalent to the observer’s latitude.
- Position Lines:
By taking sights on two or more celestial bodies, and referencing their position on a spherical map, sailors could conceptually draw lines or circles to determine their location.
- Star Position and Time of Year:
Sailors had to know which constellations were visible at different times of the year and in different hemispheres to use them effectively.
Limitations in Antiquity
- Accuracy:
Ancient instruments and charts lacked the precision of later tools like the sextant, limiting navigational accuracy.
- Longitude:
Accurately determining longitude was a significant challenge without precise timekeeping devices (chronometers), which were invented much later.
- Focus on Coastal Sailing:
While some advanced navigation occurred, much of ancient maritime activity, especially in the open ocean, was focused on more familiar waters, like the Mediterranean.’
See also my recent blogs at www.artandcivilization.tv:
- ‘The Hermetic Texts’ – in which I also enquire – ‘‘Is there a nexus between the Hermetic Texts, The Emerald Tablet, Alchemy and the Philosopher’s Stone?’
- ‘Aesthetically, could it be argued that the Giza Necropolis creates a hologram of Orion’s Belt?:
- ‘Based upon the evidence of Egyptian ship-building techniques is it possible that a ship was designed and built for disassembly transportation to the Atlantic and Indian Oceans for reassembly and seafaring?’
- ‘Did the builders of the Great Pyramid at Giza know that the Earth was round?’
Q’s and comments added:
My Q. – ‘What is Spherical Trigonometry?’
AI – ‘Spherical trigonometry is a branch of geometry that studies the properties of triangles drawn on the surface of a sphere, rather than on a flat plane. This field is essential for calculations in astronomy, navigation, and geodesy, where the curvature of the Earth or celestial bodies must be taken into account.’
My Q. – ‘Who invented i.e. was the Father of Spherical Trigonometry.’
AI – ‘The title “Father of Spherical Trigonometry” is not definitively attributed to a single individual but is often credited to Menelaus of Alexandria for his surviving text Sphaerica, or to Ibn Muadh Al-Jayyani who has also been called the “Father of Modern Spherical Trigonometry”. Other significant figures include Hipparchus (the father of trigonometry) for his early work on solving spherical triangles, and Nasir al-Din al-Tusi for making trigonometry an independent branch of mathematics with significant contributions to spherical trigonometry.
Key Figures in Spherical Trigonometry
- Hipparchus (c. 190–120 BC):
Considered the “father of trigonometry” in general, he provided methods for solving spherical triangles and compiled early trigonometric tables.
- Menelaus of Alexandria (1st century AD):
His surviving work, Sphaerica, is the first known text on spherical trigonometry, containing theorems and rules for spherical geometry.
- Ibn Muadh Al-Jayyani (10th/11th century):
An Islamic mathematician and astronomer who made significant contributions to spherical trigonometry and has also been called the “Father of Modern Spherical Trigonometry”.
- Nasir al-Din al-Tusi (13th century):
A Persian polymath who established trigonometry as a branch of mathematics independent from astronomy, providing extensive expositions on spherical trigonometry, including stating the law of sines for spherical triangles.’
So, if the builders of the Great Pyramid of Giza knew about Spherical Trigonometry, they possibly acquired or developed this knowledge over 12 thousand years before Hipparchus.
My Q. – ‘What arguments have been made by Charles Hapgood that older civilizations possessed advanced knowledge of spherical trigonometry?’
AI – ‘Charles Hapgood argued that ancient civilizations possessed advanced knowledge of spherical trigonometry, evidenced by their sophisticated world maps, such as those he studied from the Middle Ages (Portolano charts) and the Piri Reis map. He claimed these maps were compiled using a spherical projection method that ancient geographers lacked, suggesting the original mapmakers—possibly ancient Egyptians—knew how to translate spherical surfaces to flat maps with mathematical precision, an understanding lost to the Greeks.
Hapgood’s Core Argument
- Spherical Projection vs. Flat Maps:
Hapgood’s central argument was that some ancient world maps, despite their flat appearance, were based on original spherical maps.
- Mistranslations by Later Civilizations:
He proposed that ancient geographers, like the Greeks, were confronted with spherical maps and lacked the understanding of spherical trigonometry to interpret them accurately.
- Evidence in Ancient Maps:
According to Hapgood, they had to translate the complex spherical geographical data (latitudes and longitudes) into terms of a flat surface, leading to distortions.
- Ancient Egyptian Knowledge:
He concluded that the ancient Egyptians possessed advanced knowledge, including the ability to accurately translate between spherical and flat surfaces using mathematical methods, while the Greeks did not.
Evidence from Maps
- Ancient Origin:
Hapgood identified cartographic features on maps, such as the Piri Reis map and others from the Age of Discovery, that appeared to show a level of geographic accuracy and detail suggesting their origin was from an earlier, more scientifically advanced civilization.
- Missing Information:
He noted that even with distortions, the maps often contained significant geographic details, like the accurately placed mountain ranges on the Finaeus map, that implied a sophisticated understanding of the Earth’s shape and surface.
- Advanced Cartography:
The accuracy of some mapped locations, even close to the poles, led Hapgood to believe that the source maps were created using advanced cartographic and mathematical principles, including spherical trigonometry.’