Encyclopedia - Pythagoreanism: Encyclopedia - Pythagoreanism
Pythagoreanism is a term used for the esoteric and metaphysical beliefs held by Pythagoras and his followers, the Pythagoreans, who were ...   » Read the article

Encyclopedia - Pythagoreanism: Encyclopedia Ii - Pythagoreanism - Pythagorean Cosmology
Pythagorean thought was dominated by mathematics, but it was also profoundly mystical. In the area of cosmology there is less agreement a...   » Read the article

Encyclopedia - Pythagoreanism: Encyclopedia Ii - Pythagoreanism - Influence
The word 'vegetarian' was coined in 1847 when the British Vegetarian Society was formed. Before this, vegetarians were known as Pythagore...   » Read the article

Encyclopedia - Pythagoras: Encyclopedia Ii - Pythagoras - Pythagoreans
Pythagoras' followers were commonly called "Pythagoreans." For the most part we remember them as philosophical mathematicians who had an ...   » Read the article

Encyclopedia - Pythagoras: Encyclopedia - Pythagoras
Pythagoras (approximately 569 BCE – 475 BCE, Greek: Πυθαγόρας) was an Ionian mathematician and philosopher, founder of the my...   » Read the article

Encyclopedia - Tetractys: Encyclopedia - Tetractys
Using a The Tetractys, also known as the decad, is a triangular figure consisting of ten points arranged in four rows: one, two, three, a...   » Read the article

Encyclopedia - Apollonius Of Tyana: Encyclopedia - Apollonius Of Tyana
Apollonius of Tyana (13 March 2 – 98?) was a Neo-Pythagorean philosopher and teacher of Greek origin. His teaching influenced scientifi...   » Read the article

Encyclopedia - Pentagram: Encyclopedia - Pentagram
A pentagram is a five-pointed star drawn with five straight strokes. In fact, the word pentagram comes from the Greek word πεντάγρ...   » Read the article

Encyclopedia - Quincunx: Encyclopedia - Quincunx
A quincunx is the arrangement of five units in the pattern corresponding to the five-spot on dice, playing cards, or dominoes. A quincun...   » Read the article

Encyclopedia - Aristoxenus: Encyclopedia - Aristoxenus
Aristoxenus of Tarentum (4th century BC) was a Greek peripatetic philosopher, and writer on music and rhythm. He was taught first by his ...   » Read the article

Encyclopedia - Archytas: Encyclopedia - Archytas
Archytas (428 BC - 347 BC), was a Greek philosopher, mathematician, astronomer, statesman, strategist and commander-in-chief. Archytas wa...   » Read the article

Encyclopedia - Pythagorean Triple: Encyclopedia Ii - Pythagorean Triple - Generating Pythagorean Triples
An effective way to generate Pythagorean triples is based on the observation that if m and n are two positive integers with m >&#...   » Read the article

Encyclopedia - Pythagorean Theorem: Encyclopedia Ii - Pythagorean Theorem - Pythagorean Triples
A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. In other words, a Pytha...   » Read the article

Encyclopedia - Pythagorean Triple: Encyclopedia Ii - Pythagorean Triple - Generating Pythagorean Triples
An effective way to generate Pythagorean triples is based on the observation that if m and n are two positive integers with m >&#...   » Read the article

Encyclopedia - Pythagorean Triple: Encyclopedia Ii - Pythagorean Triple - Properties Of Pythagorean Triples
The properties of primitive pythagorean triples include: Exactly one of a,b is odd, c is odd. exactly one of a,b is divisible by 3 exactl...   » Read the article

Encyclopedia - Pythagorean Theorem: Encyclopedia Ii - Pythagorean Theorem - Proofs
This theorem may have a greater variety of known proofs than any other (the law of quadratic reciprocity being also a contender for that ...   » Read the article

Encyclopedia - Pythagorean Triple: Encyclopedia Ii - Pythagorean Triple - Generalizations
A set of four positive integers a, b, c and d such that a2 + b2+ c2 = d2 is called a Pythagorean quadruple. A ge...   » Read the article

Encyclopedia - Pythagoras: Encyclopedia Ii - Pythagoras - Biography
Pythagoras was born on the island of Samos, off the coast of Asia Minor. He was born to Pythais (a native of Samos) and Mnesarchus (a mer...   » Read the article

Encyclopedia - Tetractys: Encyclopedia Ii - Tetractys - Pythagorean Symbol
A prayer of the Pythagoreans shows the importance of the Tetractys (sometimes called the "Mystic Tetrad"), as the prayer was addressed to...   » Read the article

Encyclopedia - Pythagoras: Encyclopedia Ii - Pythagoras - Scientific Contributions
Some consider Pythagoras the pupil of Anaximander and some ancient sources tell of his visiting, in his twenties, the philosopher Thales,...   » Read the article

Encyclopedia - Pythagorean Theorem: Encyclopedia Ii - Pythagorean Theorem - The Pythagorean Theorem In Non-euclidean Geometry
The Pythagorean theorem is derived from the axioms of Euclidean geometry, and in fact, the Euclidean form of the Pythagorean theorem give...   » Read the article

Encyclopedia - Pythagorean Theorem: Encyclopedia Ii - Pythagorean Theorem - History
The history of the theorem called Pythagorean can be divided into three parts: knowledge of Pythagorean triples, knowledge of the relatio...   » Read the article

Encyclopedia - Pythagorean Triple: Encyclopedia Ii - Pythagorean Triple - Some Relationships
If a2 + b2 = c2 is a primitive Pythagorean triple, where a is odd, then where each fraction is reduced to lowest terms and m>n. ...   » Read the article

Encyclopedia - Pythagorean Theorem: Encyclopedia Ii - Pythagorean Theorem - Other Facts
In heraldry, the Pythagorean theorem appears as a charge in the arms of Seissenegger. The theorem is referenced in an episode of The Simp...   » Read the article

Encyclopedia - Pythagorean Tuning: Encyclopedia Ii - Pythagorean Tuning - Method
Pythagorean tuning is based on a stack of perfect fifths, each tuned in the ratio 3:2, the next simplest ratio after 2:1, which is the ra...   » Read the article

Encyclopedia - Pythagorean Triple: Encyclopedia Ii - Pythagorean Triple - A Special Case: The Platonic Sequence
The case n = 1 of the above has been known for a long time. Proclus, in his commentary to the 47th Proposition of the first book of Eucli...   » Read the article

Encyclopedia - Pythagorean Triple: Encyclopedia Ii - Pythagorean Triple - Other Formulas For Generating Triples
I: Pythagoras (c. 540 B.C.) presented this formula for generating triples: a = m : b = (m2 − 1) / 2  : c = (m2 + 1) / 2, whe...   » Read the article

Encyclopedia - Pythagorean Triple: Encyclopedia Ii - Pythagorean Triple - A Special Case: The Platonic Sequence
The case n = 1 of the above has been known for a long time. Proclus, in his commentary to the 47th Proposition of the first book of Eucli...   » Read the article

Encyclopedia - Pentagram: Encyclopedia Ii - Pentagram - History
The first uses of the pentagram we know of are found in Mesopotamian writings dating to about 3000 B.C. The Sumerian pentagrams served as...   » Read the article

Encyclopedia - Trigonometric Identity: Encyclopedia Ii - Trigonometric Identity - Pythagorean Identities
These identities are based on the Pythagorean theorem. The first is sometimes simply called the Pythagorean trigonometric identity. ...   » Read the article

Encyclopedia - Golden Mean Philosophy: Encyclopedia Ii - Golden Mean Philosophy - History Of The Golden Mean In Philosophy
Golden mean philosophy - Crete. The earliest representation of this idea in culture is probably in the mythological Cretan tale of Daed...   » Read the article

Encyclopedia - Meantone Temperament: Encyclopedia Ii - Meantone Temperament - Wolf Intervals And Extended Meantones
A whole number of just perfect fifths will never add up to a whole number of octaves, because they are incommensurable (see Fundamental t...   » Read the article

Encyclopedia - Pentagram: Encyclopedia Ii - Pentagram - Neopagan Use
Many Neopagans, especially Wiccans, use the pentagram as a symbol of faith similar to the Christian cross or the Jewish Star of David. (I...   » Read the article

Encyclopedia - Harmonic Mean: Encyclopedia Ii - Harmonic Mean - Relationship With Other Means
The harmonic mean is one of the Pythagorean means and is never larger than the geometric mean or the arithmetic mean (the other two Pytha...   » Read the article

Encyclopedia - Megapolisomancy: Encyclopedia Ii - Megapolisomancy - Fictional History Of Thibaut De Castries
"The ancient Egyptians only buried people in their pyramids. We are living in ours." - De Castries Leiber provides little background abou...   » Read the article

Encyclopedia - Megapolisomancy: Encyclopedia Ii - Megapolisomancy - Megapolisomancy Fictional Occult Pseudoscience
Megapolisomancy is a fictional occult science created by Fritz Leiber, an influential American writer of fantasy and science fiction. Fea...   » Read the article

Encyclopedia - Megapolisomancy: Encyclopedia Ii - Megapolisomancy - The Occult Science Of Megapolisomancy
According to De Castries, excessively large cities pose a clear danger to the people living in and near them by allowing mass quantities ...   » Read the article

Encyclopedia - Reptilian Humanoid: Encyclopedia Ii - Reptilian Humanoid - Modern Claims
Christine Fitzgerald, a confidante of Diana, Princess of Wales, claims that Diana told her that the Royal Family were Reptilian aliens, a...   » Read the article

Encyclopedia - Pentagram: Encyclopedia Ii - Pentagram - Geometry
A regular pentagram is the {5/2} star polygon. It is most easily drawn by drawing a regular pentagon, joining the corners with lines and ...   » Read the article

Encyclopedia - Pentagram: Encyclopedia Ii - Pentagram - Christian Use
According to Heather Child's Christian Symbols, Ancient and Modern (ISBN 0713519606), the pentagram is a symbol of the five senses. Also,...   » Read the article

Encyclopedia - Pentagram: Encyclopedia Ii - Pentagram - Flags
The pentagram appears on coats of arms and on two national flags (those of Ethiopia and Morocco). See gallery of flags by design. More co...   » Read the article

Encyclopedia - Mathematics Of Musical Scales: Encyclopedia Ii - Mathematics Of Musical Scales - Just Intonation
If we take the ratios constituting a scale in just intonation, there will be a largest prime number to be found among their prime factori...   » Read the article

Encyclopedia - Mathematics Of Musical Scales: Encyclopedia Ii - Mathematics Of Musical Scales - Temperament
Western common practice music usually cannot be played in just intonation, even when it is confined to a single key. This is because the ...   » Read the article

Encyclopedia - Reptilian Humanoid: Encyclopedia Ii - Reptilian Humanoid - Mythological References To Reptilian Humanoids
Several ancient peoples all over the world have described reptilian beings, and some have described reptilian humanoids. Common in numero...   » Read the article

Encyclopedia - Reptilian Humanoid: Encyclopedia Ii - Reptilian Humanoid - Alleged Modern Encounters
Even in modern times, some claim to have encountered reptilian humanoids. In many of these cases, a UFO is part of the encounter; alien a...   » Read the article

Encyclopedia - Reptilian Humanoid: Encyclopedia Ii - Reptilian Humanoid - Political References To Reptilians
An article appeared in the Chicago Tribune on March 1, 1995, regarding an event in which George Bushnell, head of the American Bar Associ...   » Read the article

Encyclopedia - Trigonometric Identity: Encyclopedia Ii - Trigonometric Identity - Double-angle Formulae
These can be shown by substituting x = y in the addition theorems, and using the Pythagorean formula for the latter two. Or use de Moivre...   » Read the article

Encyclopedia - Trigonometric Identity: Encyclopedia Ii - Trigonometric Identity - Half-angle Formulae
Sometimes the formulae in the previous section are called half-angle formulae. To see why, substitute x/2 for x in the power reduction fo...   » Read the article

Encyclopedia - Trigonometric Identity: Encyclopedia Ii - Trigonometric Identity - Periodicity Symmetry And Shifts
These are most easily shown from the unit circle: For some purposes it is important to know that any linear combination of sine waves of ...   » Read the article

Encyclopedia - Trigonometric Identity: Encyclopedia Ii - Trigonometric Identity - Multiple-angle Formulae
If Tn is the nth Chebyshev polynomial then de Moivre's formula: The Dirichlet kernel Dn(x) is the function occurring on both sides ...   » Read the article

Encyclopedia - Trigonometric Identity: Encyclopedia Ii - Trigonometric Identity - Inverse Trigonometric Functions
Every trigonometric function can be related directly to every other trigonometric function. Such relations can be expressed by means of i...   » Read the article

Encyclopedia - Trigonometric Identity: Encyclopedia Ii - Trigonometric Identity - Identities Without Variables
Richard Feynman is reputed to have learned as a boy, and always remembered, the following curious identity: However, this is a special...   » Read the article

Encyclopedia - Trigonometric Identity: Encyclopedia Ii - Trigonometric Identity - Calculus
In calculus the relations stated below require angles to be measured in radians; the relations would become more complicated if angles we...   » Read the article

Encyclopedia - Trigonometric Identity: Encyclopedia Ii - Trigonometric Identity - Geometric Proofs
Trigonometric identity - sinx + y = sinx cosy + cosx siny. In the figure the angle x is part of right angled triangle ABC, and the angl...   » Read the article

Encyclopedia - Trigonometric Identity: Encyclopedia Ii - Trigonometric Identity - Notation
The following notations hold for all six trigonometric functions: sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec),...   » Read the article

Encyclopedia - Perimeter: Encyclopedia Ii - Perimeter - Formulas
As a general rule, the perimeter of a polygon can always be calculated by adding all the length of the sides together. Circles: For circl...   » Read the article

Encyclopedia - Meantone Temperament: Encyclopedia Ii - Meantone Temperament - Meantone Temperaments
The term meantone temperament is sometimes used to refer specifically to quarter-comma meantone. However, systems which flatten the fifth...   » Read the article

Encyclopedia - List Of Prime Numbers: Encyclopedia Ii - List Of Prime Numbers - Lists Of Types Of Primes Alphabetical
List of prime numbers - List of Bell primes. 2, 5, 877, 27644437, 35742549198872617291353508656626642567, 35933408596862283104196018859...   » Read the article

Encyclopedia - Harmonic Mean: Encyclopedia Ii - Harmonic Mean - Examples
In certain situations, the harmonic mean provides the correct notion of "average". For instance, if for half the distance of a trip you t...   » Read the article

Encyclopedia - Harmonic Mean: Encyclopedia Ii - Harmonic Mean - Harmonic Mean Of Two Numbers
When dealing with just two numbers, an equivalent, sometimes more convenient, formula of their harmonic mean is given by: In this case...   » Read the article

Encyclopedia - Mithraism: Encyclopedia Ii - Mithraism - Parallels To Christianity
According to Martin A. Larson, in The Story of Christian Origins (1977), Mithraism and Christianity derived from the same sources, origin...   » Read the article

Encyclopedia - Pentagram: Encyclopedia Ii - Pentagram - In Literature
In the medieval romance of Sir Gawain and the Green Knight, it was on Gawain's shield along with a Madonna. The pentagram was said to be ...   » Read the article

Encyclopedia - Fermat's Last Theorem: Encyclopedia Ii - Fermat's Last Theorem - Mathematical Context
Fermat's last theorem is a generalization of the Diophantine equation a2 + b2 = c2, which is linked to the Pythagorean theorem. Ancient C...   » Read the article

Encyclopedia - Just Intonation: Encyclopedia Ii - Just Intonation - The Diatonic Scale In Just Intonation
It is possible to tune the familiar diatonic scale or chromatic scale in just intonation, in many ways, all of which make certain chords ...   » Read the article

Encyclopedia - Just Intonation: Encyclopedia Ii - Just Intonation - Why Isn't Just Intonation Used Much?
Some fixed just intonation scales and systems, such as the diatonic scale above, produce wolf intervals. The above scale allows a minor t...   » Read the article

Encyclopedia - Just Intonation: Encyclopedia Ii - Just Intonation - Non-western Tuning
In Indian music, the basic unaltered diatonic scale is considered to be 1/1, 9/8, 5/4, 4/3, 3/2, 27/16, 15/8, 2/1. This would appear prob...   » Read the article

Encyclopedia - Just Intonation: Encyclopedia Ii - Just Intonation - Western Composers Who Specified Just Intonation
Most composers don't specify how instruments are to be tuned, although historically most have assumed one tuning system which was common ...   » Read the article

Encyclopedia - Equal Temperament: Encyclopedia Ii - Equal Temperament - Explanation
The distance between each step and the next is aurally the same for any two adjacent steps; though, because steps form a geometric sequen...   » Read the article

Encyclopedia - Equal Temperament: Encyclopedia Ii - Equal Temperament - History
Vincenzo Galilei (father of Galileo Galilei) may have been the first person to advocate equal temperament (in a 1581 treatise). The first...   » Read the article

Encyclopedia - Equal Temperament: Encyclopedia Ii - Equal Temperament - Twelve-tone Equal Temperament
The ratio between two adjacent semitones can be found with a few steps: 1. Let an be the frequency of a tone n, with a12 an octave above...   » Read the article

Encyclopedia - Equal Temperament: Encyclopedia Ii - Equal Temperament - Non-12 Tet
Five and seven tone equal temperament, with 240 and 171 cent steps relatively, seem the most common outside of 12-tET. A Thai xylophone m...   » Read the article

Encyclopedia - Trigonometric Function: Encyclopedia Ii - Trigonometric Function - Series Definitions
Please note: Here, and generally in calculus, all angles are measured in radians. (See also below). Using only geometry and properties of...   » Read the article

Encyclopedia - Trigonometric Function: Encyclopedia Ii - Trigonometric Function - Definitions Via Differential Equations
Both the sine and cosine functions satisfy the differential equation i.e. each is the additive inverse of its own second derivative. W...   » Read the article

Encyclopedia - Trigonometric Function: Encyclopedia Ii - Trigonometric Function - Other Definitions
Theorem: There exists exactly one pair of real functions s, c with the following properties: For any : ... See also:Trigono...   » Read the article

Encyclopedia - Trigonometric Function: Encyclopedia Ii - Trigonometric Function - Computation
The computation of trigonometric functions is a complicated subject, which can today be avoided by most people because of the widespread ...   » Read the article

Encyclopedia - Trigonometric Function: Encyclopedia Ii - Trigonometric Function - Inverse Functions
The trigonometric functions are periodic, so we must restrict their domains before we are able to define a unique inverse. In the followi...   » Read the article

Dictionary - Pythagoreans: Spiritual - Theosophy Dictionary On Pythagoreans
Pythagoreans The school founded at Crotona, Italy in the 6th century BC by Pythagoras of Samos. Pythagoras was an initiate not only in...   » Read the article

Dictionary - Pythagorean Pentacle: Spiritual Theosophical Dictionary On Pythagorean Pentacle
Pythagorean Pentacle (Ancient Greek). A Kabbalistic six-pointed star with an eagle at the apex and a bull and a lion under the face of...   » Read the article

Dictionary - Ten Pythagorean Virtues: Spiritual Theosophical Dictionary On Ten Pythagorean Virtues
Ten Pythagorean Virtues. Virtues of Initiation, &c., necessary before admission. (See " Pythagoras ".) They are identic...   » Read the article

Dictionary - Odd Numbers: Spiritual - Theosophy Dictionary On Odd Numbers
Odd Numbers Pythagorean held they were divine. {SD 2:574}   (See also: Odd Numbers , Mysticism, Mysticism Dictionary)      » Read the article

Dictionary - Nonagous Envelope: Spiritual - Theosophy Dictionary On Nonagous Envelope
Nonagous Envelope Of Pythagoreans. {BCW 12:68n}   (See also: Nonagous Envelope , Mysticism, Mysticism Dictionary)      » Read the article

Dictionary - Music Of The Spheres: New Age Spiritual Dictionary On Music Of The Spheres
music of the spheres Celestial vibrations; Pythagorean theory of correspondences between planets and the musical scale   (See ...   » Read the article

Dictionary - Phren: Spiritual Theosophical Dictionary On Phren
Phren (Ancient Greek). A Pythagorean term denoting what we call the Kama-Manas still overshadowed by the Buddhi-Manas.   (See a...   » Read the article

Dictionary - Telesphoros: Spiritual - Theosophy Dictionary On Telesphoros
Telesphoros (Greek) [from telos end + phero bring]   Bringing to an end, completing; said of the number seven by the Pythagoreans...   » Read the article

Dictionary - Rhizomata: Spiritual - Theosophy Dictionary On Rhizomata
Rhizomata (Greek) [plural of rhizoma root, element]   The four elements forming the second quaternary in the Pythagorean system, ...   » Read the article

Dictionary - Vehicle Of Life: Spiritual - Theosophy Dictionary On Vehicle Of Life
Vehicle of Life Generally, any vehicle of vitality; specifically among ancient Pythagoreans and others, the entire constitution of the...   » Read the article

Dictionary - Monas: Spiritual Theosophical Dictionary On Monas
Monas (Ancient Greek). The same as the term Monad ; "Alone", a unit. In the Pythagorean system the duad emanates from the hi...   » Read the article

Dictionary - Quadrivium: Spiritual Theosophical Dictionary On Quadrivium
Quadrivium (Latin). A term used by the Scholastics during the Middle Ages to designate the last four paths of learning - of which ther...   » Read the article

Dictionary - Xenophilus: Spiritual Theosophical Dictionary On Xenophilus
Xenophilus. A Pythagorean adept and philosopher, credited by Lucian (de Macrob.), Pliny and others with having lived to his 170th year...   » Read the article

Dictionary - Three: Spiritual - Theosophy Dictionary On Three
Three The first odd truly manifested number in the Pythagorean system, the second in emanation from the first odd number, the unit or ...   » Read the article

Dictionary - Timaeus: Spiritual - Theosophy Dictionary On Timaeus
Timaeus (Greek) A dialogue of Plato in which the Pythagorean philosopher Timaeus gives an account of aspects of cosmogenesis and anthr...   » Read the article

Dictionary - Hyle: Spiritual - Theosophy Dictionary On Hyle
Hyle (Greek) Wood, material; primordial matter as first manifested in and from Chaos, but as yet undifferentiated; the Mother, paired ...   » Read the article

Dictionary - Hyksos, Hyksos: Spiritual - Theosophy Dictionary On Hyksos, Hyksos
Hyle (Greek) Wood, material; primordial matter as first manifested in and from Chaos, but as yet undifferentiated; the Mother, paired ...   » Read the article

Dictionary - Kol-arbas: Spiritual - Theosophy Dictionary On Kol-arbas
Kol-arbas qol-'arba` Voice of the (sacred) four, voice of the Holy Tetrad; the sacred number four in the Valentinian Gnostic system --...   » Read the article

Dictionary - Archytas Of Tarentum: Spiritual - Theosophy Dictionary On Archytas Of Tarentum
Archytas of Tarentum (flourished 400-365 BC) Greek Pythagorean philosopher, general, statesman, scientist, and mathematician, contempo...   » Read the article

Dictionary - Arithmomancy: Spiritual - Theosophy Dictionary On Arithmomancy
Arithmomancy Interpretation by means of numbers, or divination by means of numbers. The Pythagoreans and Plato used the numerical key ...   » Read the article

Dictionary - Apeiros: Spiritual - Theosophy Dictionary On Apeiros
Apeiros (Greek) The boundless, infinite; frontierless expansion. Used by Anaximander and Anaximenes, and by Plato in Philebus; the equ...   » Read the article

Dictionary - Apollonius Of Tyana: Spiritual - Theosophy Dictionary On Apollonius Of Tyana
Apollonius of Tyana First-century neo-Pythagorean, known for his ascetic life, moral teachings, and occult powers. His biography is a ...   » Read the article

Dictionary - Tetraktys: Spiritual Theosophical Dictionary On Tetraktys
Tetraktys (Ancient Greek) or the Tetrad. The sacred "Four" by which the Pythagoreans swore, this being their most binding oa...   » Read the article

Dictionary - Neopythagoreans: Spiritual - Theosophy Dictionary On Neopythagoreans
Neopythagoreans The Pythagoreans of Alexandria and other cities on the Mediterranean coast in the 1st century with whom Apollonius of ...   » Read the article