Download Mathematical Methods in Hellenistic Times

MATHEMATICAL METHODS
IN HELLENISTIC TIMES
The Hellenistic Period
 In this brief section, we show what type of
problems mathematicians solved, and the extent
of Greek Mathematics before the collapse of the
mathematical world in about 400 C.E.
EUDOXUS
 He is famous in his works in ratios and
methods of exhaustion.
 He is responsible in turning the astronomy into
a mathematical science.
 He is the inventor of the “two-sphere model”,
the modifications for the various motions of
the sun, moon and planets.
APOLLONIUS
 He proposed the following solution:
- Place the center of the sun’s orbit at a point (called
the eccenter) displayed away from the earth. If the
sun moves uniformly around the new circle (called
the deferent circle), an observer on earth will see
more than a quarter of the circle against the spring
quadrant (the upper right) than the summer
quadrant (the upper left). The distance of ED or
better the ratio of ED to DS is known as the
eccentricity of the deferent.
HIPPARCHUS AND THE
BEGINNING OF TRIGONOMETRY
 He carried out numerous observations of
planetary positions, introduced a coordinate
system for the stellar sphere, the tabulation of
trigonometric ratios.
 System Coordinate
- To deal with the positions of the stars and
planets, one needs both a unit of measure for
arcs and angles, as well as, a method of
specifying where a particular body is located
on the celestial sphere.
 He adopted the division of the circumference
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of the circle into 360 parts (degrees), along
with the sexagisimal division of degrees into
minutes and seconds.
He also used arcs of 1/24 of a circle (steps)
and 1/48 of a circle (half-steps).
Babylonians first introduced coordinates into
the sky is also known as the Ecliptic System.
The coordinate along the ecliptic is called the
longitude.
The coordinate perpendicular is called the
latitude.
basic
element
in
Hipparchus’
trigonometry was the chord subtending a
given arc (or central angle) in a circle of fixed
radius.
 The
 Hipparchus used the same measure for the
radius of the circle.
circumference = 2πR, π the sexagisimal
approximation 3;8,30
 He calculated the radius R as
60 x 360/ 2π = 6,0,0/6;17 = 57,18 = 3438’
 In a circle of radius, the measure of an angle
(length cut-off on the circumference divided
by the radius) equals its radian measure.
 To calculate the table of chords, he began with
a 60˚ angle.
Crd(α) = Crd (60) = 3438’ = 57,18
90˚ angle
Chord is equal to R√2 = 4862’ = 81, 2
 To calculate of the other angles, he used two
geometric results,
crd (180-α) = √(2R)^2 – crd^2(α)
PTOLEMY AND THE ALMAGEST
 Claudius Ptolemy
- He made numerous observations of the heavens
from locations near Alexandria and wrote several
important books.
- Mathematiki Syntaxis (Mathematical Collection),
a work in 13 books that contained a complete
mathematical description of the Greek model of the
universe with parameters for the various motions of
the sun, moon and planets.
 Chords Tables
- To construct the tables of chords of all
arcs from 1/2˚ to 180˚ (intervals of 1/2˚)
he took R = 60
 First calculation established the chord of 36˚
namely, the length of decagon inscribed in a
circle.
ROMAN MATHEMATICS
 Cicero admitted that Romans were not
interested in Mathematics.
 Vitruvius
- He wrote the book “On Architecture”,
architects needed comprehensive liberal
education (draftsmanship to astronomy).
- Geometry offers many aids to
architecture.
 Lucius Columella
- Roman Gentleman Farmer
- He wrote that one deals with fields needs
to be able to work out areas.
Basic Formulas (area of squares, rectangles,
triangles, circles) including the use of (1/3 +
1/10) s^2 for the area of triangle of side s, 22/7
= π and A = ½ (b + h)h + 1/14 (D/2)^2 for the
area of a circle segment of base b and height h.