MT 469 test 2

Hilbert

"No one shall expel us from the paradise Cantor created for us"

The Hellenistic Period

Alexandrian School: museum and library with 750,000 volumes including personal libraries of Aristotle and his successor theophrates dedicated to muses center for poets, philosphers, historians, astronomers, geogeraphers, etc

hipparchus

Father of Trigonometry division of a circle of 360 degrees diameter is divided into 120 parts, then further division by 60 for a given arc he gives the number of units in the chord sin(a) = chord (2a) / 120

Niccolo Fontana of Brescia - Tartaglia

Fior challenged Tartaglia to solve 30 cubic equations and Tartaglia did the same Fior: x^3 + mx = n (solved 30) Tartaglia: x^3 + mx^2 = n (solved all 60)

Cavaleiri

Jesuate not a jesuit theorem of cavalieri all lines of a paralellogram are twice the lines of either two triangles analytic geometry transformation of coordinates, calculus

Nicole Oresme

Latitude of Forms Geometrical verification of the Merton Rule Area = distance = A1 + 3A1 + 5A1 + ... + (2n-1)A1 = n^2A1

Cardano

Published solution to cubic in boring number of cases solution of quartic due to Luigi Ferrari x^3 + mx = n he lets t - u = n and tu = (m/3)^3

Apollonius Lost Works

Quick Delivery, Cutting off a Ratio, Cutting off an Area, on determinate section, tangencies, vergings, plane loci

Girolamo Saccheri

Saccheri Quadrilateral (two right angles at its base and equal perpendicular sides) ruled out obtuse case based on tacit assumption of infinitude of straight line

Galileo Galilei

Salviati, Sagredo, Simplicio influenced by oresme path of a projectile is a parabola cycloid (traced on paper, cut it out, weighed it and deduced that area is little less than three time the generating circle) Salviati: infinities and infentesimals divide line segment into infinite # of parts easy as dividing to finite #

Jobst Burgi

Swiss mathematics defined log 10^8(1 + 10^-4)^L = 10L approximately equal to ln

Richard Suiseth

The Calculator intensities Oresme gives ingenious graphical procedure and showed divergence of harmonic means

zenodorus

among n-sided polygons of same perimeter, the regular one has most area among regular polygons of same perimeter, the one having more sides has greater area

on determinate section

analytic geometry solution of quadratics

nichomachus

arithmetic independent of geometry arithmetic essential gives multiplication tables from 1 to 9 four perfect numbers: 6, 28, 496, 8128 sieve of eratosthenes

dipohantus

arithmetica: 13 books of which wehave 6 come from manuscript he wrote calculation of numbers 189 problems: indeterminate equations: linear, quadratic in 2 unkowns, simultaneous quadratics no use of letters, instead symbolism to represent variables and numbers

trigonometry

arose out of need for quantitative geometry created by hipparchus, menelaus, ptolemy

pappus's theorem

cat's cradle mystic hexagram duality

Kepler

classification of conics (principle of continuity) lines, hyperbolas, parabolas, ellipses, circles (F1, F2) areas of circles area of an ellipse kepler's laws (planets move about the sun in elliptical orbits with sun at one focus) used arguments involving infinitesimals

pappus

commentator work not of highest order but some worthy of note wrote the collection contains treasury of analysis

Euler

complex roots of unity alternate method of solution of x^3 + px = q let x = u - v and uv = 1/3p

alexandrian geometry

conchoid and the cissoid

Rafael Bombelli

considered the equations for x that euler came up with and assumed that x = (a + bi) + (a - bi)

Gauss

cubic can have 1 or 3 real roots

Theon

daughter was hypatia commentaries on diophantus and apollonius

Conics

definition of a cone: if a straight line passing through a fixed point is made to move about the circumference of a circle not in the same plane as the point, the moving straight line will trace out the surface of a double cone gives rise to a double branched hyperbola no longer necessary to take a plane perpendicular to an element names of conic sections

cissoid

dicoles, y^2(a+x) = (a-x)^3 used to solve the delian problem

Goldbach

didn't believe and asked Euler to factor x^4 + 72x - 20 Goldbach's Conjecture: every even # is the sum of 2 primes

Names of conic sections

ellipse - deficiency hyperbola - excess parabola - equal

alexandrian mathematics

euclid, archimedes, apollonius, eratosthenes, hipparchus, nicomedes, menelaus, heron, ptolemy, diophantus, pappus

evolute and apollonius

evolute split the plane into regions in which points in each region have the same number of normals to the conic one can draw through them

Rene Descartes

f - e + v = 2 cartesian geometry relative merits of algebra and geometry aim of his method: 1. through algebraic problems to free geometry from use of diagrams 2. to give meaning to operations of algebra fundamental principle of analytic geometry pappus's problem : 3 and 4 lines quadratic, 5 and 6 lines cubic, 7 and 8 lines results in quartic 5 lines: if not all paralell then can be constructed with straight edge and compass cartesian trident

Pierre de Fermat

fundamental principle of analytic geometry whenever in final equation two unknown quantities are found, we have a locus, the extreme of one of these describing a line construction of tangenet lines calculation of area

Gauss

fundamental theorem of algebra

Omar Khayyam

gave general solution to x^3 + Bx = C where B,C > 0 greatest Arab step in algebra

Viete

gave solution when all three roots are real irreducible case and was not solved by Cardan Trig solution: cos3A = 4cos^3A - 3cosA to solve varius equations let x= y - b/2 quadratic x = y - b/3 for cubic x = y- b/4 for quartic x = y - b/5 quintic

Pappus

generalized to n > 4 lines for locus with respect to the lines

Locus with respect to three and four lines

given three (four) lines, find the locus of a point P that moves so that the square of the distance from P to one of these is proportional to the product of the distances to the other two (or in the case of 4 lines, the product of the distances to two of them is proportional to the product of the distances to the other two

Tangencies

given three things each of which may be a point, line, circle, draw a circle that is tangent to each of these things PPP, LLL, CCC, PPL, LLP, CCP, PPC, PLC, LLC, CCL

Galois

gives necessary and sufficient conditions for polynomial to be solvable by radicals Galois theory : used in showing impossibility of greek problems

aryabhata

hindu mathematics general solution to diophantine equation ax+by=c using continued fractions

ptolemy's theorem

if ABCD is an inscribable quadrilateral, then the sum of the products of opposite sides equals the product of the diagonals computed chord 3/2 and 1/2

Euclid's 5th postulate

if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than two right angles

Lagrange and Ruffini

key is symmetric functions and permutation groups symmetric functions give relationship between roots and coefficients insolvability of the quintic

Apollonius

known as the great geometer considered himself to be a rival to archimedes numeration system, approximation of pi almost all of his works are lost, they make up majority of treasury of analysis by pappus

Geometric Interpretation of Logarithm

let AB be a line segment of length 10^7 and let P begin at A and move with variable speed decreasing in proportion to distance from B with initial velocity 10^7. let Q start from point C and move with uniform speed equal to initial velocity of P then CQ is log of PB usual properties of logs don't hold

four departments

literature, mathematics, astronomy, medicine

plane loci

locus of points the differences of the squares of whose distances from two fixed points is constant is a straight line perpendicular to the line joining the points locus of points the ratio of whose distances from two fixed points is a constant (not 1) is a circle (Circle of Apollonius)

evolutes

locus of points which are the centers of the circles of curvature of the points on a given curve circle of curvature best approximates the curve at the point

Briggs

log1 = 0 and log10 = 1 log tables with mantissa (what's in the table) and characteristic

normal lines

max min principle, avoids perpendicular to tangent since tangent is not well defined, tangent line is then defined as perpendicular to normal line uses the subnormal which is the projection of the normal line onto the axis of the conic section

conchoid

nicomedes, r = a+bsec(theta)

use of focus and directrix for Apollonius

no use, but the locus of points P for which the distance from P to a fixed point F is in constant ratio to the distance from the point to a fixed line DD' ratio for Apollonius = PF/FL = e (eccentricity) e < 1 ellipse e = 1 parabola e > 1 hyperbola

Maximum Minimum Principle

normal line from a point Q to a curve C is the line so that the distance from Q to C is a relative max or min

Scipione dal Ferro

professor of math at Bologna x^3 + mx = n

AL-hazen

proved that the 4th angle in a quadrilateral was a right angle after starting with 3 right angles assumed the 5th to prove the 5th

ptolemy

put trig in the definitive form it retained for 1000 years table of chords / first trig table construction of chords 36 and 72 side of a regular hexagon is equal to the radius and an arc of 60 degrees has length 60 and sin30=1/2 the side of an inscribed square is square root of 2 times the radius and sin45=1/120chord90 = square root 2 / 2 the side of inscribed equilateral triangle can be calculated in terms of radius which gives sin60

greatest achievement of hellenistic period

quantitative astronomical theory

N. Bernoulli

questioned eluer and said factor x^4 - 4x^3 + 2x^2 + 4x + 4

Heron

referred to as mechanicus (mechicanical engineer) rigorous mathematics and aproximate procedures/formulas of egyptians concerned with applied geometry mechanics how to dig a straight tunnel under a mountain by working simultaneously from both ends

nature of alexandrian mathematics

results in geometry were those useful in calc of area, volume, length worked with irrational numbers, approximation of pi, babylonian influence, randomly assigned numbers to things Extended arithmetic and algebra as subjects in their own right calculated centers of gravity, dealt with forces, etc..

Euler

said it couldn't be done

Lebniz

said it couldn't be done

Torricelli

shows that arc length is length of segment of tangent line to curve at its initial point calculation of normal and tangent lines to a curve

Pacioli

solution of general cubic equation is impossible

Luigi Ferrari

solution of x^4 + 6x^2 + 36 = 60x solution of x^4 + bx^3 + cx^2 + dx + e = 0

Cutting off a ratio

solutions to quadratics

Zassenhaus

solvability by radicals : x^n - 1 = 0 is solvable by radicals

Archimedes

solved x^2(c-x) = db^2

Quick Delivery

speedy methods of calculations

Menelaus

sphaerica proves theorems analogous to the theorems of euclid if abc is a spherical triangle then a + b > c equal sides subtend equal angles sum of the angles > 180 equal angles imply congruent triangles

heron's formula for area of triangle

square root of s(s-a)(s-b)(s-c)

John Napier

started with geometric sequence: 1, r, r^2, r^3,... r has to be close to 1, he chose 1 - 10^-7 definition: logN = L if N = 10^7 (1-10^-7)^L logN = L if log base 1/e (N/10^7) = L/10^7

Omar Khayyam

started with quadrilateral with 2 right angles at its base and equal perpendicular sides follows that top two angles are equal used assumptions that the angles are acute or obtuse and those result in contradictions so they must be right

Construction of normal to conic from point Q

straight-edge and compass suffices for parabola but need "solid curves" for ellipse and hyperbola plane problem = straight edge and compass solid problem= solid loci (conics) linear problem = any other curves

Descartes

tested his analytic geometry against this problem (locus with respect to the lines)

two main divisions of alexandrian math

theoretical: intellectual concepts, arithmetic and geometry applied: material concepts

Vergings

those which can be solved by straight-edge and compass

Nasir Eddin al Tusi

used a statement equivalent to the 5th

Johann Heinrich Lambert

went considerably beyond Saccheri in deducing theorems resulting from acute and obtuse angle hypotheses sum of angles in triangle <, >, = 180 for acute, obtuse, and right angle area of triangle = k(180 - sum of angles) area of triangle = k(sum of angles - 180) obtuse angle resembles spherical geometry where excess is proportional to the area

Babylonians completing the square

x^2 + b^2 = ax (EUCLID SOLVES THIS) x^2 + ax = b^2 x^2 = ax + b^2

equation for hyperbola

y^2 = lx + b^2x^2/a^2

equation for ellipse

y^2 = lx - b^2x^2/a^2

equation for parabola

y^2=lx