Riemann (and the symbolism of the 2 planes): * see Endnote 36

Riemann’s surface are part of complex analysis (investigates functions of complex numbers), a 1D complex manifold, first studied by Riemann; a deformed version of the complex plane (which is a geometric representation of the complex numbers established by the real axis & the perpendicular imaginary axis; modified Cartesian plane, with the real part of a complex number represented along the x-axis, & imaginary part represented along the y-axis).

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Kepler:

(1571-1630) German mathematician, astronomer, and astrologer; is a key figure in the 17th-century scientific revolution & best known for his laws of planetary motion which were based on using an elliptical orbit (rather than perfect circle).  He transformed the ancient cosmology into scientific astronomy by treating it as part of a universal mathematical physics; his works were the foundations for Newton's theory of universal gravitation.  He incorporated religious arguments & reasoning into his work, believing that God created the world according to an intelligible plan accessible through reason: “I am merely thinking God's thoughts after Him"

 

Newton (and religious nature):

in his own lifetime, considered an insightful and erudite theologian. He wrote many works that today would be classified as occult studies & religious tracts dealing with the literal interpretation of the Bible.

 

algebra:

the study of mathematical symbols & rules for manipulating these symbols; it is a unifying thread of almost all of mathematics; it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields.

 

Diophantus (influence of Indian, early Arabian): * see Endnote 37

Was Diophantus actually a Greek?  Florian Cajori, historian (History of Mathematics, 1893) noted that if his work had not been written in Greek it is unlikely he would be associated with Greek mathematics; there is nothing in his work that echoes earlier geometric bound Classical Greek mathematics.  He alone among the Greeks stands with a new subject and new ideas.  Hindu origins of some of his algebra is a real possibility.

 

early Arabian schools:

Until the 4th century Arabs practised polytheistic religions.  Nomadic Bedouins had distinctive practices which included fetishism, totemism & veneration of the dead; they did not consider larger philosophical questions such as the afterlife.  The urban Arabs had in a more complex pantheon of deities; the Meccans and the other settled inhabitants worshiped at permanent shrines in towns & oases.  Gods & goddesses, including Hubal & Manāt, were worshipped at shrines like the Kaaba in Mecca.  Many of the physical descriptions of the pre-Islamic gods are traced to idols, especially near the Kaaba (which contained up to 360 of them).  We have few primary sources for this period.

 

Magian:

Spengler’s term for the Arabian Culture, emerging between 0 and 300 AD and whose geographic birthplace is Armenia, S Arabia, Alexandria & Ctesiphon.

 

transvalued:

see introduction, page 24

 

Vieta:

(1540-1603), French mathematician, his new algebra was an important step towards modern algebra, due to its innovative use of letters as parameters in equations; lawyer by trade, and served as a privy councillor to both Henry III and Henry IV of France.  In 1591 he published the Art of Calculation on Symbols.  He was the first mathematician who introduced notations for the problem (and not just for the unknowns), creating the first symbolic algebra.

 

algebraic notation (of Vieta):

His 1591 text of his mathematic theory described in 3 stages how to solve problems: summarize the problem as an equation identifying known & unknown quantities; next do the analysis & finally solve using a geometric or numerical based solution.  His algebra was no longer limited to the statement of rules, but relied on an efficient algebra, the operations act on the letters & the results can be obtained at the end by a simple replacement.  Algebra had a foundation as strong as geometry & ended the Arabic algebra of procedures (of al-Jabr and Muqabala).

 

Renaissance mathematics (and Classical tendency): * see Endnote 38

Reverence for Classical sources is a prominent feature of the 16th century; Brunelleschi illustrates this; he was a student of ancient classical buildings & using the rediscovered works of Vitruvius (1st-century AD) he formulated a Renaissance style that emulated & improved on classical forms.  His major feat of engineering was building the dome of the Florence Cathedral.

 

Arabian Culture:

an alternative term for the Magian Culture

 

Attic statuary:

sculpture of the Classic age, especially in Athens (Praxiteles, Lysippus, Phidias) circa 5th century BC

 

cupola:

(aka Dome) architectural element resembling the hollow upper half of a sphere; associated with the heavens in Ancient Persia (the royal audience tents of Achaemenid) & Hellenistic world.  A dome over a square base reflected the geometric symbolism of those shapes, a circle represented perfection, eternity & the heavens while the square represented the earth.  The dome is the quintessential architectural expression of the Magian soul.

 

mosaic:

art or image made from assemblage of small pieces of colored glass, stone, or other materials, for floors, decorative or interior decoration.  Found in Mesopotamia (3rd millennium BC), Mycenaean Greece, (floor mosaics); widespread in Classical world; early Christian basilicas (4th century onwards) used wall & ceiling mosaics; flourished in the Byzantine Empire (6th to the 15th centuries), early Islamic art used it on religious buildings & palaces  (the Dome of the Rock, Jerusalem & the Umayyad Mosque, Damascus); went out of fashion in Islamic world after the 8th century.