localized cult:

Spengler uses this term to refer to cults that are place specific.  Many Greek gods were associated with a specific city: Athena with Athens, Apollo with Delphi, Zeus with Olympia, Aphrodite with Corinth; the identification of gods with specific places was fundamental to Greek religion.

 

Pythagoreans, secret doctrines of:* see Endnote 21

the doctrines of the school of Pythagoras were not public  but limited to initiates & fullest knowledge was reserved for those highest members of the hierarchy (5 levels, the top being the Pythagoreans or esoteric members).  The “esoteric” teachings (the Knowledge of The Reality) were the most secret and consisted of: the Symbols, Number (inner meaning of arithmetic & geometry), Music, Man, and Earth & the Universe.

 

theorems of regular polyhedrons: * see Endnote 22

The ancient Greeks studied the symmetrical geometric solids (regular polyhedrons) extensively.  Pythagoras discovered the tetrahedron, cube & dodecahedron; Theaetetus (contemporary of Plato) discovered the octahedron & icosahedron.  He wrote a mathematical description of all 5 and may have been responsible for the first known proof that no other convex regular polyhedra exist.

 

Descartes analysis of the infinite: * see Endnote 23

Meditations  (1641) posited that “I think therefore I am”.  Finite man can think of infinite God, therefore God must have placed the idea into him.

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theology of Reformation:

theological doctrines held in common by the 16th century Protestant reformers, (Luther, Melancthon, Zwingli, Calvin), though very heterogeneous they agreed on: opposition to the Roman Pope & RC doctrines; the sufficiency of the Bible; justification by faith alone; covenant theology; total depravity; necessity of grace of the Holy Spirit for faith.

 

theology of Counter Reformation:

comprehensive effort to reform the Roman church & reverse gains made by the Protestants, begins with Council of Trent (1545–1563), ending 1648 (close of the Thirty Years' War).  Included papal supremacy, defense of all 7 sacraments (transubstantiation, confession); reformed clericalism & monastic life; Roman Inquisition, global missionary work; the regular clergy such as the Jesuits. (taught, preached & took confession under a bishop's authority not linked to specific parish).

 

deism:

philosophical position late 17th to late 18th century; posits god does not interfere directly with the world; rejects revelation & asserts that reason & observation of the natural world can determine the existence of a single creator God; gained prominence among intellectuals especially in Britain, France, Germany where many (raised as Christians, monotheists) were disenchanted with organized religion & teachings (Trinity, infallibility of the Biblical, miracles).

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Pascal:

see Introduction page 42

 

Jansenists:

French Catholic theological movement, emphasized original sin, human depravity, the necessity of divine grace & predestination; originated from work of the Dutch theologian Cornelius Jansen, (died in 1638).  During the 17th and 18th centuries it was a strong movement within the Catholic Church; theological centre was the convent of Port-Royal-des-Champs Abbey (a haven for writers including, Pascal and Racine).

 

pietist:

a religious inclination that attempts to focus on individual holiness & living a Christian life; often led by layman or local pastors (frustrated with the perceived hypocrisy or inconsistency within the larger formal church); often outside formal organization or place of worship.

 

Voltaire:

(1694-1778) French Enlightenment writer, historian & philosopher, famous for his wit, attacks on the Catholic Church & Christianity, advocacy of freedom of religion & speech, and separation of church & state.  Prolific writer; as a satirical polemicist, he frequently made use of his works to criticize intolerance, religious dogma & French institutions of his day.   His Candide attacks the passivity inspired by Leibniz's philosophy of optimism.

 

Lagrange:

(1736-1813) Italian-French mathematician, contributed to number theory & analytic & celestial mechanics.  His “Analytic Mechanics (1788), became basis for all later work in this field.  His work on mechanistic systems revealed independent coordinates necessary for the specifications of a system of a finite number of particles, or “generalized coordinates.  From this was developed the Lagrangian equations for a classical mechanical system in which the kinetic energy of the system is related to the generalized coordinates, the corresponding generalized forces & time.

 

D’Alembert:

(1717 -1783) French mathematician, mechanician, physicist, philosopher, and music theorist, co-editor with Diderot of the Encyclopédie; known for his formula for obtaining solutions to the wave equation (d'Alembert's equation).

 

Timaeus:

(360 BC) dialogue by  Plato; a creation story with focus on rationality of god; Plato proposes the universe is the product of rational, purposive, and beneficent agency, work of a divine Craftsman, imitating an unchanging and eternal model, imposes mathematical order on a pre-existent chaos to generate the ordered universe (kosmos), the deliberate intent of his Intellect.  It contains proportion, geometric perfection, possessing order & clarity.

 

continuum:

a set of elements such that between any two of them there is a third element

 

Indian soul (and zero):

Indian mathematicians developed the concept of zero as a number (and not as a simple symbol of separation) & negative numbers.  The decimal system is derived from Hindu mathematics; the Bakhshali manuscript (oldest extant mathematical manuscript in South Asia, 7th century AD) employs a decimal place value system with a dot for zero. 

 

magnitudes:

size of a mathematical object, determines whether the object is larger or smaller than other objects of the same kind.  The Greeks defined several types of magnitude, including: positive fractions, lines (ordered by length), plane figures (ordered by area), solids (ordered by volume), angles (ordered by angular magnitude); did not consider negative magnitudes to be meaningful, & even today magnitude is chiefly used in contexts where zero is either the smallest size or less than all possible sizes.

 

Archytas:

(428–347 BC) Greek philosopher, mathematician; in geometry, solved the problem of doubling the cube by construction in solid geometry using the intersection of a cone, a sphere & a cylinder.  A second generation of Pythagorean, sought to combine empirical observation with Pythagorean theory.

 

Eudoxus:

(395-342) Greek mathematician & astronomer, of Cnidus, substantially advanced proportion theory, contributed to the identification of constellations & to the development of observational astronomy in the Greek world; established first sophisticated, geometrical model of celestial motion; wrote on geography & contributed to philosophical discussions in Plato’s Academy; his description of the constellations and the 12 signs of the Zodiac show close similarities to Babylonian astronomy;  none of his work survive, his contributions are known from many discussions throughout antiquity.

 

second and third powers:

The 2nd power of a number is that number squared (erg 7 to the second power, 7^2 = 72 or 49); the area or space inside a square is equal to the length of the side of the square to the second power.  The 3rd power of a number is that number cubed (e.g. 7 to the third power, 7^3= 73 or 343); the volume of a cube is equal to the length of one of its sides taken to the third power.