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Iamblichus, quadratures, trisections and the lacuna of the cycloid

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Today Syria has been turned into a hellhole by the unmāda-traya. However, just before the irruption of the second Abrahamism to end the late Classical world, it was home to great men like Iamblichus. Hailing form a clan for priest-chiefs, Iamblichus taught at Apameia, a site which is today being ravaged by Mohammedan vandals aided and abetted by the sister Abrahamisms. After all one point where the Abrahamisms see eye to eye it is the destruction of the heathens and their sites. Iamblichus was a key link in a great chain of tradition from the yavana sage Pythagoras. Indeed, as Gregory Shaw pointed out in his paper “Platonic Siddhas”, Iamblichus could be seen as a siddha among the yavana-s. In the Lives of Philosphers, Eunapius Sardianus states that he performed certain secret rituals. Regarding that a rumor is thus provided:
“O master, most inspired, why do you thus occupy yourself in solitude, instead of sharing with us your more perfect wisdom? Nevertheless a rumor has reached us through your slaves that when you pray to the gods you soar aloft from the earth more than ten cubits to all appearance; that your body and your garments change to a beautiful golden hue; and presently when your prayer is ended your body becomes as it was before you prayed, and then you come down to earth and associate with us.” [Translated by WC Wright]

His epithet “most inspired” is clearly related to his status as the cognate of the siddha-s among the yavana-s. This is reflected in his ontology of the gods which can be compared in its theoretical foundations to that of tantra-s of the Hindu siddha-s. On one hand, he was the intellectual successor of Porphyry that famous critic of the second Abrahamism who had been thrashed by a mob of Christians in Palestine. On the other he was the teacher of Aedesius who in turn was the teacher of emperor Julian, who for a brief moment in history almost reversed the flow of the second Abrahamism. Iamblichus along with Proclus, that last sage among the yavana-s, preserved traditions of Pythagoras that mark the pinnacle of yavana knowledge, a height not scaled by other peoples of the ancient world. From the fragments of their work we hear of the yavana heroics in solving that ancient problem which goes back to the common ancestor of the yavana-s and the ārya-s the squaring of the circle or the quadrature of the circle. One of the last yavana heathen philosophers Simplicius, who was hounded by the śavapūjaka Justinian had to flee along with his fellow heathen yavana-s to the Sassanian court of Kushru. At the treaty of 533 CE concluded between the Iranians and the śavapūjaka-s it was enjoined that they should be allowed to return to their homes and practice their heathen rituals. However, we hear nothing of them thereafter suggesting they were either silenced by continued persecution or killed. It was this Simplicius who preserved a fragment of Iamblichus wherein we hear of the methods of the yavana mahāpuruṣa-s in achieving the famed quadrature.

Iamblichus points out that a method for squaring the circle was first presented by Sextus the Pythagorean and handed down to his successors. He says that Nicomedes achieved the same using a curve known as the quadratrix. The same was achieved by Archimedes using the Archimedean spiral. Whereas, Apollonius is said to have used the curve he termed the sister of the conchoid, which is the same curve as the quadratrix used by Nicomedes. Further, he mysteriously states that Carpus used a curve arise from double motion. Simplicius in his commentary adds that some mechanical devices were invented for this but not a theoretical proof. This indicates that between Iamblichus to Simplicius under the destruction of Classical knowledge by the śavapūjaka-s the know-how of these mechanical devices was already lost. In his commentary Proclus confirms that Nicomedes performed the quadrature using the quadratrix. On the other hand Pappus one of the last in the tradition of yavana mathematics states that the use of the quadratrix in quadrature was first due Dinostratus, the disciple of Plato who had performed the construction of the doubling the cubical altar of Apollo during the Delian plague sent by the god. The quadrature by Dinostratus using the quadratrix also links it to another classical yavana problem the trisection of the angle. Thus it can be used to simultaneous trisect an angle and square the circle. This construction has a certain magical quality to quality to it – a wonder in itself that shows why the yavana philosophers and our ārya ancestors linked it to religion.

quadratrix_squaring

The quadratrix deployment in the Dinostratus construction goes thus:
1) In the quadrant of the circle with center at A divide the vertical radius into n equal parts.
2) Divide the quadrant into the same number of equal parts n.
3) Mark the points of intersection of the radii dividing the quadrant into n equal parts and the horizontal parallel line dividing the vertical radius into n equal parts. The locus of these points is the quadratrix.
4) To trisect an \angle CAC' let it cut the quadratrix at point H.
5) Then draw a parallel line to AC through H. It cuts the vertical radius at G.
6) Trisect the \overline{AG} and mark the lower \frac{1}{3} of it, \overline{AJ}.
7) Draw a parallel line to AC through J to cut the quadratrix and draw segment AT by connecting A to the this point of intersection with the quadratrix.
8) The \angle CAT is \frac{1}{3} \angle CAC'

1) To perform the quadrature mark the point where the quadratrix intersect the horizontal radius AC.
2) Draw the \overline{KL}=radius of circle perpendicular to AC.
3) Draw the tangent line passing through C to the circle to be squared.
4) Draw the line AL and mark the point where it intersect the above tangent.
5) Double the \overline{CM} along the tangent to get point N.
6) Create \overline{CE}= radius of circle to be squared along the same tangent in the opposite direction.
7) Bisect the \overline{EN} to obtain point O and construct a circle with O as center.
8) Extend AC to meet the above circle a R to deploy the geometric mean theorem.
9) CR is the side of the square CRSP with same area as the starting circle with center at A.

Continued…


Filed under: Heathen thought, History, Scientific ramblings Tagged: cycloid, Geometric construction, geometry, Greek, Greek thought, Iamblichus, recreational geometry

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