Cusanus:

(1401-1464), (aka Nicholas of Cusa) German philosopher, theologian & astronomer; early proponent of humanism & a conciliarist at Council of Basle; known for his mystical writings about Christianity, particularly on the possibility of knowing God; in Vision of God (1453) says “unless God were infinite, he would not be the end of desire.” reversing Aristotle’s supposition that god cannot be infinite; his astronomical observations are based on abstract speculations not observation; suggests the earth is a star like other stars, not fixed in place, not center of the universe; the “outermost sphere” of the Aristotelian & Ptolemaic world picture is not a boundary, the universe is therefore “infinite,” in the sense of physically unbounded.

 

de Vinci ( Leonardo da Vinci):

(1452-1519) Italian Renaissance polymath (contributed to painting, sculpting, music, literature, architecture, science, mathematics, engineering, invention, anatomy, geology, astronomy, botany, writing, history & cartography; father of paleontology, ichthyology & architecture, one of the greatest painters of all time; credited with invention of parachute, helicopter & tank; in his Notebooks (40 years before Copernicus), he wrote in large letters:

"IL SOLE NO SI MUOVE," or "The sun does not move."  & states: "The earth is not in the centre of the Sun's orbit nor at the centre of the universe."

 

treatise on grains of sand:

(aka The Sand Reckoner) Archimedes set out to determine an upper bound for the number of grains of sand that fit into the universe. In order to do this, he had to estimate the size of the universe according to the contemporary model, and invent a way to talk about extremely large numbers.

 

aether:

the material that fills the region of the universe above the terrestrial sphere; referenced by Plato (Timaeus) & Aristotle, who noted that crystalline spheres made of aether held the celestial bodies, this led to Aristotle's explanation of the observed orbits of stars and planets in perfectly circular motion in crystalline aether.

 

Eudoxus:

(390-337 BC) Greek astronomer, mathematician, student of Plato; all work lost- fragments preserved in Hipparchus; his planetary system is a convoluted & complex explanation of the wandering motions of the planets, using combinations of uniform circular motions centred on a spherical Earth; his importance to Greek astronomy is considerable, as he was the first to attempt a mathematical explanation of the planets.

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Apollonius: * see Endnote 28

(late 3rd- early 2nd centuries BC) aka Apollonius of Perga; Greek geometer & astronomer known for theories on conic sections; using Euclid & Archimedes he was only superseded by analytic geometry; his definitions of the terms ellipse, parabola & hyperbola still in use. Little of his work survives; his erroneous hypothesis of eccentric orbits (explaining aberrant motion of the planets) accepted in Middle Ages, disproved in Renaissance.

 

loci:

a curve or other figure formed by all the points satisfying a particular equation of the relation between coordinates, or by a point, line, or surface moving according to mathematically defined conditions.

 

co-ordinates:

a group of numbers used to indicate the position of a point, line, or plane.

 

Fermat:

(1607-1665) French lawyer & mathematician, credited with developments leading to infinitesimal calculus, including his technique of adequality (from Diophantus) to find maxima for functions & tangent lines to curves, similar to differential calculus.  While reading Diophantus he found the solution for an equation for which the ancient claimed no solutions existed (Fermat's Last Theorem).  This initiated advances in number theory & the further study of Diophantine equations.  Also contributed to analytic geometry, probability & optics (Fermat's principle for light propagation)

 

exhaustion method Archimedes: * see Endnote 29

a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape.

 

Eratosthenes:

(276-194 BC) Greek mathematician (invented sieve of Eratosthenes, an efficient method of identifying prime numbers), astronomer & father of geography; famous for his accurate calculation of the circumference of the Earth; first to calculate tilt of the Earth's axis; calculated distance from the Earth to the Sun, invented the leap day & created first map of the world, using parallels & meridians; established scientific chronology & was chief librarian at library of Alexandria.

 

quadrature of the parabola:

treatise on geometry by Archimedes, 3rd century BC; a letter to his friend Eratosthenes, presents 24 propositions regarding parabolas, culminating in a proof that the area of a parabolic segment (the region enclosed by a parabola and a line) is 4/3 that of a certain inscribed triangle.

 

inscribed rectangles: * see Endnote 30

to compute the area under a curve (a parabola) we can approximate by using rectangles inscribed in the curve & circumscribed on the curve. The total area of the inscribed rectangles is the lower sum, & the total area of the circumscribed rectangles is the upper sum.

 

Riemann’s integral: * see Endnote 31

in real analysis mathematics, this was the first rigorous definition of the integral of a function on an interval; created by Riemann, presented to the faculty at the University of Göttingen in 1854 (published 1868); for many functions & practical applications it can be evaluated by the fundamental theorem of calculus or approximated by numerical integration.

 

quadratures: * see Endnote 32

archaic historical term which means determining area; Greek mathematicians, according to Pythagorean doctrine, understood determination of area of a figure as the process of geometrically constructing a square having the same area (squaring), thus the name quadrature for this process.  Quadrature problems served as one of the main sources of problems in the development of calculus, and introduce important topics in mathematical analysis. 

 

bounding function: * see Endnote 33

(aka bounded function) in mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. In other words, there exists a real number M such that:                               for all x in X.    A function that is not bounded is said to be unbounded.

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cubic style (early Egyptians): * see Endnote 34

The Archaic (aka Early Dynastic) Period of Egypt (3100-2686 BC) immediately followed unification of Upper & Lower Egypt, to include 1st & 2nd dynasty.  Funeral practices for the rich involved the construction of mastabas (meaning "house for eternity"), a flat-roofed, rectangular structure with inward sloping sides, constructed out of mud-bricks (from the Nile River), about 4 times as long as wide, many rose to 30 feet in height; marked the burial sites of eminent Egyptians.

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