Pythagoras (theorum): * see Endnote 12

Pythagoras & his disciples believed that the world could be explained using mathematics, based on whole numbers & their ratios; reality was mathematical.  All relationships could be reduced to numbers in order to account for geometrical properties.  One major advances was the formalizing of mathematics with idea of axioms (statements regarded as true), from which theorems of geometry were logically  proven, famously the Pythagorean theorem.

 

Descartes (Cartesian coordinates):* see Endnote 13

Cartesian coordinate system revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra.   It allowed geometric shapes (such as curves) to be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape.  It is the foundation of analytic geometry, allows geometric interpretations of linear algebra, complex analysis, differential geometry, multivariate calculus, group theory.

 

vivify:

to give life to; animate; quicken.

 

thoroughbass:

(aka figured bass) musical notation in which numerals & symbols (often accidentals) indicate intervals, chords & non-chord tones that a musician playing piano, harpsichord, organ, lute (any instrument using chords) play in relation to the bass note that these numbers and symbols appear above or below; associated with basso continuo, improvised accompaniment used in almost all genres of music in the Baroque period (c.1600–1750).

 

perspective: * see Endnote 14

Linear perspective represents the light that passes from a scene through an imaginary rectangle (the plane of the painting), as if an artist was looking through a window & painting what he sees directly onto the windowpane. Each painted object is a flat, scaled down version of the object on the other side of the window.  It uses a horizon line (often implied) directly opposite the viewer's eye, representing objects infinitely far away.  Any perspective representation of a scene that includes parallel lines has 1 or more vanishing points.  It employs foreshortening.

 

fugue:

a contrapuntal compositional technique in music, using 2 or more voices, built on a subject (a musical theme) introduced at the beginning in imitation (repetition at different pitches) & which recurs frequently in the course of the composition; comprised of 3 sections: an exposition, a development, and a final entry that contains the return of the subject in the fugue's tonic key.

 

Riemann:

(1826-1866) German mathematician, contributed to analysis, number theory & differential geometry.  In real analysis, he is famous for the first rigorous formulation of the integral (the Riemann integral); in complex analysis his Riemann surfaces broke new ground.  His Riemann hypothesis progressed analytic number theory and his differential geometry laid the foundations of the mathematics of general relativity.  He established Riemannian geometry by finding the correct way to extend into n dimensions the differential geometry of surfaces.