Arabian algebra: * see Endnote 68

The Muslim Abbasid caliph al-Mamun (809–833) ordered the translations of Greek works including Ptolemy into Arabic.  So began a cultural awakening; Arab research in mathematics & the sciences increased.  The father of algebra is Al-Khwarizmi (died circa 850 AD); demonstrated solving polynomials up to the second degree; introduced transposition of subtracted terms to the other side of an equation (or cancellation of like terms on opposite sides of the equation); employed rhetorical algebra, where numbers were spelled in words although eventually replaced by numeric notation (only starting in the 13th century did Arabs mathematician adopted symbolic algebra).

 

Indian trigonometry:
In mathematics, Bhaskara I's sine approximation formula is a rational expression in one variable for the computation of the approximate values of the trigonometric sines discovered by Bhaskara I ( 600 - 680), an Indian mathematician.  This formula is given in his treatise Mahabhaskariya; it is not known how he arrived at this. The formula is elegant, simple & enables one to compute reasonably accurate values of trigonometric sines without using any geometry whatsoever.
 

Classical mechanics: * see Endnote 69

Aristotle (4th century BC) established the system of Aristotelian physics (which was later largely disproved).  He believed in logic & observation & was the first to propose that abstract principles govern nature.  He argued that terrestrial bodies rise or fall to their "natural place"; he mistakenly claimed that an object twice as heavy as another would fall to the ground from the same height in half the time.  Aristotle was influenced by Plato's teachings on the perfection of the circular uniform motions of the heavens.  Thus he believed in a natural order where the motions of the heavens were necessarily perfect, in contrast to the terrestrial world of changing elements & imperfection.

 

Theory of Aggregates:

an archaic word for infinite sets such as those considered by Georg Cantor; sometimes also used to refer to a finite or infinite set in which multiplicity is ignored.

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nexus:

means of connection; tie; link; a connected series or group; the core or center, as of a matter or situation

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art of the gems: * see Endnote 70

Spengler uses this to illustrate the principle of proportion which he sees as integral to magnitude.

transformations of groups:

In semi group theory, a transformation is a function f that maps a set X to itself, i.e. f : X → X.  A translation, or translation operator, is an affine transformation of Euclidean space which moves every point by a fixed distance in the same direction.  Examples include linear transformations and affine transformations, rotations, reflections and translations. These can be carried out in Euclidean space, particularly in 2D & 3D. They are also operations that can be performed using linear algebra, and described explicitly using matrices.

Tema con Variazioni (18th century orchestral form):

(Italian for Theme with Variations) formal musical technique where material is repeated in an altered form. The changes may involve melody, rhythm, harmony, counterpoint, timbre, orchestration or any combination of these.

 
congruence theorems of Euclid:

Triangles are congruent if they have all 3 sides equal;2 sides and the angle between them equal SAS), or 2 angles and a side equal from Euclid’s Book I, propositions 4, 8 & 26). Triangles with 3 equal angles are similar, but not necessarily congruent and triangles with 2 equal sides and an adjacent angle are not necessarily equal or congruent.

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