1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.
• Lunar tides, which alternate about every 6 hours.
• Days and nights, which take roughly 12 hours each.
• Lunar months, with a lunar cycle completing in roughly 30 days.105
• 12 lunar months of 30 days each approximate a 360 day solar year.106
Figure 18 - Decimal and Sexagesimal Counting
Telling Time
Within sexagesimal mathematics, the circle symbolizes time, with no beginning and no end. The Sun’s 360° path across the zodiac takes approximately 365 days to complete, with a difference of 5 festival days. The clock goes about its business with no regard for whether or not the world actually runs in integer multiples or divisions of 12, but Base-60 provides a reasonably close approximation to natural phenomena.
Figure 19 - Triangulating Against the Heavenly Bodies
Sumerian and Babylonian astronomers measured the passage of time by triangulating on the sun’s daily, monthly, and annual position, noting that position in relation to the background stars. They learned to calculate the sun’s changing angle by using various trigonometric functions; like the tangent function that compares the length of a measuring rod, with the length of the shadow it casts (Figure 19). The oldest trigonometric table ever discovered (Figure 30a) used the Pythagorean theorem to determine the length of the third side and then used the cosecant function to determine the angle θ (see Figure 29). In Chapter 5, we will examine indisputable archeological evidence that trigonometry and the Pythagorean theorem existed long before the birth of Pythagoras.107
Imagine yourself as part of a race of vastly superior pre-Sumerians who had recently emerged from the shelter of cave life. The celestial dome would certainly have been much more interesting than the dome of your cave. But, even as part of an advanced race you would have taken a considerable amount of time to simply observe the heavens before understanding how and why to construct the first star observatory. The first religious temples were archeoastronomical devices used to measure cyclical events in the sky. The most well known temple/observatory of this genre is Stonehenge, built in England around 3,000 BCE (Figure 20a). But, what is religious about Stonehenge? How can telling time by observing the sun’s position in the sky possibly relate to religion?
Figure 20a - Stonehenge in England (circa 3000 BCE)
The oldest of these, Göbekli Tepe (literally: Navel Mountain; circa 10,000 BCE), was discovered in 1994 during excavations in southeast Turkey by German and Turkish archeologists (see Figure 20b); a few hundred miles from Mount Ararat, where Noah and his family were said to emerge from the Ark.
Figure 20b - Göbekli Tepe (lit. Navel Mountain)
Concentric Circles in the World’s Oldest Temple (circa 10,000 BCE)
Until the 1994 discovery, the oldest temples discovered had also been circular temple/observatories. There were ten circular stone foundations dating to the Halaf period (circa 5500 BCE) at Tell Arpachiya, which archeologist Jack Finegan suggests were “the earliest structures in northern Mesopotamia to have been erected for religious purposes.” Finegan also considered 18 circular shrines unearthed in Eridu as “the earliest shrines in southern Mesopotamia.”108 Temple/observatories were also discovered at Karahundj in the Armenian highlands (circa 5000 BCE; Figures 20c - f), and in Nabta Playa, Egypt, as well as a number of other important archeological sites.
Think of Aryan priestly astronomers observing the skies from the ancient stone circle of Karahundj in the Armenian Highlands, ca. 5000 BCE. After constructing a device with the accuracy of a gun sight (Figure 20f), it would be much easier to quantify their observations. Of course, there are no written records going back this far, so we need to imagine what a race, possibly superior to our own, might have accomplished through careful observation? Figures 20b - f suggest that quantifiable observations of the heavens could have occurred 2000 years or more before the invention of writing. In the following passage, however, even noted scholar Joseph Campbell seems to accept the notion that sophisticated mathematics could not have come into existence before the invention of writing:
The most important and far-reaching cultural mutation of this kind in the history of the human race was that which occurred in Mesopotamia about the middle of the fourth millennium B.C., with the rise in the lower reaches of the twin rivers Tigris and Euphrates, of a constellation of city-states governed by kings according to a notion of cosmic order and law derived from a long continued, systematic observation of the heavens ... members of a new type of highly specialized , heavenward gazing priesthood invented, ca. 3200 B.C., writing, mathematical notation (both sexagesimal and decimal), and the beginnings of a true science of exact astronomical observation.109
Figures 20c - Karahundj: Armenia’s Stonehenge (circa 5000 BCE)
Figure 20d, e & f - Stones Line Up Like a Gunsight
The existence of Göbekli Tepe and Karahundj as star observatories suggest the possibility that the origins of geometry and trigonometry were earlier than most scholars are prepared to acknowledge. Neugebauer admits that he expends little effort to explore the ancient origins of the fully developed science found within Old Babylonian period cuneiform tablets (circa 1800 BCE). There are other scholars, however, like Jöran Friberg, who search for the beginnings of mathematics in 4th millennium Sumerian script during the transition period between pictographs and cuneiform phonetics.110 As we have already mentioned, Kramer goes so far as to credit the superior pre-Sumerian race with all of the great advances of Sumerian civilization. In light of Kramer’s observations, perhaps we should at least reexamine our prejudice regarding the IQ and accomplishments of prehistoric man. We should take care not to underestimate the capabilities of a race that may well have been more biologically evolved than our own. Is it possible that mathematics and science preceded writing? Perhaps time will tell.
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