Ataraxia:

Greek, meaning a lucid state of robust equanimity characterized by on-going freedom from distress & worry, used initially by philosopher Pyrrho, later Epicurus & the Stoics; achieving ataraxia is a common goal for all 3 schools but the role & value of ataraxia, how to achieve it & the mental disturbances which prevent it, all differ between the 3.

 

Analysis (impressionistic mathematic):

the study of infinite processes, dealing with limits & related theories, such as calculus (differentiation, integration), measure, infinite series & analytic functions; usually studied in the context of real & complex numbers & functions.  It grew out of calculus, which involves the elementary concepts and techniques of analysis.  It is distinguished from geometry but can be applied to any space of mathematical objects that has a definition of nearness (topological space) or specific distances between objects (metric space).

 

number ("bodies"):

Spengler is contrasting the Apollonian prime symbol (bodies) with Faustian maths, which is highly abstract, lacking any corporal sense.  Faustian mathematical objects are anything that has been (or could be) formally defined & with which one may do deductive reasoning & mathematical proofs; it is studied in philosophy of mathematics.  A few commonly encountered mathematical objects include: numbers, integers, integer partitions; sets, set partitions, functions, relations; points, lines, line segments; topological spaces & manifolds.

 

aggregates:

aka sets; in mathematical context a collection of objects classed together; set theory is a branch of mathematical logic that studies sets.  Although any type of object can be collected into a set, it is usually applied to objects relevant to mathematics.  The language of set theory can be used to define nearly all mathematical objects.

 

multidimensional geometry:

Euclidean geometry was overturned by the revolutionary discovery of non-Euclidean geometry in the works of Bolyai, Lobachevsky & Gauss who demonstrated that Euclidean space is only 1 possibility.  A much broader vision of geometry was expressed by Riemann (On the hypotheses on which geometry is based, 1867).  Euclidean geometry allowed 3 dimensions; for 200 years Faustian mathematicians have used higher dimensions:  Minkowski space (4-D) approximates the universe without gravity; superstring theory uses 10 dimensions; super gravity and M-theory employ 11 dimensions, and the state-space of quantum mechanics is an infinite-dimensional function space.  High-dimensional spaces frequently occur in Faustian mathematics & sciences as parameters or configuration spaces; they are abstract spaces, independent of the physical world. 

 

physics (mass-points):

Quantum mechanics is a fundamental theory in physics which describes nature at the energy levels of atoms & subatomic particles.  It differs in 3 ways from classical Newtonian physics: energy, momentum, angular momentum & other quantities of a bound system are restricted to discrete values (quantization); objects can be both particles & waves & the precision by which quantities can be measured is limited (uncertainty principle).  In 1900 its originator, Max Planck, claimed that any energy-radiating atomic system can be divided into a number of discrete "energy elements".  In 1905, Einstein explained the photoelectric effect by postulating that light (all electromagnetic radiation), can be divided into a finite number of "energy quanta" that are localized points in space.

 

variable efficients: * see EndNote<A>

aka coefficients, in physics a quantitative expression of a specific property of matter or a phenomenon;  number (constant) for a given substance, body, or process under certain specified conditions, serving as a measure of one of its properties

 

Pietism:

movement within Lutheranism; it has a non-Scholastic, non-Philosophical approach to Faith, which de-emphasized formal aspects (dogma, doctrine, membership) & the sacraments, placing greater emphasis on Bible studies by the laity, it was rooted in the personal experience, not the institution.  In the late 17th century the German Lutheran theologian Philipp Spener established Pietism; he emphasised personal transformation through spiritual rebirth & individual devotion.  It spread from Germany to Switzerland & the greater German-speaking Europe, Scandinavia, the Baltics & the rest of Europe.  Europeans immigrants took it to the USA where it influenced other Protestants denominations & contributed to the 18th-century foundation of evangelicalism. 

 

microcosm:

a little world; a world in miniature (opposed to macrocosm).

 

Titian:

Chapter III page 108 Chapter VII pages 221, 226, 243, 244, 249, 252.

 

Corot:

see Chapter VII page 246

 

Menzel:

see above page 271

 

Baroque music (thematic not melodic): * see EndNote<B>

Melody was not the prime focus for Baroque composers.  They worked & developed many other areas of music.  Baroque composers employed elaborate musical ornamentation,  by adding notes non-essential  for the melody.  They might use different keys on a single melody.  They developed new instrumental playing techniques & expanded the size, range & complexity of instrumental performance.  The developed new forms, for mixed vocal/instrumental (e.g. opera, cantata and oratorio) and for pure instrumental (e.g. solo concerto and sonata).  A popular approach was dense, complex polyphonic music, often with multiple independent melody lines performed simultaneously (for example the fugue).

 

Baroque music (imitative tone-picture): * see EndNote<C>

imitative music employs the repetition of a melodic phrase at a different pitch or key from the original or in a different voice part; the repetition of a melody in a polyphonic texture shortly after its first appearance in a different voice.  Imitative writing was employed heavily in the highly polyphonic compositions of the Renaissance and Baroque eras.

and see above- Baroque music (thematic not melodic)

 

Wagner (leitmotiv): * see EndNote <D>

a short, constantly recurring musical phrase associated with a particular person, place, or idea; a short musical idea, melodic, harmonic, or rhythmic, or all three; a musical fragment or succession of notes that has some special importance in or is characteristic of a composition,  closely related to the musical concepts of idée fixe or motto-theme.  In orchestral music it was used as early as 17th century (L'Orfeo by Monteverdi).  In late 18th century French opera (such as the works of Gluck, Grétry and Méhul), the "reminiscence motif" can be identified, which may recur at a significant juncture in the plot.  Prior to Wagner, in the early 19th century composers of Romantic opera (Carl Maria von Weber) use recurring themes in association with specific characters.  The man most closely linked to the leitmotif is Wagner, especially his Der Ring des Nibelungen.  He was not its originator & did not employ the word in connection with his work.

and See Chapter I page 35, 45, Chapter III pages 97, 111 and Chapter VII pages 220, 222

 

Goethe's "Urfaust":

The earliest forms of Faust, known as the Urfaust, writen 1772-75; it has 22 scenes, 1 in prose, 2 largely prose & the remaining 1,441 lines in rhymed verse.  Lost until re-discovered in 1886, many years after the publication of Faust part 2 (1832).  Compared to the published version, Urfaust is representative of the Sturm und Drang (“Storm & Stress”) period, more an exploration of the human soul than intellect.  It includes elements of Romanticism later eliminated from the final version.

 

Holderlin (his last poems): * see EndNote<E>

In the years after his return from Bordeaux (after 1802), he completed some of his greatest poems Once finished, he returned to them repeatedly, creating new & stranger versions sometimes in several layers on the same manuscript.  Some of these are fragmentary, but they have astonishing intensity; he considered the fragments, with unfinished lines & incomplete sentences, to be poems in themselves. This obsessive revising & stand-alone fragments reflect his mental disorder, but proved very influential on later poets (Celan).  In his later years of madness, Hölderlin would occasionally pencil ingenuous rhymed quatrains [a type of stanza, or a complete poem, consisting of 4 lines] sometimes of a childlike beauty, which he would sign with fantastic names (most often "Scardanelli") and give fictitious dates from previous or future centuries.

 

Copernicus:

see Chapter II page 68

 

Columbus:

see Chapter IV page 148

 

ornament-language:

Here Spengler means Impressionism