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18 Cards in this Set
- Front
- Back
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Archimedes viewed his most important work as:
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sphere inside a cylinder has volume = 2/3 that of the cylinder and surface area = 2/3 that of the cylinder. The height of the cylinder = 2x radius of the sphere.
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How do we know about Archimedes's work?
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Most of Archimedes work is passed down through a Palimsest, not discovered until 1906 (a painting over the manuscript had to be removed carefully).
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Name some famous problems solved by Archimedes
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-Theorem of the broken chord
-Crown problem -relation between length of lever arms and weights in balance -Quadrature of the parabola -Area of a spiral |
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What is Appolonius famous for?
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A contemporary of Archimedes, he described the plane conic sections: parabola, ellipse, hyperbola.
He did this using slices from a double cone. |
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Describe the results of Archimede's theorem of the broken chord.
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A chord is broken into two parts AB and BC with BC greater than AB, as shown. F is on BC such that MF is perpendicular to BC and arc AM equals arc MC. Therefore F is the midpoint of the broken chord.
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Describe the crown problem
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He told the king (while still in the nude) that he could submerge gold equal to what the crown should have weighed and measure the water displaced. Then measure the water displaced by the crown. If they were not equal, then the crown was not all gold.
Origin of phrase "eureka" |
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What was the meaning of Appolonius's "symptoms"
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The relationship of a given conic section (ellipse, parabola etc) to the x-y plane it is sitting on (ie, the orientation of ellipse to a given plane).
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Archimedes's burning mirror
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A big copper mirror. Light hitting the mirror is directed to the focus. Shining a strong flame from the focus to a spot on the mirror can burn a hole somewhere far away (moving the focus beam to a different part of the mirror burns a hole somewhere else, hence, aiming.)
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Give the old-school definition of a parabola
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The locus of points equidistant from the focus (a point) and a directrix (a line not sharing any points with the focus).
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Describe Archimedes's observations about lever arms and weights.
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He found that the weights were inversely proportional to the distances - when the lever is balanced, a/b then w2/w1.
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What did Archimedes calculate for pi?
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He also proved that 3 + 10/71 (3. 1408) less than C/d is less than 3 + 1/7 (3.1429) where C is the circumference of a circle and d is the diameter
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How did the Greeks view the heavens?
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As spheres (planets) mixed up with lots of stars. Have constellations that correspond to astrology.
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What did greeks identify about the Earth?
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Equator, north pole, south pole, and the ecliptic.
ecliptic: great circle that describes the sun's travel around earth (over the course of a year: poles are summer and winter solstace.) Equator-great circle and ecliptic intersections are the vernal and autumnal equinoxes. |
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Angle of ecliptic to equator
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before euclid: 24 degrees between equator and eliptic
23deg51min-ptolemy 23 deg30min-now |
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Ptolemy, what did he do with geometry
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Wanted to do right triangle trig (and more).
Used chords on a circle. Chord described by radius and angle of chord. Showed that chordalpha/2R = sin(alpha/2) or, chord alpha = 2Rsin(alpha/2) |
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To do trig, Ptolemy had to do what?
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Approximate square-roots using 45,45,90 triangle. sqrt(2a^2) = hypotenuse.
Then, take length sqrt(2), construct perpendicular of length 1, have side with length sqrt(3). Clever girl! |
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How did Ptolemy approximate pi? What did he use this to do?
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-3.1418
-Construct tables of chord length down to 7.5 degree chords. (note: chord tables in base 60) |
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What identities did ptolemy prove?
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sin^2 a + cos^a = 1,
angle sum forumlae, half angle formulae, all using chord functions. |
