assgmnt geometri
TRANSCRIPT
-
7/31/2019 assgmnt geometri
1/24
ESB 4144 MODERN GEOMETRICS
INTRODUCTION
GEOMETRY
Geometry is a branch of mathematics that deals with the measurement, properties, and
relationships of points, lines, angles, and two- and three-dimensional figures. "Geometry,"
meaning "measuring the earth," is the branch of math that has to do with spatial relationships.
In other words, geometry is a type of math used to measure things that are impossible to
measure with devices.
The word geometry Originates from the Greek words (geo meaning world, metri meaning
measure) and means, literally, to measure the earth. It is an ancient branch of mathematics,
but its modern meaning depends largely on context; however, geometry largely encompasses
forms of non-numeric mathematics, such as those involving measurement, area and perimeter
calculation, and work involving angles and position. It was one of the two fields of pre-
modern mathematics, the other being the study of numbers.
In modern times, geometric concepts have been generalized to a high level of abstraction and
complexity, and have been subjected to the methods of calculus and abstract algebra, so that
many modern branches of the field are barely recognizable as the descendants of early
geometry. For example, no one has been able take a tape measure around the earth, yet we are
pretty confident that the circumference of the planet at the equator is 40,075.036 kilometers
(24,901.473 miles) . How do we know that? The first known case of calculating the distance
around the earth was done by Eratosthenes around 240 BCE. What tools do you think current
scientists might use to measure the size of planets? The answer is geometry.
THE GEOMETERS SKETCHPAD
The Geometer's Sketchpad is a popular commercial interactive geometry software program
for exploring Euclidean geometry, algebra, calculus, and other areas ofmathematics. It was
created by Nicholas Jackiw. It is designed to run on Windows 95 orWindows NT 4.0 or later
and Mac OS 8.6 or later (including Mac OS X). It also runs on Linux underWine with few
bugs.
Geometer's Sketchpad includes the traditional Euclidean tools of classical Geometric
constructions; that is, if a figure (such as the pentadecagon) can be constructed with compass
1
http://en.wikipedia.org/wiki/Geometryhttp://imagine.gsfc.nasa.gov/docs/ask_astro/answers/970401c.htmlhttp://imagine.gsfc.nasa.gov/docs/ask_astro/answers/970401c.htmlhttp://en.wikipedia.org/wiki/Eratostheneshttp://dictionary.sensagent.com/Interactive_geometry_software/en-en/http://dictionary.sensagent.com/Euclidean_geometry/en-en/http://dictionary.sensagent.com/Algebra/en-en/http://dictionary.sensagent.com/Calculus/en-en/http://dictionary.sensagent.com/Mathematics/en-en/http://dictionary.sensagent.com/Windows_95/en-en/http://dictionary.sensagent.com/Windows_NT_4.0/en-en/http://dictionary.sensagent.com/Mac_OS_8/en-en/http://dictionary.sensagent.com/Mac_OS_X/en-en/http://dictionary.sensagent.com/Linux/en-en/http://dictionary.sensagent.com/Wine_(software)/en-en/http://dictionary.sensagent.com/Pentadecagon/en-en/http://en.wikipedia.org/wiki/Geometryhttp://imagine.gsfc.nasa.gov/docs/ask_astro/answers/970401c.htmlhttp://imagine.gsfc.nasa.gov/docs/ask_astro/answers/970401c.htmlhttp://en.wikipedia.org/wiki/Eratostheneshttp://dictionary.sensagent.com/Interactive_geometry_software/en-en/http://dictionary.sensagent.com/Euclidean_geometry/en-en/http://dictionary.sensagent.com/Algebra/en-en/http://dictionary.sensagent.com/Calculus/en-en/http://dictionary.sensagent.com/Mathematics/en-en/http://dictionary.sensagent.com/Windows_95/en-en/http://dictionary.sensagent.com/Windows_NT_4.0/en-en/http://dictionary.sensagent.com/Mac_OS_8/en-en/http://dictionary.sensagent.com/Mac_OS_X/en-en/http://dictionary.sensagent.com/Linux/en-en/http://dictionary.sensagent.com/Wine_(software)/en-en/http://dictionary.sensagent.com/Pentadecagon/en-en/ -
7/31/2019 assgmnt geometri
2/24
ESB 4144 MODERN GEOMETRICS
and straight-edge, it can also be constructed using this program. However, the program also
allows users to employ transformations to "cheat," creating figures impossible to construct
under the traditional compass-and-straight-edge rules (such as the regularnonagon). You can
animate objects. Also, you are able to find the midpoint and mid segments of objects.
Geometer's Sketchpad also allows to measure lengths ofsegments, measures
ofangles, area,perimeter, etc. Some of the tools one can use include; construct function,
which allows the user to create objects in relation to selected objects. The transform function
allows the user to create points in relation to objects, which include distance, angle, ratio, and
others. With these tools, one can create numerous different objects, measure them, and
potentially figure out hard-to-solve math problems.
PYTHAGOREAN THEOREM
The Pythagorean Theorem states that, in a right triangle, the square of a (a) plus the square
of b (b) is equal to the square of c (c): a2 + b2 = c2. In a right angled triangle the square of
the long side (the "hypotenuse") is equal to the sum of the squares of the other two sides. It is
stated in this formula: a2 + b2 = c2.
Years ago, a man named Pythagoras found an amazing fact about triangles: If the triangle
had a right angle (90) ...... and you made a square on each of the three sides, then ...... the
biggest square had the exact same area as the other two squares put together! The longest
side of the triangle is called the "hypotenuse". In a right angled triangle the square of the
hypotenuse is equal to the sum of the squares of the other two sides.So, the square of a (a)
plus the square of b (b) is equal to the square of c (c): a2+ b2= c2. The theorem that the
sum of the squares of the lengths of the sides of a right triangle is equal to the square of the
length of the hypotenuse.
2
http://dictionary.sensagent.com/Nonagon/en-en/http://dictionary.sensagent.com/Midpoint/en-en/http://dictionary.sensagent.com/Line_segment/en-en/http://dictionary.sensagent.com/Angle/en-en/http://dictionary.sensagent.com/Area/en-en/http://dictionary.sensagent.com/Perimeter/en-en/http://dictionary.sensagent.com/Nonagon/en-en/http://dictionary.sensagent.com/Midpoint/en-en/http://dictionary.sensagent.com/Line_segment/en-en/http://dictionary.sensagent.com/Angle/en-en/http://dictionary.sensagent.com/Area/en-en/http://dictionary.sensagent.com/Perimeter/en-en/ -
7/31/2019 assgmnt geometri
3/24
-
7/31/2019 assgmnt geometri
4/24
ESB 4144 MODERN GEOMETRICS
HISTORY OF PYTHAGORAS
Pythagoras of Samos is often described as the first pure mathematician. He is an extremely
important figure in the development of mathematics yet we know relatively little about his
mathematical achievements. Unlike many later Greek mathematicians, where at least we have
some of the books which they wrote, we have nothing of Pythagoras's writings. The society
which he led, half religious and half scientific, followed a code of secrecy which certainly
means that today Pythagoras is a mysterious figure. Pythagoras lived in the 500s BC, and
was one of the first Greek mathematical thinkers. Pythagoreans were interested in
Philosophy, especially in Music and Mathematics. The statement of the Theorem was
discovered on a Babylonian tablet circa 1900 1600 B.C. Professor R. Smullyan in his book
5000 B.C. and Other Philosophical Fantasies tells of an experiment he ran in one of his
geometry classes. He drew a right triangle on the board with squares on the hypotenuse and
legs and observed the fact the the square on the hypotenuse had a larger area than either of
the other two squares. Then he asked, Suppose these three squares were made of beaten
gold, and you were offered either the one large square or the two small squares. Which would
you choose? Interestingly enough, about half the class opted for the one large square and
half for the two small squares. Both groups were equally amazed when told that it would
make no difference.
We do have details of Pythagoras's life from early biographies which use important original
sources yet are written by authors who attribute divine powers to him, and whose aim was to
present him as a god-like figure. What we present below is an attempt to collect together the
most reliable sources to reconstruct an account of Pythagoras's life. There is fairly good
agreement on the main events of his life but most of the dates are disputed with different
scholars giving dates which differ by 20 years. Some historians treat all this information as
merely legends but, even if the reader treats it in this way, being such an early record it is of
historical importance.
Pythagoras's father was Mnesarchus, while his mother was Pythais and she was a native of
Samos. Mnesarchus was a merchant who came from Tyre, and there is a story that he brought
corn to Samos at a time of famine and was granted citizenship of Samos as a mark of
gratitude. As a child Pythagoras spent his early years in Samos but travelled widely with his
father. There are accounts of Mnesarchus returning to Tyre with Pythagoras and that he was
4
-
7/31/2019 assgmnt geometri
5/24
ESB 4144 MODERN GEOMETRICS
taught there by the Chaldaeans and the learned men of Syria. It seems that he also visited
Italy with his father.
Little is known of Pythagoras's childhood. All accounts of his physical appearance are likely
to be fictitious except the description of a striking birthmark which Pythagoras had on his
thigh. It is probable that he had two brothers although some sources say that he had three.
Certainly he was well educated, learning to play the lyre, learning poetry and to recite Homer.
There were, among his teachers, three philosophers who were to influence Pythagoras while
he was a young man. One of the most important was Pherekydes who many describe as the
teacher of Pythagoras.
The other two philosophers who were to influence Pythagoras, and to introduce him tomathematical ideas, were Thales and his pupil Anaximanderwho both lived on Miletus. In it
is said that Pythagoras visitedThales in Miletus when he was between 18 and 20 years old.
By this time Thales was an old man and, although he created a strong impression on
Pythagoras, he probably did not teach him a great deal. However he did contribute to
Pythagoras's interest in mathematics and astronomy, and advised him to travel to Egypt to
learn more of these subjects.Thales's pupil, Anaximander, lectured on Miletus and
Pythagoras attended these lectures. Anaximander certainly was interested in geometry
andcosmology and many of his ideas would influence Pythagoras's own views.
In about 535 BC Pythagoras went to Egypt. This happened a few years after the tyrant
Polycrates seized control of the city of Samos. There is some evidence to suggest that
Pythagoras and Polycrates were friendly at first and it is claimed that Pythagoras went to
Egypt with a letter of introduction written by Polycrates. In fact Polycrates had an alliance
with Egypt and there were therefore strong links between Samos and Egypt at this time. The
accounts of Pythagoras's time in Egypt suggest that he visited many of the temples and took
part in many discussions with the priests. According toPorphyryPythagoras was refused
admission to all the temples except the one at Diospolis where he was accepted into the
priesthood after completing the rites necessary for admission.
It is not difficult to relate many of Pythagoras's beliefs, ones he would later impose on the
society that he set up in Italy, to the customs that he came across in Egypt. For example the
secrecy of the Egyptian priests, their refusal to eat beans, their refusal to wear even cloths
made from animal skins, and their striving for purity were all customs that Pythagoras would
5
http://win1%28%27../Glossary/homer',350,200)http://win1%28%27../Glossary/homer',350,200)http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Thales.htmlhttp://win1%28%27../Glossary/anaximander',350,200)http://win1%28%27../Glossary/anaximander',350,200)http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Thales.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Thales.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Thales.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Thales.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Thales.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Thales.htmlhttp://win1%28%27../Glossary/cosmology',350,200)http://win1%28%27../Glossary/cosmology',350,200)http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Porphyry.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Porphyry.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Porphyry.htmlhttp://win1%28%27../Glossary/homer',350,200)http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Thales.htmlhttp://win1%28%27../Glossary/anaximander',350,200)http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Thales.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Thales.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Thales.htmlhttp://win1%28%27../Glossary/cosmology',350,200)http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Porphyry.html -
7/31/2019 assgmnt geometri
6/24
ESB 4144 MODERN GEOMETRICS
later adopt. Porphyry in says that Pythagoras learnt geometry from the Egyptians but it is
likely that he was already acquainted with geometry, certainly after teachings
fromThales and Anaximander.
In 525 BC Cambyses II, the king of Persia, invaded Egypt. Polycrates abandoned his alliance
with Egypt and sent 40 ships to join the Persian fleet against the Egyptians. After Cambyses
had won the Battle of Pelusium in the Nile Delta and had captured Heliopolis and Memphis,
Egyptian resistance collapsed. Pythagoras was taken prisoner and taken to
Babylon. Iamblichus writes that Pythagoras :-
... was transported by the followers of Cambyses as a prisoner of war. Whilst he
was there he gladly associated with the Magoi ... and was instructed in their
sacred rites and learnt about a very mystical worship of the gods. He also
reached the acme of perfection in arithmetic and music and the other
mathematical sciences taught by the Babylonians...
In about 520 BC Pythagoras left Babylon and returned to Samos. Polycrates had been killed
in about 522 BC and Cambyses died in the summer of 522 BC, either by committing suicide
or as the result of an accident. The deaths of these rulers may have been a factor in
Pythagoras's return to Samos but it is nowhere explained how Pythagoras obtained his
freedom. Darius of Persia had taken control of Samos after Polycrates' death and he would
have controlled the island on Pythagoras's return. This conflicts with the accounts
ofPorphyry and Diogenes Laertius who state that Polycrates was still in control of Samos
when Pythagoras returned there.
Pythagoras made a journey to Crete shortly after his return to Samos to study the system of
laws there. Back in Samos he founded a school which was called the semicircle. Iamblichus
writes in the third century AD that:-
... he formed a school in the city [of Samos], the 'semicircle' of Pythagoras,
which is known by that name even today, in which the Samians hold political
meetings. They do this because they think one should discuss questions about
goodness, justice and expediency in this place which was founded by the man
who made all these subjects his business. Outside the city he made a cave the
private site of his own philosophical teaching, spending most of the night and
daytime there and doing research into the uses of mathematics...
6
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Porphyry.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Thales.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Thales.htmlhttp://win1%28%27../Glossary/iamblichus',350,200)http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Porphyry.htmlhttp://win1%28%27../Glossary/diogenes_laertius',350,200)http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Porphyry.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Thales.htmlhttp://win1%28%27../Glossary/iamblichus',350,200)http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Porphyry.htmlhttp://win1%28%27../Glossary/diogenes_laertius',350,200) -
7/31/2019 assgmnt geometri
7/24
ESB 4144 MODERN GEOMETRICS
Pythagoras left Samos and went to southern Italy in about 518 BC (some say much earlier).
Iamblichus gives some reasons for him leaving. First he comments on the Samian response to
his teaching methods:-
... he tried to use his symbolic method of teaching which was similar in all
respects to the lessons he had learnt in Egypt. The Samians were not very keen
on this method and treated him in a rude and improper manner.
This was, according to Iamblichus, used in part as an excuse for Pythagoras to leave Samos:-
... Pythagoras was dragged into all sorts of diplomatic missions by his fellow
citizens and forced to participate in public affairs. ... He knew that all the
philosophers before him had ended their days on foreign soil so he decided to
escape all political responsibility, alleging as his excuse, according to some
sources, the contempt the Samians had for his teaching method.
Pythagoras founded a philosophical and religious school in Croton (now Crotone, on the east
of the heel of southern Italy) that had many followers. Pythagoras was the head of the society
with an inner circle of followers known as mathematikoi. The mathematikoi lived
permanently with the Society, had no personal possessions and were vegetarians. They were
taught by Pythagoras himself and obeyed strict rules. The beliefs that Pythagoras held were:
(1) that at its deepest level, reality is mathematical in nature,
(2) that philosophy can be used for spiritual purification,
(3) that the soul can rise to union with the divine,
(4) that certain symbols have a mystical significance, and
(5) that all brothers of the order should observe strict loyalty and secrecy.
Both men and women were permitted to become members of the Society, in fact several later
women Pythagoreans became famous philosophers. The outer circle of the Society was
known as the akousmatics and they lived in their own houses, only coming to the Society
during the day. They were allowed their own possessions and were not required to be
vegetarians.
Of Pythagoras's actual work nothing is known. His school practised secrecy and
communalism making it hard to distinguish between the work of Pythagoras and that of his
7
-
7/31/2019 assgmnt geometri
8/24
ESB 4144 MODERN GEOMETRICS
followers. Certainly his school made outstanding contributions to mathematics, and it is
possible to be fairly certain about some of Pythagoras's mathematical contributions. First we
should be clear in what sense Pythagoras and the mathematikoi were studying mathematics.
They were not acting as a mathematics research group does in a modern university or other
institution. There were no 'open problems' for them to solve, and they were not in any sense
interested in trying to formulate or solve mathematical problems.
Rather Pythagoras was interested in the principles of mathematics, the concept of number, the
concept of a triangle or other mathematical figure and the abstract idea of a proof. As
Brumbaugh writes in :-
It is hard for us today, familiar as we are with pure mathematical abstractionand with the mental act of generalization, to appreciate the originality of this
Pythagorean contribution.
In fact today we have become so mathematically sophisticated that we fail even to recognise
2 as an abstract quantity. There is a remarkable step from 2 ships + 2 ships = 4 ships, to the
abstract result 2 + 2 = 4, which applies not only to ships but to pens, people, houses etc.
There is another step to see that the abstract notion of 2 is itself a thing, in some sense every
bit as real as a ship or a house.
Pythagoras believed that all relations could be reduced to number relations.
AsAristotle wrote:
The Pythagorean ... having been brought up in the study of mathematics, thought
that things are numbers ... and that the whole cosmos is a scale and a number.
This generalization stemmed from Pythagoras's observations in music, mathematics andastronomy. Pythagoras noticed that vibrating strings produce harmonious tones when the
ratios of the lengths of the strings are whole numbers, and that these ratios could be extended
to other instruments. In fact Pythagoras made remarkable contributions to the mathematical
theory of music. He was a fine musician, playing the lyre, and he used music as a means to
help those who were ill.
8
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Aristotle.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Aristotle.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Aristotle.html -
7/31/2019 assgmnt geometri
9/24
ESB 4144 MODERN GEOMETRICS
Pythagoras studied properties of numbers which would be familiar to mathematicians today,
such as even and odd numbers,triangular numbers,perfect numbers etc. However to
Pythagoras numbers had personalities which we hardly recognize as mathematics today:
Each number had its own personality - masculine or feminine, perfect or
incomplete, beautiful or ugly. This feeling modern mathematics has deliberately
eliminated, but we still find overtones of it in fiction and poetry. Ten was the very
best number: it contained in itself the first four integers - one, two, three, and
four[1+ 2 + 3 + 4 = 10] - and these written in dot notation formed a perfect
triangle.
Of course today we particularly remember Pythagoras for his famous geometry theorem.
Although the theorem, now known as Pythagoras's theorem, was known to the Babylonians
1000 years earlier he may have been the first to prove it. Proclus, the last major Greek
philosopher, who lived around 450 AD wrote:
After[Thales, etc.] Pythagoras transformed the study of geometry into a liberal
education, examining the principles of the science from the beginning and
probing the theorems in an immaterial and intellectual manner: he it was who
discovered the theory ofirrationaland the construction of the cosmic figures.
AgainProclus, writing of geometry, said:-
I emulate the Pythagoreans who even had a conventional phrase to express what
I mean "a figure and a platform, not a figure and a sixpence", by which they
implied that the geometry which is deserving of study is that which, at each new
theorem, sets up a platform to ascend by, and lifts the soul on high instead of
allowing it to go down among the sensible objects and so become subservient to
the common needs of this mortal life.
Heath gives a list of theorems attributed to Pythagoras, or rather more generally to the
Pythagoreans.
1. The sum of the angles of a triangle is equal to two right angles. Also the Pythagoreans
knew the generalization which states that a polygon with n sides has sum of interior
angles 2n - 4 right angles and sum of exterior angles equal to four right angles.
9
http://win1%28%27../Glossary/triangular_number',350,200)http://win1%28%27../Glossary/triangular_number',350,200)http://win1%28%27../Glossary/perfect_number',350,200)http://win1%28%27../Glossary/perfect_number',350,200)http://ref%28%27%20k%20von%20fritz%2C%20biography%20in%20%3Ci%3Edictionary%20of%20scientific%20biography%3C/i%3E%20(New%20York%201970-1990).%20',1)http://ref%28%27%20k%20von%20fritz%2C%20biography%20in%20%3Ci%3Edictionary%20of%20scientific%20biography%3C/i%3E%20(New%20York%201970-1990).%20',1)http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Proclus.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Proclus.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Thales.htmlhttp://win1%28%27../Glossary/irrational',350,200)http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Proclus.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Proclus.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Heath.htmlhttp://win1%28%27../Glossary/triangular_number',350,200)http://win1%28%27../Glossary/perfect_number',350,200)http://ref%28%27%20k%20von%20fritz%2C%20biography%20in%20%3Ci%3Edictionary%20of%20scientific%20biography%3C/i%3E%20(New%20York%201970-1990).%20',1)http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Proclus.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Thales.htmlhttp://win1%28%27../Glossary/irrational',350,200)http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Proclus.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Heath.html -
7/31/2019 assgmnt geometri
10/24
ESB 4144 MODERN GEOMETRICS
2. The theorem of Pythagoras - for a right angled triangle the square on
the hypotenuseis equal to the sum of the squares on the other two sides. We should note
here that to Pythagoras the square on the hypotenuse would certainly not be thought of as
a number multiplied by itself, but rather as a geometrical square constructed on the side.
To say that the sum of two squares is equal to a third square meant that the two squares
could be cut up and reassembled to form a square identical to the third square.
3. Constructing figures of a given area and geometrical algebra. For example they solved
equations such as a (a -x) =x2 by geometrical means.
4. The discovery of irrationals. This is certainly attributed to the Pythagoreans but it
does seem unlikely to have been due to Pythagoras himself. This went against
Pythagoras's philosophy the all things are numbers, since by a number he meant the ratio
of two whole numbers. However, because of his belief that all things are numbers it
would be a natural task to try to prove that the hypotenuse of an isosceles right angled
triangle had a length corresponding to a number.
5. The five regular solids. It is thought that Pythagoras himself knew how to construct
the first three but it is unlikely that he would have known how to construct the other two.
6. In astronomy Pythagoras taught that the Earth was a sphere at the centre of the
Universe. He also recognized that the orbit of the Moon was inclined to the equator of the
Earth and he was one of the first to realize that Venus as an evening star was the same
planet as Venus as a morning star.
Primarily, however, Pythagoras was a philosopher. In addition to his beliefs about numbers,
geometry and astronomy described above, he held:
... the following philosophical and ethical teachings: ... the dependence of the
dynamics of world structure on the interaction of contraries, or pairs of
opposites; the viewing of the soul as a self-moving number experiencing a form
of metempsychosis, or successive reincarnation in different species until its
eventual purification (particularly through the intellectual life of the ethically
rigorous Pythagoreans); and the understanding ...that all existing objects were
fundamentally composed of form and not of material substance. Further
Pythagorean doctrine ... identified the brain as the locus of the soul; and
prescribed certain secret cultic practices.
In their practical ethicsare also described:-
10
http://win1%28%27../Glossary/hypotenuse',350,200)http://win1%28%27../Glossary/hypotenuse',350,200)http://win1%28%27../Glossary/locus',350,200)http://win1%28%27../Glossary/ethics',350,200)http://win1%28%27../Glossary/ethics',350,200)http://win1%28%27../Glossary/hypotenuse',350,200)http://win1%28%27../Glossary/locus',350,200)http://win1%28%27../Glossary/ethics',350,200) -
7/31/2019 assgmnt geometri
11/24
ESB 4144 MODERN GEOMETRICS
In their ethical practices, the Pythagorean were famous for their mutual
friendship, unselfishness, and honesty.
Pythagoras's Society at Croton was not unaffected by political events despite his desire to
stay out of politics. Pythagoras went to Delos in 513 BC to nurse his old teacher Pherekydes
who was dying. He remained there for a few months until the death of his friend and teacher
and then returned to Croton. In 510 BC Croton attacked and defeated its neighbour Sybaris
and there is certainly some suggestions that Pythagoras became involved in the dispute. Then
in around 508 BC the Pythagorean Society at Croton was attacked by Cylon, a noble from
Croton itself. Pythagoras escaped to Metapontium and the most authors say he died there,
some claiming that he committed suicide because of the attack on his Society. Iamblichus in
quotes one version of events:-
Cylon, a Crotoniate and leading citizen by birth, fame and riches, but otherwise
a difficult, violent, disturbing and tyrannically disposed man, eagerly desired to
participate in the Pythagorean way of life. He approached Pythagoras, then an
old man, but was rejected because of the character defects just described. When
this happened Cylon and his friends vowed to make a strong attack on
Pythagoras and his followers. Thus a powerfully aggressive zeal activated Cylon
and his followers to persecute the Pythagoreans to the very last man. Because of
this Pythagoras left for Metapontium and there is said to have ended his days.
This seems accepted by most but Iamblichus himself does not accept this version and argues
that the attack by Cylon was a minor affair and that Pythagoras returned to Croton. Certainly
the Pythagorean Society thrived for many years after this and spread from Croton to many
other Italian cities. Gorman argues that this is a strong reason to believe that Pythagoras
returned to Croton and quotes other evidence such as the widely reported age of Pythagoras
as around 100 at the time of his death and the fact that many sources say that Pythagoras
taught Empedokles to claim that he must have lived well after 480 BC.
The evidence is unclear as to when and where the death of Pythagoras occurred. Certainly the
Pythagorean Society expanded rapidly after 500 BC, became political in nature and also spilt
into a number of factions. In 460 BC the Society :-
... was violently suppressed. Its meeting houses were everywhere sacked and
burned; mention is made in particular of "the house of Milo" in Croton,
11
-
7/31/2019 assgmnt geometri
12/24
ESB 4144 MODERN GEOMETRICS
where 50 or60 Pythagoreans were surprised and slain. Those who survived took
refuge at Thebes and other places
Pythagorean Theorem Theory
A
B
C
The Pythagorean Theorem: The sum of the areas of the two squares on the legs (a and b)
equals the area of the square on the hypotenuse (c).
In mathematics, the Pythagorean Theorem orPythagoras' Theorem is a relation
inEuclidean geometry among the three sides of a right triangle (right-angled triangle). In
terms of areas, it states:
In any right triangle, the area of the square whose side is the hypotenuse(the side opposite
the right angle) is equal to the sum of the areas of the squares whose sides are the two legs
(the two sides that meet at a right angle).
The theorem can be written as an equation relating the lengths of the sides a, b and c, often
called thePythagorean equation: where c represents the length of the hypotenuse,
and a and b represent the lengths of the other two sides.
The Pythagorean Theorem is named after the GreekmathematicianPythagoras, who by
tradition is credited with its discovery andproof, although it is often argued that knowledge
of the theorem predates him. There is evidence that Babylonian mathematicians understood
the formula, although there is little surviving evidence that they used it in a mathematical
framework.
12
http://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Euclidean_geometryhttp://en.wikipedia.org/wiki/Euclidean_geometryhttp://en.wikipedia.org/wiki/Right_trianglehttp://en.wikipedia.org/wiki/Hypotenusehttp://en.wikipedia.org/wiki/Hypotenusehttp://en.wikipedia.org/wiki/Hypotenusehttp://en.wikipedia.org/wiki/Right_anglehttp://en.wikipedia.org/wiki/Theoremhttp://en.wikipedia.org/wiki/Equationhttp://en.wikipedia.org/wiki/Greekshttp://en.wikipedia.org/wiki/Pythagorashttp://en.wikipedia.org/wiki/Pythagorashttp://en.wikipedia.org/wiki/Mathematical_proofhttp://en.wikipedia.org/wiki/Mathematical_proofhttp://en.wikipedia.org/wiki/Babylonian_mathematicshttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Euclidean_geometryhttp://en.wikipedia.org/wiki/Right_trianglehttp://en.wikipedia.org/wiki/Hypotenusehttp://en.wikipedia.org/wiki/Right_anglehttp://en.wikipedia.org/wiki/Theoremhttp://en.wikipedia.org/wiki/Equationhttp://en.wikipedia.org/wiki/Greekshttp://en.wikipedia.org/wiki/Pythagorashttp://en.wikipedia.org/wiki/Mathematical_proofhttp://en.wikipedia.org/wiki/Babylonian_mathematics -
7/31/2019 assgmnt geometri
13/24
ESB 4144 MODERN GEOMETRICS
The theorem has numerousproofs, possibly the most of any mathematical theorem. These are
very diverse, including both geometric proofs and algebraic proofs, with some dating back
thousands of years. The theorem can be generalized in various ways, including higher-
dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles,
and indeed, to objects that are not triangles at all, but n-dimensional solids. The Pythagorean
theorem has attracted interest outside mathematics as a symbol of mathematical abstruseness,
mystique, or intellectual power; popular references in literature, plays, musicals, songs,
stamps and cartoons abound.
As pointed out in the introduction, ifc denotes the lengthof the hypotenuse
and a and b denotes the lengths of the other two sides, the Pythagorean Theorem can be
expressed as the Pythagorean equation:
If the length of both a and b are known, then c can be calculated as follows:
If the length of hypotenuse c and one leg (a orb) are known, then the length of the other leg
can be calculated with the following equations:
or
The Pythagorean equation relates the sides of a right triangle in a simple way, so that if the
lengths of any two sides are known the length of the third side can be found. Another
corollary of the theorem is that in any right triangle, the hypotenuse is greater than any one of
the legs, but less than the sum of them.
A generalization of this theorem is the law of cosines, which allows the computation of the
length of the third side of any triangle, given the lengths of two sides and the size of the angle
between them. If the angle between the sides is a right angle, the law of cosines reduces to the
Pythagorean equation.
13
http://en.wikipedia.org/wiki/Mathematical_proofhttp://en.wikipedia.org/wiki/Lengthhttp://en.wikipedia.org/wiki/Lengthhttp://en.wikipedia.org/wiki/Law_of_cosineshttp://en.wikipedia.org/wiki/Mathematical_proofhttp://en.wikipedia.org/wiki/Lengthhttp://en.wikipedia.org/wiki/Law_of_cosines -
7/31/2019 assgmnt geometri
14/24
ESB 4144 MODERN GEOMETRICS
PROOF
A. Proof using similar triangles
H
A
B C
This proof is based on theproportionality of the sides of two similartriangles, that is, upon
the fact that theratio of any two corresponding sides of similar triangles is the same
regardless of the size of the triangles. LetABCrepresent a right triangle, with the right angle
located at C, as shown on the figure. We draw the altitude from point C, and callHits
intersection with the sideAB. PointHdivides the length of the hypotenuse c into
parts dand e. The new triangleACHis similarto triangleABC, because they both have a
right angle (by definition of the altitude), and they share the angle atA, meaning that the third
angle will be the same in both triangles as well, marked as in the figure. By a similar
reasoning, the triangle CBHis also similar toABC. The proof of similarity of the triangles
requires the Triangle postulate: the sum of the angles in a triangle is two right angles, and is
equivalent to theparallel postulate. Similarity of the triangles leads to the equality of ratios of
corresponding sides:
The first result equates the cosine of each angle and the second result equates thesines.
These ratios can be written as:
Summing these two equalities, we obtain
Which, tidying up, is the Pythagorean Theorem:
14
http://en.wikipedia.org/wiki/Proportionality_(mathematics)http://en.wikipedia.org/wiki/Proportionality_(mathematics)http://en.wikipedia.org/wiki/Similarity_(geometry)http://en.wikipedia.org/wiki/Similarity_(geometry)http://en.wikipedia.org/wiki/Ratiohttp://en.wikipedia.org/wiki/Ratiohttp://en.wikipedia.org/wiki/Altitude_(triangle)http://en.wikipedia.org/wiki/Similarity_(geometry)http://en.wikipedia.org/wiki/Triangle_postulatehttp://en.wikipedia.org/wiki/Parallel_postulatehttp://en.wikipedia.org/wiki/Cosinehttp://en.wikipedia.org/wiki/Sinehttp://en.wikipedia.org/wiki/Sinehttp://en.wikipedia.org/wiki/Proportionality_(mathematics)http://en.wikipedia.org/wiki/Similarity_(geometry)http://en.wikipedia.org/wiki/Ratiohttp://en.wikipedia.org/wiki/Altitude_(triangle)http://en.wikipedia.org/wiki/Similarity_(geometry)http://en.wikipedia.org/wiki/Triangle_postulatehttp://en.wikipedia.org/wiki/Parallel_postulatehttp://en.wikipedia.org/wiki/Cosinehttp://en.wikipedia.org/wiki/Sine -
7/31/2019 assgmnt geometri
15/24
ESB 4144 MODERN GEOMETRICS
The role of this proof in history is the subject of much speculation. The underlying question
is why Euclid did not use this proof, but invented another. One conjecture is that the proof by
similar triangles involved a theory of proportions, a topic not discussed until later in
theElements, and that the theory of proportions needed further development at that time.
B. Euclids Proof
B C
A
In outline, here is how the proof inEuclid'sElements proceeds. The large square is divided
into a left and right rectangle. A triangle is constructed that has half the area of the left
rectangle. Then another triangle is constructed that has half the area of the square on the left-
most side. These two triangles are shown to be congruent, proving this square has the same
area as the left rectangle. This argument is followed by a similar version for the right
rectangle and the remaining square. Putting the two rectangles together to reform the square
on the hypotenuse, its area is the same as the sum of the area of the other two squares. The
details are next.
LetA,B, Cbe the vertices of a right triangle, with a right angle atA. Drop a perpendicular
fromA to the side opposite the hypotenuse in the square on the hypotenuse. That line divides
the square on the hypotenuse into two rectangles, each having the same area as one of the two
squares on the legs.
15
http://en.wikipedia.org/wiki/Euclidhttp://en.wikipedia.org/wiki/Euclidhttp://en.wikipedia.org/wiki/Euclid's_Elementshttp://en.wikipedia.org/wiki/Euclid's_Elementshttp://en.wikipedia.org/wiki/Vertex_(geometry)http://en.wikipedia.org/wiki/Euclidhttp://en.wikipedia.org/wiki/Euclid's_Elementshttp://en.wikipedia.org/wiki/Vertex_(geometry) -
7/31/2019 assgmnt geometri
16/24
ESB 4144 MODERN GEOMETRICS
For the formal proof, we require four elementarylemmata:
1. If two triangles have two sides of the one equal to two sides of the other, each to each,
and the angles included by those sides equal, then the triangles are congruent (side-
angle-side).
2. The area of a triangle is half the area of any parallelogram on the same base and
having the same altitude.
3. The area of a rectangle is equal to the product of two adjacent sides.
4. The area of a square is equal to the product of two of its sides (follows from 3).
Next, each top square is related to a triangle congruent with another triangle related in turn to
one of two rectangles making up the lower square.
K
G
F
D E
I
H
A
CB
L
16
http://en.wikipedia.org/wiki/Lemma_(mathematics)http://en.wikipedia.org/wiki/Lemma_(mathematics)http://en.wikipedia.org/wiki/Side_angle_sidehttp://en.wikipedia.org/wiki/Side_angle_sidehttp://en.wikipedia.org/wiki/Lemma_(mathematics)http://en.wikipedia.org/wiki/Side_angle_sidehttp://en.wikipedia.org/wiki/Side_angle_side -
7/31/2019 assgmnt geometri
17/24
ESB 4144 MODERN GEOMETRICS
K
BC
A
D
F
G
L
The proof is as follows:
1. Let ACB be a right-angled triangle with right angle CAB.
2. On each of the sides BC, AB, and CA, squares are drawn, CBDE, BAGF, and ACIH, in
that order. The construction of squares requires the immediately preceding theorems in
Euclid, and depends upon the parallel postulate
3. From A, draw a line parallel to BD and CE. It will perpendicularly intersect BC and DE
at K and L, respectively.
4. Join CF and AD, to form the triangles BCF and BDA.
5. Angles CAB and BAG are both right angles; therefore C, A, and G are collinear.
Similarly for B, A, and H.
6. Angles CBD and FBA are both right angles; therefore angle ABD equals angle FBC,
since both are the sum of a right angle and angle ABC.
7. Since AB is equal to FB and BD is equal to BC, triangle ABD must be congruent to
triangle FBC.
8. Since A-K-L is a straight line, parallel to BD, then parallelogram BDLK has twice the
area of triangle ABD because they share the base BD and have the same altitude BK,
i.e., a line normal to their common base, connecting the parallel lines BD and AL.
(lemma 2)
9. Since C is collinear with A and G, square BAGF must be twice in area to triangle FBC.
10. Therefore rectangle BDLK must have the same area as square BAGF = AB 2.
17
http://en.wikipedia.org/wiki/Line_(geometry)#Collinear_pointshttp://en.wikipedia.org/wiki/Line_(geometry)#Collinear_pointshttp://en.wikipedia.org/wiki/Line_(geometry)#Collinear_points -
7/31/2019 assgmnt geometri
18/24
ESB 4144 MODERN GEOMETRICS
11. Similarly, it can be shown that rectangle CKLE must have the same area as square
ACIH = AC2.
12. Adding these two results, AB2 + AC2 = BD BK + KL KC
13. Since BD = KL, BD BK + KL KC = BD(BK + KC) = BD BC14. Therefore AB2 + AC2 = BC2, since CBDE is a square.
This proof, which appears in Euclid'sElements as that of Proposition 47 in Book 1,
demonstrates that the area of the square on the hypotenuse is the sum of the areas of the other
two squares. This is quite distinct from the proof by similarity of triangles, which is
conjectured to be the proof that Pythagoras used.
Algebraic proofs
The theorem can be proved algebraically using four copies of a right triangle with
sides a, b and c, arranged inside a square with side c as in the top half of the diagram.[16] The
18
http://en.wikipedia.org/wiki/Pythagorean_theorem#cite_note-rotate-15http://en.wikipedia.org/wiki/File:Pythagoras_algebraic2.svghttp://en.wikipedia.org/wiki/Pythagorean_theorem#cite_note-rotate-15 -
7/31/2019 assgmnt geometri
19/24
ESB 4144 MODERN GEOMETRICS
triangles are similar with area , while the small square has side b a and area (b a)2.
The area of the large square is therefore
But this is a square with side c and area c2, so
A similar proof uses four copies of the same triangle arranged symmetrically around a square
with side c, as shown in the lower part of the diagram. This results in a larger square, with
side a + b and area (a + b)2. The four triangles and the square side c must have the same area
as the larger square,
Giving,
A related proof was published by former U.S. President James A. Garfield. Instead of a
square it uses a trapezoid, which can be constructed from the square in the second of the
above proofs by bisecting along a diagonal of the inner square, to give the trapezoid as shown
in the diagram. The area of the trapezoid can be calculated to be half the area of the square,
that is
The inner square is similarly halved, and there are only two triangles so the proof proceeds as
above except for a factor of which is removed by multiplying by two to give theresult.
Proof using differentials
One can arrive at the Pythagorean theorem by studying how changes in a side produce a
change in the hypotenuse and employing calculus.
19
http://en.wikipedia.org/wiki/James_A._Garfieldhttp://en.wikipedia.org/wiki/Trapezoidhttp://en.wikipedia.org/wiki/Trapezoid#Areahttp://en.wikipedia.org/wiki/Calculushttp://en.wikipedia.org/wiki/James_A._Garfieldhttp://en.wikipedia.org/wiki/Trapezoidhttp://en.wikipedia.org/wiki/Trapezoid#Areahttp://en.wikipedia.org/wiki/Calculus -
7/31/2019 assgmnt geometri
20/24
ESB 4144 MODERN GEOMETRICS
The triangleABCis a right triangle, as shown in the upper part of the diagram, withBCthe
hypotenuse. At the same time the triangle lengths are measured as shown, with the
hypotenuse of lengthy, the sideACof lengthx and the sideAB of length a, as seen in the
lower diagram part.
Ifx is increased by a small amount dx by extending the sideACslightly toD, theny also
increases by dy. These form two sides of a triangle, CDE, which (withEchosen so CEis
perpendicular to the hypotenuse) is a right triangle approximately similar toABC. Therefore
the ratios of their sides must be the same, that is:
This can be rewritten as follows:
This is a differential equation which is solved to give
And the constant can be deduced fromx = 0,y = a to give the equation
This is more of an intuitive proof than a formal one: it can be made more rigorous if proper
limits are used in place ofdx and dy.
20
http://en.wikipedia.org/wiki/Differential_equationhttp://en.wikipedia.org/wiki/File:Pythag_differential_proof.svghttp://en.wikipedia.org/wiki/Differential_equation -
7/31/2019 assgmnt geometri
21/24
ESB 4144 MODERN GEOMETRICS
APPLICATION
The Pythagorean theorem has far-reaching ramification in other fields (such as the arts), as
well as practical application. The theorem is invaluable when computing distances between
two points, such as in navigation and land surveying. Another important application is in the
design of ramps. Ramp designs for handicap-accessible sites and for skateboard parks are
very much in demand.
The most widely quoted "practical" application of the Pythagorean theorem is actually anapplication of its converse. The theorem of Pythagoras says that if a triangle has sides of
length a, b and c and the angle between the sides of length a and b is a right angle, then a^2 +
b^2 = c^2. The converse says that if a triangle has sides of length a, b and c and a^2 + b^2 =
c^2 then the angle between the sides of length a and b is a right angle. Such a triple of
numbers is called a Pythagorean triple, so 3,4,5 is a Pythagorean triple and so are 6, 8, 10 and
5, 12, 13.
The application is in construction. It is very important when starting a building to have a
square corner, and a Pythagorean triple provides an easy and inexpensive way to get one.
Drive a stake at the desired corner point and another stake 3 meters from the corner along the
line where you want one wall of the building. Then position a third stake so that its distance
from the corner is 4 meters and the third side of the triangle formed by the three stakes is 5
meters. Since 3, 4, 5 is a Pythagorean triple the angle at the corner is a right angle.
The Pythagorean theorem is a starting place for trigonometry, which leads to methods, forexample, for calculating the heights of mountains. The Pythagorean theorem is also an
21
-
7/31/2019 assgmnt geometri
22/24
ESB 4144 MODERN GEOMETRICS
example of the somewhat rare situation where both the theorem and its converse are true. It is
also useful in calculating distances.
EASE AND PROBLEM OF USE GEOMETRY SKETCHPAD
Geometer's Sketchpad is a wonderful mathematical program that combines the use of
technology with geometry. This program has several benefits to its use. First of all, students
are able to explore and learn on their own the meaning of several important definitions in
geometry. Students, by simply clicking on the mouse, learn the many parts that make up
geometry. The students are learning for them, and therefore are likely to retain far much
more information than they would in a traditional classroom. In addition to learning
definitions, all of the student's work is "saved." Therefore, students can compare differenttrails with one another, instead of simply loosing the information. Also, because information
is preserved, students have the benefit to examine and compare several similar cases in
seconds, rather than hours it might take to draw the figures accurately. This, of course, will
lead to generalizations and patterns far quicker than in a traditional geometric classroom.
Frequent problems experienced in a traditional geometric classroom are often obsolete with
the use of Geometer's Sketchpad. Students will be learning at a quicker pace and therefore
will able to accomplish more. Instead of "leaving things out," which happens far too often in
traditional classrooms, students are encouraged to learn as much as possible. Also, students
who might often find it difficult to stay on task and understand what is being discussed will
have the option of the "help menu." This tool is used by so many people today. Again, with
this tool those students can also move beyond the Euclidean plane (2 D) into a far more
complex geometrical world. The program also contains polar coordinate capabilities, often
discussed in calculus classes. In fact, Geometer's Sketchpad is not only for math, it has been
used in Art, Science, and a variety of other subjects across the curriculum. The program's
versatility is amazing. We truly believe it is one of the best innovations for the mathematical
world.
Another great benefit to this program is its ability to graph data collected while using it. As
anyone whose studied geometry in detail knows, there is quite a bit of trial and error before
the correct answer is reached. By using Geometer's Sketchpad, students have the benefit of
recalling information collected by simply clicking a button instead of fumbling through heaps
of papers. These graphs could also become a way of completing assignments because they
22
-
7/31/2019 assgmnt geometri
23/24
ESB 4144 MODERN GEOMETRICS
can be labeled, include captions, and perform many of the other techniques used in other
computer based programs like excel.
The lists of the possibilities of how this program could be used in a classroom are endless.
Teachers and students can work together through the use of an overhead, having different
members of the class come up and "try" certain things. Or, if computers are available in the
classroom, students could simply be given a problem and asked to solve. If that is not
possible, this program would be an excellent source for an enrichment activity to those
students who find it interesting.
With Geometer's Sketchpad, students are able to take their learning into their own hands.
Students will not only be forced to understand the material in order to work the program, butthey will most likely discover new things to them, they will want to go beyond what is
expected of them. The Key Curriculum Press feels that this program could be the program
that would help to make "award-winning mathematical discoveries.
With all advantages, there are disadvantages as well. Geometer's Sketchpad is a very
complicated tool to master. There are many steps that need to be followed, as well as, an
understanding of what all the "buttons" do and why they do it.
Beside that the disadvantage of geometers sketchpad is we need to be familiar and
comfortable with the use of the computer per secondly familiar with whatever software that is
supposed to be the treatment for the Pythagoras theorem.
We also have the opportunity to explore the software, the subject will be overly anxious into
wanting to concentrate on too many new things at the same time.
We need to focus and concentrate is supposed to be learned. We also need to focus and
concentrate on the content that is supposed to be learned. The geometers sketchpads need
using the computer software how to use it. It means the students must be having the computer
while using the sketchpads.
23
-
7/31/2019 assgmnt geometri
24/24
ESB 4144 MODERN GEOMETRICS