The Greeks
History of Mathematics
Greece
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In 800-650 BC ,Greece consisted of Italy all the way to the edge of the
Black sea. This colonization affected Greek mathematics in that all early
Greek mathematics came from Asia minor, southern Italy, & Africa.
Greece
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Coinage affected the Greeks - stimulated trade, gave rise to a
money economy based not only on agriculture but on movable
goods, created a leisure class.
Geography of Greece affected politics - no single ruler because
each state separated by mountains or sea. Alexander eventually
united Greece. (He died in 323 BC ruling over 2 million square miles
– gave rise to the Hellenistic age
Greek mathematics differ from the Babylonians and the Egyptians more profound, more abstract, and more rational. Notion of seeking
knowledge for its own sake.
Greek civilization begins in Iron Age, flourishing in the 4th and 5th
centuries BC. Most of the productive mathematics done in 350-200
BC.
Thales (625-547 BC)
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Thales is credited with inventing the modern proof.
Thales is known as the first of the Seven Sages of Greece; coined the
maxim “Know thyself” and when asked what was the strangest thing he
had ever seen: “An aged tyrant.”
Thales presumably got interested in geometry when he visited Egypt and
wanted to measure the pyramids by means of shadows. Thales also
determined how far a ship at sea is from the shore.
It is said that Thales predicted a solar eclipse but this would not have
been possible because astronomical records were not precise enough at
that time.
Pythagoras
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Pythagoras (580-500 BC) followed Thales. We know that he lived on the
Aegan island of Samos.
The distinctive feature of the school founded by Pythagoras was its aims
were political, philosophical, and religious.
The 300 or so students at the school studied the Quadrivium (number
theory, music, geometry, and astronomy).
What set the Pythagoreans apart from other sects was their belief that
knowledge is the greatest purification and to them knowledge meant
mathematics.
Mathematics played an essential part in life & religion. We saw last time
that music provided the Pythagoreans with the best instance of their
principle that “number was the cause of everything in nature.”
Pythagoreans
Pythagoreans Celebrate Sunrise
by Fyodor Bronnikov
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Nichomachus’ Introductio Arithmeticae is the most complete
exposition of Pythagoras and his successors work.
The Pythagoreans did not have any symbols so it’s likely that
they used pebbles to think of numbers geometrically – triangle
numbers, square numbers, etc.
The most important achievements of the Pythagorean
school in its influence of the number concept was the
discovery of the “irrational.”
Commensurable
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This idea caused great consternation among the Pythagoreans for it
challenged the adequacy of their philosophy that number was the
essence of all things. They called these numbers “unutterable.
Eventually Eudoxus (408-355BC) resolved this crisis. He revised
the theory of proportion applicable to incommensurable as well
commensurable quantities
Hippias(b. c. 460BC) is credited with discovering that the side of a
square and its diagonal are not commensurable.
i.e., no matter how small one goes, there is no magnitude dividing
evenly into both the square’s side and its diagonal. This discovery
shattered all Pythagorean proofs using commensurability of all
segment.
Examples…
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Eudoxus
(408-355
BC)
Eudoxus eventually gave proofs about similar triangles that did not rely
on this concept. Also it had an unsettling impact upon the
supremacy of whole numbers, for if not all quantities were
commensurable then, then whole numbers were somehow inadequate
to represent geometric lengths. This firmly established the supremacy
of geometry over arithmetic in all subsequent Greek mathematics.
For example, in a square of side length 1, with diagonal root 2, then root
2 arithmetic character is mysterious so better to think of magnitudes as
geometric entries!
Eudoxus is best known for 3 things: his general theory of proportion,
golden ratio, & method of exhaustion, which Archimedes refines.
Approximating
root 2
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The Greeks were not the first to consider numerical values for root 2.
The Babylonians approximated the square root of 2.
Here the Babylonian clay tablet YBC 7289 (c. 1800–1600 BCE) gives an
approximation of in four sexagesimal figures, which is about six decimal
figures:1 + 24/60 + 51/(60)^2 + 10/(60)^3 = 1.41421296
Hippocrates(460-380 BC)
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The mathematician that dominated the 2nd half of the 5th C BC was
Hippocrates.
Hippocrates began his life as a merchant but ended his life as a teacher
(like Thales) . He was robbed of his money (he was robbed by either pirates
or fraudulent customs officials) and went to Athens to prosecute the
offenders. He stayed and attended the lecturers of many philosophers. He
may have come in contact with some Pythagoreans.
The major accomplishment of Hippocrates is that he was the first to write a
systematically organized geometry textbook, called Stoicheia Elements,
that is, basic theorems, or building blocks of mathematical theory.
From then on, mathematicians from all over the ancient world could, at
least in principle, build on a common framework of basic concepts,
methods, and theorems, which stimulated the scientific progress of
mathematics.
Special Problems from Antiquity
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Three problems that Hippocrates faced when he arrived in Athens:
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Quadrature of the circle: Is it possible to construct a square whose
area is equal to that of a given circle? Equivalent to constructing a line
segment whose length is root pi times the radius of the circle.
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Duplication of the cube: Find the edge of a cube, having twice the
volume of a given cube.
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Trisection of an angle.
Add a 4th problem: Construction of a regular heptagon.
Plato insisted 3 things with these special problems: the task be
performed with straightedge and compass only.
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With the straight edge, a line can be drawn through 2 points and
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With the compass, a circle with a given center and radius can be
drawn.
Squaring a Circle
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Surprisingly, the impossibility proofs for the three problems: doubling the
cube, trisecting an angle, and constructing a regular heptagon, all fall into
the same framework. Each depends on showing that roots of certain third
degree polynomial equations are not constructible.
The problem of squaring of a circle found itself in a completely different
category. Carl Louis Ferdinand von Lindemann (1852 – 1939) was a
German mathematician who was the first to prove that π is transcendental
in 1882. Since π is NOT algebraic, it is automatically not constructible.
The area of a circle with radius 1 is exactly π . If a square with such area
were constructible, so would be the number itself. The areas of this
square and this circle are equal. In 1882, it was proven that this figure
cannot be constructed in a finite number of steps with a compass and
straightedge.
Statue of
Euripides in
Louvre, Paris
Doubling the Cube
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Doubling the Cube, the most famous of the collection, is often referred to
as the Delian problem due to a legend that the Delians had consulted
Plato on the subject.
In another form, the story asserts that the Athenians in 430 B.C.
consulted the oracle at Delos in the hope to stop the plague ravaging
their country. They were advised by Apollo to double his altar that had the
form of a cube. As a result of several failed attempts to satisfy the god,
the pestilence only worsened and at the end they turned to Plato for
advice.
According to a letter from the mathematician Eratosthenes(276-195 BC)
to King Ptolemy of Egypt, Euripides mentioned the Delian problem in
one of his (now lost) tragedies.
Euripides (480 BC – 406 BC) was the last of the three great tragedians of classical Athens
(the other two were Aeschylus and Sophocles).
Eratosthenes
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To "double the cube" means to be given a
cube of some side length s and volume V, and
to construct a new cube, larger than the first,
with volume 2V and therefore side length
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The problem is known to be impossible to
solve with only compass and straightedge,
because the cube root of 2≈ 1.25992105 is
not a constructible number.
Trisecting an Angle
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That there may be a difficulty trisecting
an angle is rather surprising. Bisecting is
easy! The same easy approach applies
to bisecting both straight line segments
and circular arcs.
The problem of trisecting a line segment
is no more difficult than finding its n-th
part for an arbitrary n. However, the
general problem of trisecting an arbitrary
angle is not solvable in a finite number of
steps.
Two points must be made.
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First, some angles are trisectable with a ruler and a compass. An obvious
example is supplied by the three quarters of the straight angle - 67.5o. (The
angle itself is constructible as it is obtained by two consecutive angle
bisections. Its third is obtained along the way.) Angles of 30o (draw a right
triangle with a side 1 and hypotenuse 2) and 45o (bisect the right angle) are
both constructible. Therefore, the latter also admits a classical trisection.
However, and this is where the difficulty lies, an arbitrary angle can't be
trisected with a ruler and a compass, as stated above.
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Secondly, consider the geometric series 1/4 + 1/16 + 1/64 + ... that adds up to
1/3. Once we know how to bisect an angle, we may also find its 2-n-th part, for
any n. In particular, one may construct 1/4, 1/16, ... of any angle and, in
principle, find its third after an infinite number of steps. This solution is
universal but requires a forbidden (infinite) number of steps.
The problem was settled in 1837 by Pierre Laurent Wantzel (1814-1848) who
had proven that there was no way to trisect a 60o angle in the classical framework.
Constructing a Regular Heptagon
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3-, 4-, 5-, and 6-gons (i.e., regular triangles, squares, pentagons, and
hexagons) are all easily constructible. Octagons are constructible on the
heels of squares with a single angle bisection. All polygons obtained
from the above four by doubling the number of sides are also
constructible. Not so for a heptagon, a 7-sided polygon.
In 1796, at the age of 19, Gauss showed that a regular heptadecagon (a
17-sided polygon) is constructible. Gauss showed that the heptadecagon
was only a particular case of a family of constructible regular polygons.
Any N-gon, where N is in the form 2np1...pm, where pi's are distinct
Fermat primes (i.e. primes in the form 22k + 1), is constructible. Wantzel
later proved that no other regular polygon has this property.
The Lune of Hippocrates
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Some apparent partial solutions gave false hope in the “Squaring the
Circle” Problem for a long time. In this figure, the shaded figure is the
Lune of Hippocrates. Its area is equal to the area of the triangle ABC.
This is not a complete solution of the task (obviously since the
complete solution is proven to be impossible).
Explanation….