Cavemen painted designs on cave walls.
Papyrus describes Egyptian geometry.
Babylonian mathematicians used Pythagorean theorem to find distances.
He lived -750 to -690 and constructed circles from rectangles and squares from circles.
Sophists invented geometric proofs and studied circles as many-sided regular polygons. They tried to square circle, trisect angle, and double cube using only straightedge and compass.
He lived -470 to -410 and wrote first geometry text, first calculated curved area using rectilinear figures {quadrature, Hippocrates}, and first proved theorems using earlier theorems {pyramiding theorems}. He invented method of proving something by disproving its opposite {indirect proof, Hippocrates}.
He lived -408 to -355. He studied limits, used infinite polygons to find curved-figure areas and volumes {exhaustion method, Eudoxus}, and developed explicit axioms.
Proportion is magnitude or length. He showed how to prove that two different integer ratios, which make real numbers, are equal or not equal. Proportions are magnitude or length ratios. To compare ratios, find integer pairs such that product of first integer and numerators and product of second integer and denominators makes numerators greater than denominators. If successful, first ratio is greater than second, because new ratio, first/second, is less than first ratio and greater than second ratio. If unsuccessful, find integer pairs such that product of first integer and numerators and product of second integer and denominators makes numerators less than denominators. If successful, first ratio is less than second, because new ratio, first/second, is greater than first ratio and less than second ratio. If not successful, ratios are equal. You can thus approach any real number and so can work with irrational-number square roots of positive integers.
Planetary orbits are nested spheres. He measured year length.
He lived -325 to -265 developed Euclid's theorem and Euclid's algorithm. He studied perpendicular, parallel, superposition, arc, and prime numbers. He used exhaustion method, rather than infinitesimals, to study curves. He systematized plane geometry, number proportions and ratios, prime numbers, and solid geometry.
Book 1 is about congruence, parallel lines, Pythagorean theorem, simple constructions, constructions with equal areas, and parallelograms {rectilinear figure, Euclid}. Sum of two triangle-side lengths is greater than or equal to third-side length. Book 2 is about geometric algebra, using areas and volumes to find products and quadratic equations, and adding line segments to add. Book 3 is about circles, chords, tangents, secants, central angles, and inscribed angles. Book 4 is about figures inscribed in, or circumscribed around, circles. Book 5 is about proportion by magnitudes, commensurable magnitudes, and incommensurable magnitudes. Book 6 is about similar figures, using proportions. Book 7 is about number theory, Euclidean algorithm, and numbers as line segments. Book 8 is about geometric progressions. Book 9 is about square and cubic numbers, plane and solid numbers, geometric progressions, and the theorem that number of primes is infinite. Book 10 classifies incommensurable magnitudes. Book 11 is about convex solids and generation of solids. Book 12 is about curved-surface areas and volumes, using exhaustion method and indirect proof. Book 13 is about regular polyhedrons in spheres and regular polygons in circles.
He lived -276 to -194 and found circumference of Earth.
He lived -262 to -185 and was Neo-Pythagorean and mystic. He invented a systematic theory of parabolas, ellipses, and hyperbolas, based on eccentricity, directrix, and focus. He studied right circular cones, oblique circular cones, hyperbolas, parabolas, ellipses, conjugate diameters, tangents, asymptotes, foci, conic intersections, maximum and minimum conic lengths, conic normals, similar and congruent conics, and conic segments. Two conic tangents meet at poles, and sides are polars. Given three points, lines, or circles, construct a circle tangent to or including the points, lines, or circles {Apollonian problem}.
Ethics
Simple life is best.
Mind
Mind and body are separate realities.
He lived 87 to 150, invented maps with longitude and latitude, discovered Ptolemy's theorem, and invented epicycles to describe planetary motions.
He lived 260 to 350 and proved Pappus' theorem {Guldinus theorem}.
He lived 940 to 997, used secant and cosecant, and constructed using straightedges and circles.
He lived 1201 to 1274 and used tangent and secant. He invented devices to resolve linear motion into sum of two circular motions {Tusi-couple, Nasir-Eddin}.
He lived 1540 to 1603 and invented sine law, cosine law, and Napier's rule.
He lived 1548 to 1620 and used decimal fractions and force parallelograms.
He lived 1598 to 1647, invented Cavalieri's theorem, and studied indivisibles method [1629].
He lived 1591 to 1661, invented Desargue's theorem, and studied projective geometry, involution, harmonic point sets, and poles and polar theory.
He lived 1713 to 1765, studied space curves [1731], invented Clairaut's equation, and determined Earth's shape.
She lived 1718 to 1799 and published discussion of cubic witch of Agnesi curve [1948].
He lived 1746 to 1818, studied developable surfaces, rediscovered projective geometry [1768], and was the "father of descriptive geometry".
He lived 1788 to 1867, rediscovered projective geometry, and studied affine geometry, differential geometry, and harmonic point sets.
He lived 1802 to 1860 and used substitute parallel axiom [1823], applied to intersecting and non-intersecting lines, to make non-Euclidean geometry [1833].
He lived 1801 to 1868 and studied trilinear coordinates and line coordinates.
He lived 1801 to 1883 and invented Plateau's problem.
He lived 1798 to 1867 and analyzed projective geometry without metric and without congruence.
He lived 1826 to 1866. He studied non-Euclidean geometry, differential geometry, complex functions, multiple-valued functions, mapping, prime-number theorems, analytic number theory, and singularities. He invented Riemann surfaces, Riemann-Darboux integral, Riemann zeta function, Riemann mapping theorem, and Riemann hypothesis. Riemann integrals are sums over infinity of step functions. All closed line segments have the same number of points. All points, in plane touching Riemann sphere at South Pole, map to sphere points, with points at infinity mapping to North Pole. Compact-plane points can thus map to limited, closed, and bounded surfaces.
He lived 1838 to 1926.
He lived 1924 to ? and ascribed fluctuations to discontinuous effects and to trends. He studied fractals, self-symmetry, 1/f noise, and 1/f squared noise. Fractal curves have non-integral dimensions. 1/f noise is like Cantor sets. In continuous intervals, continually removing inner third of each remaining continuous interval still leaves infinitely many points, and total empty distance is interval length {Cantor set, Mandelbrot}. Cantor sets are the same at all scales.
He lived 1945 to ?.
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Date Modified: 2022.0225