Circular Proportions

The trisection of an angle is a famous problem used to encourage innovative thinking. However recently , since the algebraic proof of impossibility, it has been used to brainwash vulnerable mathematicians into a hopeless conformity.

The solution was clearly found by the ancient Sumerian and Akkadian peoples , the Dravidian and Harrapan Indus Valley civilisations and the Mongol chinese steppe and Plain civilisations, all of whom had the wheel and the 60 modulo arithmetics.

The issue is a pragmatic metrical one, and relies on skillful Neusis, as well as expertise with circles

Such an expertise is now called sacred geometry, but it is a science of spherical and circular relations. Of all the forms we have explored it is the circle that encodes proportion in its simplest form: one perimeter to 1 diameter!

We all accept that a circle can be patterned by six petals formed by 6 overlapping circles. We accept the number 6 because we see symmetry. This means we cannot distinguish the 6 forms we see in the pattern by any known or used measurement; accept by calculus! In that branch of ” precision” we find pi to be not 3 but 3.1415… Because our calculation is not based on observation but by a division process!

The difference is profound. Do we trust our eyes or our formal calculation process? Both, because as it turns out our ancestors did not need precision. 6 was good enough for them even though we know that it should be 6.28…

When your compasses do not quite meet the diameter we are taught to do it again until it does, because the radius must step 6 times into the circumference! It does not, but by convention we say it does.

The pattern of 6 is so compelling , we want 6 equilateral triangles as a constructed Constant of space. Construct them in a circle and they fit, construct them in a tessellation and they fit, but they do not precisely fit a circle anymore! The construction in the circle has slightly distorted the plane forms.

Using a constant radius we can construct the sacred geometrical flower pattern. That is when we can start to set out proportions . While we ca crowd 6 around a centre of a circle with 1 radius we can crowd 9×6 around a circle with 3 times the radius! This means we can trisect the diamond made of 2 equilateral triangles . But we have to use the chord length of the 1/3 rd circle to step these off on the larger( 3 x ) circle.

This proportion exists in this set up because circles are proportions. Without a rigid measure it is fiddly to do it is much simpler with a set of measuring tools that can retain and transfer these lengths to the proper positions

There exists a circle for which this length is the precise chord which trisect the arc into 3 similar sectors. Finding it by trial and error can be made easier by using the sacred geometry to narrow the search down. The Neusis becomes simpler and more precise.

Draw an angle and make the limbs or rays long enough to step off 3 radii. The radius is the semi circle drawn at the vertex of the angle, extend your compass to 3 radii and draw the semi circle,

Using a pair of rigid divider measure the chord of the angle in the smaller semi circle.

Step this off on the larger arc until step 3( which is too small ) and step 4 (which is too large).

Leaving the dividers fixed , now use the intersection of the upper ray between steps 3 and 4 with the semi circle as the centre for a circle that has a radius given by the displacement to step 4. Retaining that radios go to step 3 and mark an intersect toward 4 as the centre of a second circle through 3

Now using the point of intersection of these circles with the ray draw a semi circle from the vertex. Using the dividers step off to point 2 along this arc. Setting your compass to the displacement from the point on the ray cut by this arc( the same as that cut by circles at 3 and 4) now draw a circle that intersects circles 3 and 4

The circles are a probability space . Where they intersect is probably the point for the radius of a semi circle which can be trisected by the dividers precisely.
There may be 2 intersections that are clear. Choose the one that fits best.
Where these circles cut the upper ray is a point which was used to draw a semi circle. That semi circle will be unable to contain more than two steps within the arc of the angle.

The demonstration relies on neusis so be as accurate as possible.

The empirical deduction is that the 3 x radius semi circle is going to present an arc( the angle) which being less curved will be too big, by a proportion . Points 3 and 4 are used to narrow the space that the sought for circular arc must pass through. By using the 4th point as a radius displacement and drawing a circle the bounds can be seen to decrease until the circle cuts the upper ray. Thus any circle drawn from the vertex passing through that circle has a high probability of approaching the required radius from above.

The second circle passing through point 3 from a marked centre has the probability of a circle approaching the correct circle from below. Thus both bound a circle which will likely contain 3 to 4 steps

The second circle will bound a circle that is likely to contain 2 to 3 steps

Where they intersect has a high probability of being the correct circle requiring precisely 3 steps to equal the angle.

Clearly the dividers must be kept rigid and the stepping off done as accurately as possible.

Should the result not be “perfect” then the two guiding semi circles, just drawn , can be used to repeat the method.

You will find that if the angle is a 60° or 90° or some multiple of those the circles at 3 and 4 will very nearly coincide. Do not neglect to differentiate the points of intersection.

At first I thought this was a method of approximation relying on proportions un related to sacred geometry, but when I saw that the circle count was 4:3 for the 120° I could see then the sacred geometrical pattern peeking through. The first radius would cut the smaller circle into 6, but the chord was cutting the 3 x circle into approximately 12. The circle I sought would be cut precisely into 9 by that same chord.

These are empirical findings, the sort every geometer should be looking for as a matter of professional expertise!



We mathematicians have become obsolete.

We were doomed to die out as soon as “we” were born in the medieval period as a class of astrologers whose main task was to calculate the positions of planets and stars and the length breadth weights and measures of the empire.

Prior to that only Astrologers were given that respect and position within society, based on the Sumerian and Egyptian empires and their need for insight from the heavenly gods as to the opportune time for any and every venture. Such shamans using various practices were called sages and seers , whose visions were highly sought after and whose interpretations were regarded with awe.

The quest was to see, to visualise how any event would carry out its due process within the heavenly cycles.

It was the Pythagorean school who developed the ancient arts in Astrology and shamanism into a single or Monas-tic tradition. They collected together the worlds knowledge of Astrology and systematised it gradually into a coherent system based on the Mosaics of their Mousaion or temples to the Musai. 

Such temples collected all the traditions and curated them , bringing together a vast worldly knowledge of Astrology and the Arts. To study at one of these Monasteries was to devote ones life to ceaseless research and curation, finding thought patterns that gave you mastery over the worlds astrological and Artistic knowledge, finding those patterns by a gift of the Muses!

The Aristotelian Lyceum replaced that whimsical tradition by hard logical and taxonomic learnings. For that reason alone the position of Mathematikos was doomed!

Prior to Plato extending the reach of the Pythagorean school and praxis into Athens Greece , Pythagoras was reputed to have established a Mousaion in Sicily under the Patronage of a local chieftain who provided the “Monks” with protection in return for their public services of education. Those that came to listen soon divided into the Akoustia and the devotees. 

The Akoustia came to listen to the public teachings but the devotees came to learn and to be inspired of a Muse so as to teach and express the gift of that Muse.

As such there was no formal curriculum as in the Taxonomically driven Aristotelian Lyceum , instead discourses on topics were given by the Mathematikos, those qualified to teach, and debate and discussion ensued. The Monk or devotee was set Koans, test designed to bring them into close vicinity of the Muses. The Musai would then whimsically impart gifts to the Devotee. What they may be would become apparent as the gift was expressed.

One became a Mathematikos, not by examination and jumping over hurdles, but rather by Merit, that is by the acknowledgement of other Mathematikoi and even non Mathematikoi that one had the gift of a Muse, and particularly the gift of astrology and it’s counterpart geometry.

A Mathematikos was thus a gifted person who could reason by the astrological and geometrical patterns of the heavenly Musai, in particular circular or periodic proportions.

The development of the Arithmoi which was the distinguished Pythagorean name for what we now call mosaics was key to all astrology and geometry, for by these fundamental patterns all proportions were distinguished .

When Aristotle failed to attain the Status of Mathrmatikos, it was not because he was not gifted, but rather because he did not merit it among the Pythagoreans. He openly disputed with them over the logic of their principles and while he remained a firm Platonist he broke away from some key Pythagorean teachings. 

Nevertheless others found him to be remarkably gifted and Learned and Philip of Macedonia employed him to tutor his 2 sons Alexander the great and Phillip. Alexander was very taken by Aristotle, regarding him as his Secret diviner, for him alone. Thus he was annoyed to hear that Aristotle had established an Academy to rival the Athenian Platonic outpost of Pythagoreanism.

The fates of empires lead to the dissolution of his academy while the Platonic academy survived until Roman times and even beyond in some meagre way enamoured by enthusiasts. Aristotles one salvation was the Arabs and Islamist rulers who defeating the Hellenists found Aristotles ideas to have been spread by Alexander everywhere. Thus they adopted them as clearly important to developing a strong Empire. 

Later when Rome sacked Athens and the Platonists fled, Pythagorean literature came into the possession of the Islamic scholars, who mistaking it for a version of Aristotelianism mishandled its provenance and bound together two formerly incompatible traditions!

The Pythagoreans were moved to display their philosophy, the Aristotelians were moved to taxonomise it! What we call geometry was and is a counterpart to Astrology, but the principles of both arose out of the Artisanship of the temple builders and mosaic layers, the Tekne or Mechanics in Greek society .

Because of their low status, but absolutely essential skills they were a distinct slave class or worker class. As such they were employed by the wealthy with reasonable respect to their skills, but no high borne would deign to learn such skills! That is until they became a Pythagorean!

In India the Temple builders were again a distinct group and they handed down trade secrets in Ganitas and Sutras that were the aphoristic stock in trade of the Indian temple Astrologers

The Arab empire firmly combined these 2 streams of similar wisdoms into Geometry and Algebra . Between the 2 the Arithmoi sat as a peculiar  skill which eventually in the Arabic universities became Arithmetic of Numbers, and Number theory. It was also called Kaballah by certain foreigners from mQuaballah to calculate, and even called Gematria( from geometria) or later Numerology.

This is the background to what Medieval scholars of the 14 th century began to call Mathematics. From arithmetical arts to arithmetical arts mainly employing the Algebraic rhetoric to mathematics seems to be the uneven course of the spread of this idea that 2 dysfunctional systems can be combined into one subject area.

Finally in the 1800 the crisis was reached, and geometry and mathematics was about to collapse. Algebra came to the rescue and from it analysis, and the problem limb Geometry was left to wither away!

However Hermann Grassmann thought this was a great injustice. He separated the geometry from Arithmetic and pointed out that that just left Formal expertises which were now inter communicant with a vibrant Geometry, Kinematics and Phorometry and Mechanics.

It is the Mechanics who in Archimedes time displayed the circular proportions on the Antikythera mechanism. It is the mechanics who displayed the polynomial calculations on mechanical computers and it is the mechanics and now Technologists who have refined the mechanisms to display the proportions on these mosaic screens,just as the Pythagoreans did.

The movement of the astrological bodies need only to be proportionately displayed, that was the goal of every Pythagorean. By so doing all wisdoms and knowledge can be organised and cohered or adhered to the great cycle and periods of the heavenly gods, and the opportune time for every venture clearly displayed, where possible.

We have no need of Mathematics so called now, because we can see on a computer monitor all we wish to envision, visualise and more!

What we require are those gifted individuals and mechanics who can build curate and maintain our new temples of the Musai.