Sympathetic Vibratory Physics

also Squaring of the Circle.

Keely
The vapor from the liberator, registered at 20,000 lbs. per square inch has a range of atomic motion of 1333 1/3 the diameter of the atmospheric molecule with constant rotary vibratory action. At 10,000 lbs., 666 2/3, at 5,000, 333 1/3, at 2500, 166 2/3, at 1250, 83 1/3, at 625, 41 2/3. The higher the range of atomic motion the greater its tenuity and pressure. The very evolution on the negative shows a vacuum of a much higher order than was ever produced before confounding all theory to analyze. The highest vacuum known is 17.999999, or not quite 30 inches, but Keely produced etheric vacuums repeatedly of 50 to 57 inches ranging down to 30 inches or 57 lbs. All operations of nature have for their sensitizing centers of introductory action, triple vacuum evolutions. These evolutions are centered in atomic triple revolutions, highly radiophonic in their character and thoroughly independent of all outside forces in their spheres of action. No conceivable power, however great, can break up their independent centers. These triple centers are the foundation of the universe, and mathematically considered, the respective and relative motion of these atomic triplets, gravitating to and revolving around each other, is about one and one-third of their circumference. The problem of this action, when analyzed mathematically, (taking it as the quadrature of the circle) would baffle mathematical science to bring it to a numerical equation. Every revolving body is impressed by nature with certain laws making it susceptible of the operation of force, which being applied, impels motion. These bodies never can approach nearer than a certain limit, nor farther than a certain point. They are, at some mean point, made perfectly equal, and may therefore be considered as one force and as one element. It matters not that other and disturbing forces exist outside or inside the space these bodies revolve in, because if this force must be considered as acting uniformly, applying itself to each of these bodies in a way to produce a perfect equation on all, it is as if this outside force were nonexisting. [Snell Manuscript - The Book, page 2]

Treatise on the geometry and mathematics of Squaring of the Circle: Quadrature of the Circle

"In QUADRATURE OF THE CIRCLE Revealing THE TRUE VALUE OF Pi, we explore and expand upon Mr. Parker's findings. In clear, up to date English the central principles of the circle are revealed which allow the true and correct value of Pi to be calculated. Several important characteristics of the the arc, straight line, circle, triangle and regular polygon which have not been given due recognition are explored. Methods for constructing each geometric shape are given and the great importance of the Pythagorean or 'right' Triangle described. General formulas are developed allowing any regular shape to be made equal to another by area, perimeter, or proportion." Quadrature of the Circle George Hull's book

Cubing the Sphere - 3D version of squaring the circle

Ramsay
The ratio of 1:2 belongs to neither proximate nor differential periods. It corresponds to unity; and, like the square and the circle, it admits of no varieties. [Scientific Basis and Build of Music,page 106]

Alchemy
Explanation
Whoever delves into this subject with diligence, seeking to attain the quadrature of the circle, and strives as much as possible to grasp the true secrets of the wise among common philosophers, may he know that these are but the surface of hermetic knowledge that can be found. For only those possessing true knowledge of the Magistery and knowing how to proceed correctly in their practice, from the beginning of the work until its perfect completion, have the understanding necessary for comprehending the ancient wisdom of the sages.
They bring together the two opposites, the fire and the water, which appear irreconcilable and separate. Yet, in their perfection, they unite into one primordial principle.
Those who desire to unlock the secrets of the circle and to square nature
Should follow this faithful guide presented here.
From fire and water, friends, comes the sun and the moon.
Water comes from fire, and from air springs the earth.
The circle in its trinity will become squared.
When you have mastered the balancing of opposites, both above and below, you will see all and understand all.
These teachings, once hidden, will be revealed to the wise through alchemical means, and the circle, which seemed insurmountable, will appear as a straightforward path."
"What does this page show regarding all the magic of these figures? They are drawn in a mirror or on a tablet, which reveals their spirit or their hidden essence, but which the eye perceives in their figures.
The first point in the mystery is indicated here by the table, which has a detailed history of creation.
However, consider in this figure the four elements drawn to form sympathetic qualities and how they remarkably agree in their influence toward their judicial vision.
The seal of the echoes: air, hot and humid; water, humid and cold; fire and dry earth.
This seal should be heated by air and fire."
"The mutual heat that can be extracted from them, as it were, hopes for an admirable effect.
The same fire is mixed in the same way by its dryness with that of the earth; the humidity of the air with that of water, the cold of the air with that of water, and that of fire with the dryness of the earth.
These elements are magically converted one into another by their corresponding qualities: dry, cold, humid, and warm. Forms are then accurately manifested within the limits of their philosophical principles, which encompass a remarkable amount of wisdom that extends even to the details of the stone.
This stone maintains its nature within the "Fire of the Hermaphrodite," which governs the order.
This gives birth to infinite influence."
"To examine everything that the body offers, this is the goal of the Elixir.
For in our elixir, nothing should be left behind, all must be done by fire and water, but with a constant moisture that is achieved by air and the fluid of the dew.
The four elements do not form a single world, but are accompanied by the stars of the moon, which will manifest our magical work under the benign influence of the constellations of Aries, Taurus, Gemini, Libra, Sagittarius, and Scorpio.
From them, one learns the art of the double-headed magic, which encloses itself in the song of the heart, always conserving its nature through the path it takes."
"The vulgar courses are finished, but the bridgework sent by the lapwing of the Sun to the divine reconciliation, after having wisely treated their approaching spirits, made it possible to relate tender loves of which I cannot speak further here, nor explain them in more detail.
However, we note that Mercury, old negotiator of the philosophical adept, seeks the men of the sages as I have told you.
A flexible, living spirit, warm, acidic, and humid,
Again, what appeared to Lotham astonished him.
But here Mercury is abstracted from subtle bodies
by a quiet and vulgar man, full of weaknesses.
A spirit so named by men, Mercury of Mercury,
more subtle, hot, and tender, that is dematured.
Vulcan by fire secretly sanctifies the philosophical globe, imposing on it the fire, and their authors the fine works of the sages." [“Seven Keys of Honoratus Marinier”, c. 1785, MS. Wellcome 3442]

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Quadrature of the Circle