The fundamental polarities of hyle or Nous correspond to frequencies of vibration which exceed by even or odd numbers, the exact multiples of 9 Hz, or cycles per second. The basic unit of 9 Hz represents an intrinsic period of expression. The period is defined as the shortest interval of time during which any phenomenon moves through one cycle of manifestation. For example, a sine wave of 9 Hz will complete a single cycle of expression (rise from the baseline to a peak positive amplitude, return to the baseline and onto a peak negative amplitude, and finally return to the baseline awaiting a second cycle) in one ninth of a second. A 1 Hz sine wave will complete a single expression of itself in one second (**Figure 6–Components of a Periodic Wave**).

As an aside, it is interesting to speculate upon how the ancient Babylonians derived the second. The ancients were well aware of the 365 1/4 day yearly cycle and the 29 ½ day synodic cycle of the moon. The ratio of the solar ‘year’ to the lunar ‘year’ is twelve to the nearest integer. Using the law of correspondence the following equation should hold: solar cycle : lunar cycle :: big daily cycle : little daily cycle. The little daily cycle would represent 1/12th of the day-night cycle, or since the length of the sunlight period is about equal to the length of the moonlight period, a unit of time equal to 1/24th of the big daily cycle is obtained. This is our present division of time called the hour.

Since the Babylonian number system was based upon the unit 1/60 rather than 1/10 of the later Arabic system, it is reasonable to suppose that they divided the hour into sixtieths to define the minute and into sixtieths again to define the second. Therefore, they were able to relate time measures for the macrocosm, or universe, to time measures for the microcosm, or man. The heart beats once every second, a man breathes once every 3 1/3 seconds (ratio of exhalation to inhalation is 3:2).

Likewise, all observable events in our universe pass through cycles of expression with certain specific periods. They have a beginning, progress to an evolutionary height and eventually come to a distinct endpoint. Both the beginning and end represent the endpoints of ongoing cycles of advancement or decay. The perfect cycle of ascending or descending harmonics is pictured in mystical literature by the spiral or entwined serpent.

Contrary to today’s science, the ancient esoteric schools classified vibrations as even or odd. Their method of determining whether a vibration was even or odd is provided through the agency of the nine-sided figure called the enneagram (see figure 7). Looking closely, you will notice that the enneagram is formed by combination of three other symbols. The outermost symbol is the circle representing cosmic unity, the All, the totality of existence, something without beginning or end. The circle is numerically equivalent to the number nine. Enclosed within the totality of the circle is the triangle with its three points. The triangle is numerically equivalent to the number three and represents the three creative forces (the affirming, the denying and the reconciling) supporting all phenomena. The points of the triangle (numbers 3, 6, and 9) divide the circle into a triad. Also enclosed within the circle is an unnamed, six-sided figure formed by joining the numerical points 1, 4, 2, 8, 5, 7, 1. The number six represents perfected expression, the joining of the triangles of the cosmic and material planes,. Gurdjieff used this figure to represent the law of the octave, the law of seven or the law of Heptaparaparshinokh (**Figure 7–Enneagram**).

Seven occurs by counting the points around the figure starting and stopping with the number one. Mystically, the ancient Egyptians associated the number seven with the goddess of measure, Seshat, who was the female counterpart of the male god, Thoth. Seshat’s emblem was the seven-petaled flower and is found on the of inscriptions accompanying the foundation stones of the temples. Seven represented harmony, growth the number of stages in a manifestation.

Although the enneagram was known to the ancient Egyptians and Sumerians, it was not until after the Arabs discovered the uniqueness of base ten as a counting system and its adoption by the Sufi schools of Central Asia that the enneagram reached its height. Prior to this time, these ancient laws were utilized using a system ascribed to Pythagoras (about 600 BC) of theosophic reduction, also known as the method of residues in mathematics.

In base ten, every number can be reduced to a single integer by success addition of the digits composing the number. For example, 792 can be written as 7 + 9 + 2 or 18 or 1 + 8 or 9, 1356 as 1 + 3 + 5 + 6 or 15 or 1 + 5 or 6. Expanding this concept further, we see that every number in base ten can be written by the formula 9n + r, where n and r = 1, 2, 3, …. 9. This result can best be seen in **figure 7**.

The ancients determined the evenness or oddness of a vibration as shown below. Even vibrations were those having two pulses, or beats; odd vibrations had three pulses, or beats. For example, a vibratory rate of 942 has an even, theosophic residual of 6 since 942 can be factored into the expression (9 x 104 + 6); a vibratory rate of 790 has an odd, theosophic residual of 7 since 790 can be factored into the expression (9 x 87 + 7).

The actual reason why the ancients divided vibrations into even and odd quantities is unknown. Moreover, some division is that the ancients were trying to explain the electric charge called by particles with mass, positive, negative and neutral. Modern science has yet to state its understanding of these ancients concepts.