IV.4.4 The Sexagesimal System

Since the Sumerians divided the celestial orbs into twelve zodiac signs, they divided the orb into 360 degrees, so each zone spans 30 degrees from the celestial orb. If we consider a circle whose diameter equals one meter, the length of the thirty-degrees arc will be equal to one Cubit: . This calculation, however, is not accurate, because the Meter should be defined according to the Cubit and not the opposite, since the definition of the Meter suffered slight variations as we have seen in section I.2.2 above. The problem is that we don t have any precise value of the length of Cubit used in ancient systems, because various civilizations used slightly different cubits. The Egyptian Royal Cubit is the earliest attested standard measure, but the Cubit rods that have survived range from  to  meters.

Nevertheless, even with the current conventional measurement units, when we make a pendulum with one meter length and we displace its mass with a small angle or distance, it will take almost exactly one second to return to its original equilibrium position, because: , where  is the gravitational acceleration due to gravity, as shown in Figure IV.11.

Figure IV.11: The Cubit, the Meter and the Second are related through the motion of the Pendulum. A circle whose radius is one Meter, its circumference equals twelve Cubits, and when the length of a Pendulum is one Meter and the mass is displaced, it will take one Second to return to its equilibrium position. These definitions are NOT exact because these standard units have suffered various alterations throughout history, and it is not known exactly what is the ultimate reference.

The Meter was originally defined in 1793 as one ten-millionth of the distance from the Equator to the North Pole. In 1799, it was redefined in terms of a prototype Meter bar, which was changed in 1889, and in 1960, it was redefined in terms of the wavelength of a certain emission line of krypton-86.

Before that, in 1670-5, the Meter was suggested as the length of a pendulum with a one-second period, but the French Academy of Sciences selected the meridional definition over the pendular definition because the force of gravity varies slightly over the surface of the Earth, which affects the period of a pendulum.

The current definition was adopted in 1983 as the length of the path traveled by light in a vacuum in  seconds.

It may appear that the original 1670 definition of the meter, as the length of a pendulum with a one-second period, is purely conventional, and so it has been replaced in 1793 by the meridional definition, as one ten-millionth of the distance from the Equator to the North Pole, which also may appear to be purely conventional. However, whether the scientists who proposed these definitions knew it or not, this standard measurement system is fundamentally related to the ancient sexagesimal system and the physical properties of the Earth, through the pendulum parameters, as Figure IV.11.

So clearly there is a very strong correlation between the sexagesimal system, which is still being used in measuring time, and the length of Meter and the gravity of the Earth. It is also evident that these and other fundamental numbers, such as  and , the golden ratio, have been clearly encoded in the dimensions and position of the Pyramids of Giza, such as the latitude of the Grand Gallery in the Great Pyramid of Giza () which coincides exactly with the value of the value of the speed of light in vacuum.

The sexagesimal numeral system, with sixty as its base, was originated by the Sumerians, and continued to be used by the ancient Babylonians. In modern times, the decimal system is used for general computations, but the sexagesimal is still used in measuring angles, geographic coordinates, and time.

The number 60, is a superior highly composite number. It has twelve numbers distinctive factors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60, which means that many fractions involving sexagesimal numbers can be simplified when expressed in the sexagesimal system. That is why it is still quite convenient in expressing time, because one hour can be divided evenly into durations of 30, 20, 15, 12, 10, 6, 5, 4, 3, 2, and 1 minutes. In many ancient texts, the sexagesimal is used most uniformly and consistently in mathematical tables of data, because of its mathematical advantages for writing and calculating fractions. Ptolemy used the base 60 to express the fractional parts of numbers in his Almagest. In particular, his table of chords, which was essentially the only extensive trigonometric table for more than a millennium, has fractional parts in base 60.

Infinity can be described as a recursive loop, that recurs indefinitely, such as the squaring or integrating curves. This concept is like confining an unlimited spirit into a body. Contemporary computer science uses this kind of recursion to model the fractal structures in nature, such as Koch Snowflake fractal in Figure I.2.

Figure IV.12: The Babylonian sexagesimal system is based on dividing the circle into 360 degrees, and each degree into 60 seconds. The circle is also divided into 12 zones, each of 30 degrees span.