mathematic (of the Egyptians):

mathematics used from the Old Kingdom until Hellenistic Egypt (3000-300 BC), utilized a numeral system for counting & solving written mathematical problems, involving multiplication & fractions; surviving texts show that Egyptians understood concepts of geometry (determining the surface area & volume of 3-D shapes useful for architectural engineering) & algebra, such as the false position method and quadratic equations.

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algebra-astronomy (of the Babylonian culture):

In the 7th & 8th centuries BC, Babylonian astronomers (whom the Greeks called Chaldeans) developed a new approach; they considered the ideal nature of the universe & applied internal logic to the planetary systems.  They were the first to make refined mathematical descriptions of astronomical phenomena & recognize astronomical phenomena as periodic & apply mathematics to predictions.  They used mathematics to determine the variation in daylight over a solar year.  They used the base 60 numeral system; from this is derived our “time” (60 seconds in a minute, 60 minutes in an hour) & our 360 degrees circle.

 

ecliptic co-ordinate system (of Babylonian astronomy):

Fragments of Babylonian astronomy survived (the ephemerides or tables) giving the positions of naturally occurring astronomical objects at a given time & recording historical positions with date & time.  They studied eclipses, retrograde motion/planetary stations, planetary ingresses, sidereal time, positions for the mean & true nodes & phases of the moon and the positions of planets & minor celestial bodies such as Chiron.   To make these calculations they used basic arithmetic & a coordinate system based on the ecliptic, the path in the heavens that the sun & planets travel through.

 

Alexandrine mathematic: * see Endnote 17

Greek mathematics & astronomy reached its most advanced stage during the Hellenistic period; Alexandria, its most important centre of learning, attracted scholars from across the Hellenistic world, mostly Greek & Egyptian, but also Jewish & others, represented by scholars such as Hipparchus, Apollonius, Ptolemy, and Diophantus (see below).  During this period Greek language & culture flooded the Middle East & Greek became the language of scholarship.  Greek mathematics merged with Egyptian & Babylonian to give rise to a Hellenistic mathematics.  

 

Persian-Babylonian schools: * see Endnote 18

The Persian-Babylonian schools were part of the Sasanian Empire (or Neo-Persian Empire) which was the last period of the Persian Empire before Islam; named after the House of Sasan who ruled from 224 to 651 AD; successor to the Parthian Empire; for over 400 years it was recognized as a leading world power alongside its neighboring arch-rival the Roman-Byzantine Empire.  These urban schools flourished between the 5th and early 7th century AD under the enlightened rulers of this empire, patrons of letters & philosophy. 

 

Edessa:

capital city of the historic Syriac kingdom located in Upper Mesopotamia, semi-autonomous 132 BC to 216 AD becoming a Roman province from 216–608 AD & from 318 AD part of the Diocese of the East.  By 5th century AD it was a center of Syriac literature & learning.  The School of Edessa was a theological school of great importance to the Syriac-speaking Assyrian world, founded in the 2nd century by the kings of the Abgar dynasty. When Nisbis fell to the Persians in 363 AD, St. Ephrem the Syrian, with his disciples/teachers, left for Edessa, where he re-established his school. It grew in importance, housed many monks as well as Ephrem, the latter interpreting Scripture & composing poetry & hymns and teaching in the school.  Following the Nestorian schism (489 AD) the school closed.

 

Gundisapora: *  see Endnote 19

(aka Gundeshapur) founded as garrison town 260 AD (SW Iran) by Sassanid king Shapur I following the defeat of the Roman Emperor Valerian; in the 5th century it became the intellectual center of the Sassanid Empire, home of the Academy of Gundishapur.

 

Ctesiphon:

ancient city located on the E. bank of Tigris (near Baghdad); from 2nd century BC to 7th century AD, main capital of Parthians and Sassanids (the 2 successor dynasties of the Persian empire); developed into a rich commercial metropolis, merging with the surrounding cities along both shores of the Tigris to include the Hellenistic city of Seleucia; its environs were therefore sometimes referred to as "The Cities".  In the late 6th & early 7th century, it was one of the largest cities in the world.  During the Roman–Parthian Wars, it fell 3 times to the Romans, & later twice during Sasanian rule. It was also the site of the Battle of Ctesiphon (363 AD).  It was 1 of 3 Sassanid Empire's centers of education along with the Academy of Gondishapur & Resaina.

 

Zenodorus:

(200-140 BC)  Greek mathematician, wrote On Isometric Figures, a treatise on isoperimetric figures and plane figures of equal perimeter but differing areas, and solid figures of equal surface but differing volume.  Origins unknown, possibly an Athenian.

 

Serenus:

300 – c. 360 AD) Greek mathematician of the RomanEmpire, from Antinouplis, a city in Egypt; wrote 2 works: On the Section of a Cylinder & On the Section of a Cone; these works were connected to Apollonius' Conics, on which he commented.

  

Hypsicles:

(190-120 BC) Greek living in Alexandria, mathematician & astronomer, author of Book XIV of Euclid's Elements, concerned with the inscription of regular solids in a sphere and re-affirmed earlier proofs regarding volume. He also wrote On the Ascension of Stars, the first work in which the ecliptic is divided into 360 parts or degrees, an idea borrowed from Babylonia, reflecting links between Greek & Babylonian astronomy in the 2nd century B. C.

 

Chaldean circle division: * see Endnote 20

Classical Greek & Latin sources frequently use the term "Chaldean" for the astronomers of Mesopotamia, priest-scribes specializing in astrology & other forms of divination.  The division of the ecliptic into the zodiac signs originates in Babylonian ("Chaldean") astronomy during the first half of the 1st millennium BC.  The zodiac draws on stars in earlier Babylonian star catalogs, compiled around 1000 BC. Some of the constellations can be traced  back to Bronze Age (Old Babylonian) sources, including Gemini "The Twins", "The Great Twins", and Cancer "The Crab", among others.

 

Diophantus:

(210-285 AD) Hellenistic mathematician, father of algebra; lived in Alexandria, possibly a Greek, Hellenized Egyptian or Babylonian, Jewish or Chaldean; first Greek(?) mathematician who recognized fractions as numbers allowing positive rational numbers for coefficients & solutions; his approximations are an important part of mathematical research.  Some of his problems from his Arithmetica (writings) have been found in Arabic sources; highly influential in 16th century Europe.

 

Arameans:

ancient NW Semitic Aramaic-speaking tribal confederation, origins in region known as Aram (present-day Syria) 11th to 8th centuries BC; established a patchwork of independent Aramaic kingdoms in the Levant and seized parts of Mesopotamia; never became a unified state, but formed small independent kingdoms, chiefly across the coastal land between S. Turkey and N. Sinai. Their political influence was confined to these small entities which were absorbed into the Neo-Babylonian Empire (9th century BC).  At this point, Chaldeans, Aramaeans, Suteans & Babylonians became indistinguishable.

 

Syriac:

Emerging in 5th century BC Assyria, it was a spoken dialect across much of the Near East, Asia Minor and Eastern Arabia; in the 1st century AD, in Edessa, became a major literary language & spread throughout the Middle East 4th- 8th centuries; it preserved a large body of Syriac literature (90% of the extant Aramaic literature).  There were substantial efforts to translate Greek texts into Syriac & many Greek works survive only in Syriac translation

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Arabian-Islamic thinkers (mathematics):

emerges with Al-Khwarizmi (780-850 AD) who with Diophantus developed Algebra.  He deals with ways to solve for the positive roots of first & second degree (linear and quadratic) polynomial equations, introduces the method of reduction, and gave general solutions for these equations. His rhetorical equations (written as full sentences) was transitioned to symbolic algebra by later scholars.  It was a revolutionary move away from the Greek concept of mathematics based on geometry.  Algebra was a unifying theory, allowed rational & irrational numbers, geometrical magnitudes, all be treated as "algebraic objects"; a much broader concept, allowed mathematics to be applied in a new way.

 

harmony (Pythagorean, origins in music):

Musical principles were vital for the Pythagoreans.  They discovered the precise relation between the pitch of the musical note & length of the string that produces it; they found that the string lengths were in simple ratios of each other, one was half the size of the first, another was 2/3 the size, and so on.  They established the foundations of the study of harmonics—how strings and columns of air vibrate, how they produce overtones, how the overtones are related arithmetically to one another.  Thus musical notes could be translated into mathematical equations reflecting proportion as well as showing that number was at the root of all things. 

  

polyphony:

one type of musical texture – texture being the way that melodic, rhythmic & harmonic aspects of a composition are combined to shape overall sound & quality; in particular it consists of 2 or more simultaneous lines of independent melody (as opposed to 1 voice- monophony or 1 dominant melodic voice accompanied by chords- homophony).  In Western music, it refers to music of the late Middle Ages & Renaissance.  Baroque forms such as the  fugue are described as contrapuntal, not as polyphony.

 

harmony:

the study & analysis of individual sounds by hearing, usually, this means simultaneously occurring frequencies, pitches (tones, notes), or chords; it involves chords and their construction and chord progressions and the principles of connection that govern them; called the "vertical" aspect of music, as distinguished from melodic line, or the "horizontal" aspect (e.g. Counterpoint, based on relationships between melodic lines & polyphony or simultaneous sounding of separate independent voices)

 

chaos:

a Greek neuter noun, means "yawning" or "gap"; for Hesiod Chaos was personified enough to have borne children, but was also a place, far away, underground and "gloomy", beyond which lived the Titans.  For the Roman poet Ovid, it was unformed mass, where all the elements were jumbled up together in a "shapeless heap"

 

cosmos:

Pythagoras was the first person to call the universe a kosmos (Greek- an equal presence of order & beauty).  The universe is a cosmos because the phenomena of nature embody geometrical form and proportion, allowing things to unfold and function in elegant and efficient ways which gives rise to beauty; it shows all things are related through whole–part and proportional relationships.  Number leads to proportion, and proportion gives harmony (fitting together); the kosmos itself is a harmony in which all of the parts are proportionally bound together.