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    Science in Ancient Greece!

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    Science in Ancient Greece! Empty Science in Ancient Greece!

    Post by Hikikomori Mon Jun 15, 2009 2:35 pm

    Hello chobots Smile

    Another interesting topic for you.
    Let's find out about the Science in Ancient Greece study
    The prize is... V-Flag!!!
    We'll prefer answers with your drawings Wink
    Be creative!!!
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    Science in Ancient Greece! Empty About Jellyfish

    Post by dude144 Mon Jun 15, 2009 3:17 pm

    Jellyfish is present and found in every ocean in the world. The lifetime of a jellyfish is at the most three or six months.They have two body forms through their life cycle - the polyp stage and the medusa stage. In the polyp stage, they are in the form of a sessile stalk with their mouth and tentacle facing upwards. In this stage, they catch passing food. The second stage of the jellyfish' body structure is more popular. During this stage, they have an umbrella shaped body called the bell.
    Science in Ancient Greece! Jallyf13


    Last edited by dude144 on Mon Jun 15, 2009 3:32 pm; edited 1 time in total
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    Post by rickyglam Mon Jun 15, 2009 3:18 pm

    Public water work were one of the greatest influences in the ancient greece they bring clean water there. Oh and almost forgot my drawing
    Science in Ancient Greece! Water10
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    Post by bube077 Mon Jun 15, 2009 4:18 pm

    The Greeks were very interested in science as a way of organizing the world and making order out of chaos, and having power over some very powerful things like oceans and weather. From about 600 BC, a lot of Greek men spent time observing the planets and the sun and trying to figure out how astronomy worked. They must have gotten their first lessons from the Babylonians, who were very good at astronomy and also very interested in it.
    By the 400's BC, Pythagoras was interested in finding the patterns and rules in mathematics and music, and invented the idea of a mathematical proof. Although Greek women usually were not allowed to study science, Pythagoras did have some women among his students. Socrates, a little bit later, developed logical methods for deciding whether something was true or not.
    In the 300's BC, Aristotle and other philosophers at the Lyceum and the Academy in Athens worked on observing plants and animals, and organizing the different kinds of plants and animals into types. Again, this is a way of creating order out of chaos.

    After Aristotle, using his ideas and also ideas from Egypt and the Persians and Indians, Hippocrates and other Greek doctors wrote important medical texts that were used for hundreds of years.




    https://2img.net/h/oi42.tinypic.com/28lv37p.jpg

    i dint know how to put a picture i draw Sad
    its about discovering planets!

    by:bube07
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    Science in Ancient Greece! Empty ALL MY INFORMATION ON A DIFFERENT PAGE

    Post by Mimja Mon Jun 15, 2009 4:44 pm

    ALL MY INFORMATION IS ON THE INVENTIONS THAT INFLUENCED HISTORY PAGE
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    Post by patobranco1 Mon Jun 15, 2009 5:12 pm

    Science in Ancient Greece

    Thales of Miletus is regarded by many as the father of science; he was the first Greek philosopher to seek to explain the physical world in terms of natural rather than supernatural causes.

    Science in Ancient Greece was based on logical thinking and mathematics. It was also based on technology and everyday life. The arts in Ancient Greece were sculptors and painters. The Greeks wanted to know more about the world, the heavens and themselves. People studied about the sky, sun, moon, and the planets. The Greeks found that the earth was round. sunny

    Eratosthenes of Alexandria, who died about 194 BC, wrote on astronomy and geography, but his work is known mainly from later summaries. He is credited with being the first person to measure the Earth's circumference.
    alien

    Science in Ancient Greece! Greece11
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    Science in Ancient Greece! Empty Archimedes Screw!

    Post by XxMcho04xX Mon Jun 15, 2009 5:40 pm

    Hello there! Whos Archimedes and why does he pertain to Ancient Greece you may be wondering? Well enjoy the pictures and little snippets of his work. Aside from Archimedes screw, Archimedes also has the Archimedes principal! Have you learned that in math yet? I did Very Happy Well it's time to learn! I know I did researching these topics Very Happy

    Science in Ancient Greece! Archim10

    ~Mcho04~
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    Post by turtle11 Mon Jun 15, 2009 5:45 pm

    Science in Ancient Greece was based on logical thinking, mathematics, technology, and everyday life. The Greeks wanted to know more about the world. They studied about the sky, sun, moon, and the planets. The Greeks found that the Earth was round. A person from Greece made the first accurate measurement of the Earth's diameter. The Ancient Greek philosophers were amazed by volcanoes and earthquakes. Another guy speculated that earthquakes resulted from winds within the Earth caused by the Earth's own heat and heat from the sun. Volcanoes, he thought, marked the points at which these winds finally escaped from inside the Earth into the atmosphere.

    https://i.servimg.com/u/f61/14/01/35/54/ancien12.jpg

    -turtle11
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    Post by Bakugon Mon Jun 15, 2009 5:59 pm

    The Greeks were very interested in science as a way of organizing the world and making order out of chaos, and having power over some very powerful things like oceans and weather. From about 600 BC, a lot of Greek men spent time observing the planets and the sun and trying to figure out how astronomy worked. They must have gotten their first lessons from the Babylonians, who were very good at astronomy and also very interested in it.
    By the 400's BC, Pythagoras was interested in finding the patterns and rules in mathematics and music, and invented the idea of a mathematical proof. Although Greek women usually were not allowed to study science, Pythagoras did have some women among his students. Socrates, a little bit later, developed logical methods for deciding whether something was true or not.


    In the 300's BC, Aristotle and other philosophers at the Lyceum and the Academy in Athens worked on observing plants and animals, and organizing the different kinds of plants and animals into types. Again, this is a way of creating order out of chaos.

    After Aristotle, using his ideas and also ideas from Egypt and the Persians and Indians, Hippocrates and other Greek doctors wrote important medical texts that were used for hundreds of years.


    Last edited by Bakugon on Mon Jun 15, 2009 6:06 pm; edited 1 time in total (Reason for editing : Messed up)
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    Post by Kool_Man Mon Jun 15, 2009 6:53 pm

    Although the Greeks were the first Europeans to consider questions of astronomy, mathematics, physics and biology, it was not until the time of Aristotle that they recognised science as a discipline distinct from philosophy. Nevertheless, they made some astounding discoveries and their names live on.
    Pythagoras (570-500BC) not only pioneered the study of mathematics in the western world, but was also a reputed miracle worker. His famous theorem for calculating the length of the hypotenuse of a right-angled triangle is well known. Less well known is his mystical theory of the transmigration of souls.
    Hippocrates (460-390BC), a physician and medical writer, is the father of modern medicine. He established a renowned school of medicine on the island of Cos, where students learned to diagnose illness through observation rather than theory. It was from this school that the first version of the Hippocratic oath derived.
    Archimedes (287-211BC) is most famous for running through the streets shouting “Eureka!” when he discovered the principle of specific gravity while sitting in his bath. But we can also credit him with the invention of the Archimedean screw – a device still used to draw water upwards – and many important theories of geometry.
    Euclid (Greek: Εὐκλείδης — Eukleídēs), fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician and is often referred to as the "Father of Geometry." He was active in Hellenistic Alexandria during the reign of Ptolemy I (323–283 BC). His Elements is the most successful textbook and one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics (especially geometry) from the time of its publication until the late 19th or early 20th century.[1][2][3] In it, the principles of what is now called Euclidean geometry were deduced from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory and rigor.
    Masters of Science, Ancient Greece made Science!


    Last edited by Pinkgirlcpst on Tue Jun 16, 2009 5:12 am; edited 1 time in total
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    Post by mondobob Mon Jun 15, 2009 7:17 pm

    in our science class, we are learning (yes we are wtill in school for 7 more days) how mass ÷ vlume = density. thats how archimedes solved the crown problem. he was too see if the crown was pure gold or added silver WITHOUT harming the crown. He was hoping to calculate the density by melting the gold/silver and using a much older version of the graduated cylender, but that smashed his hopes. I am sure you have heard of the bath tale with him, and that is true. he discovered that when he stepped in the yub, the water rose (an old man playing in a bath tub lol! ) he then discovered that objects of different density diplaces more water than others, like gold is more dense than silver. he used this to discover then density. "he then ran in the streets unclothed (thats a better name) shouting eureka! meaning i found it!" i got that quote form bill-nye the science guy, lol.

    I will show you how to do the displacement techniche tomorrow using pics
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    Science in Ancient Greece! Empty _blockhead_ 's Submition : Hades

    Post by _blockhead_ Mon Jun 15, 2009 8:38 pm

    Hades Is A mytholgical Greece God Of The UnderWorld. He Was Said To be An Evil God Who Ruled Over The Dead, He Was Very Greedy And Became Very Wealthy From The Rocks Mined From Inside The Earth. And His Wife Was Persephone The Queen Of The Underworld.
    Science in Ancient Greece! Hades11

    https://i.servimg.com/u/f89/13/89/84/01/hades10.png
    BTW i didnt Copy Any Of The Details Here I just Googled The Names So i Could spell Them Correct. I learned All This In Reading Smile
    And I Drew The Pic On Chobots


    Last edited by _blockhead_ on Mon Jun 15, 2009 8:41 pm; edited 1 time in total (Reason for editing : Adding text)
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    Post by Gmailchatter Mon Jun 15, 2009 10:31 pm

    My topic thingy is on Greek Geometry.

    Greeks gave us one more topic to study about in math. They had already given us tons of subjects, but they felt that we needed to learn 1 more and thats geometry.Geometry comes from the greek words geo(meaning earth) and metron(meaning measure)Geometry was important to Greeks because it helped them with surveying, astronomy, navigation, and building. It was written in Ancient Greece by Euclid, Pythagoras, Thales, Plato and Aristotle just to mention a few. Geometry is actually known as Euclidean geometry. Geometry is the study of angles, shapes, perimeter, area, and volume. It's not an easy topic, ecpecially with vocabulary. There's line, point, line segment, ray, angle,including (acute, obtuse, and right) and plane. The shapes include circles, squares, triangles, rectangles, and any other shape you can think of! Geometry has differnt formula's too. Like width times lenght = perimeter. Enjoy my facts and hope u learned somehting!And here r some shapes for the pic. Wink
    Science in Ancient Greece! 2ytzdid
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    Post by Rashood Mon Jun 15, 2009 11:35 pm

    The Greeks were very interested in science as a way of organizing the world and making order out of chaos, and having power over some very powerful things like oceans and weather. From about 600 BC, a lot of Greek men spent time observing the planets and the sun and trying to figure out how astronomy worked. They must have gotten their first lessons from the Babylonians, who were very good at astronomy and also very interested in it.
    By the 400's BC, Pythagoras was interested in finding the patterns and rules in mathematics and music, and invented the idea of a mathematical proof. Although Greek women usually were not allowed to study science, Pythagoras did have some women among his students. Socrates, a little bit later, developed logical methods for deciding whether something was true or not.

    In the 300's BC, Aristotle and other philosophers at the Lyceum and the Academy in Athens worked on observing plants and animals, and organizing the different kinds of plants and animals into types. Again, this is a way of creating order out of chaos.

    After Aristotle, using his ideas and also ideas from Egypt and the Persians and Indians, Hippocrates and other Greek doctors wrote important medical texts that were used for hundreds of years.
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    Post by Rashood Mon Jun 15, 2009 11:39 pm

    The weather on Earth happens because of the evaporation and condensation of water, and because of the spinning of the earth and the tilt of the earth on its axis. Does it seem strange that such quiet things could cause big thunderstorms and huge waves and blizzards? Here's why.

    Because the Earth is tilted on its axis, the northern and southern parts of the Earth are sometimes closer to the Sun and sometimes farther away. This causes the seasons - spring, summer, winter, and fall. In the winter, if you are far enough north or south, it gets cold enough for rain to fall as snow. In the summer, if you are close to the equator, it gets warm enough to warm up the ocean and cause hurricanes and typhoons.

    Because the Sun's rays hit the Equator [View map] straight on, and only hit the North and South Poles at a slant, the Earth was always colder up near the North Pole [View map] and down near the South Pole [View map] , and it was always warmer near the Equator, just as it is today. Near the Equator, the hot air rises, because hot air is lighter than cold air. As the hot air rises, there's an empty area near the surface of the land or ocean. So cold air from the North and South Poles flows to the Equator to fill in that empty area. This movement of air is what we call "wind".

    Because the Earth is spinning at the same time, the air doesn't flow straight north or south, but twists to the east or west. Air that is going towards the Equator twists to the west. Sailors call these winds the Trade Winds, and they're the winds that Columbus used to sail to North America.

    Further north, or further south, the winds blow mainly to the east, instead: sailors call these winds the "westerlies", because they come from the west. Sailors used these westerlies to sail back to Europe [View map] from North America [View map] , or from Europe to Australia [View map] and New Zealand [View map] . Then at the North Pole and the South Pole, the winds blow mainly to the west again, and sailors call them "easterlies."

    To find out more about weather, check out these books from Amazon.com or from your library:
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    Post by Rashood Mon Jun 15, 2009 11:43 pm

    The first molecules of water formed out in space, about 14 billion years ago, as part of the nebula left over after an early star exploded in a supernova. There were a lot of hydrogen and oxygen atoms floating around in these nebulae, and when they stuck together that made water molecules. Because hydrogen and oxygen were both very common atoms, they made a lot of water molecules.

    Hydrogen and oxygen atoms stuck together because hydrogen atoms were able to share their electrons with oxygen atoms, and oxygen atoms wanted two more electrons than they had. We call this a covalent bond. When two hydrogen atoms shared their electrons with an oxygen atom, it made a molecule that was stronger than any of the atoms were on their own, so most of the oxygen in space is formed into water molecules.

    There is still plenty of water all over space and other planets. There's water in nebulae today, and in comets, and on the Moon, and on Mercury, Mars, and Neptune, for instance. But the water in all of these places is frozen into ice. Earth is the only place we know of that has liquid water ( possibly Mars does too).

    We don't know how water got to Earth, or whether it was here from the beginning when the planet first formed, but there has been liquid water on Earth for about four billion years. Water's an important part of heating and cooling the Earth, and also water is where the first living cells got started. All plants and animals need water to survive.

    When water gets too cold, or the pressure gets too high, it turns into ice. When water gets too hot, or the pressure gets too low, it turns into steam.
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    Post by nuwfall9 Tue Jun 16, 2009 12:49 am

    there was a science in aceint greece he was thinking how the earth created people before did not take him for real then they did they called it the big ban theroy

    if i win my name is nuwfall9
    -nuwfall9
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    Post by nuwfall9 Tue Jun 16, 2009 1:20 am

    i mean the big bang theroy
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    Post by nuwfall9 Tue Jun 16, 2009 1:26 am

    it was made by little thing coming thogether then a big explogen in the sapce in years that made some thing made of dust that was gravetey then a huge fire ball it was the sun then small thing that we are all made off then the earth the erath took years to take it shape as now then animals started to come like now the first was a dinsore and did you know a crocadile is a dinosore he lived when the dinsores where there and they are still a life.
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    Science in Ancient Greece! Empty Botany

    Post by Bubu1028 Tue Jun 16, 2009 1:59 am

    my name is bubu1028 and im gonna talk about Botany Exclamation Exclamation
    Greek influence on agriculture was the establishment of the science of botany. Botany is the study of all aspects of plant life, including where plants live and how they grow. The Greek philosopher Aristotle, who lived during the 300's BC, collected information about most of the plants known at that time in the world. He also studied other sciences and math.
    [img]Science in Ancient Greece! Th_bot[/img]
    bounce His student Theophrastus classified and named these plants. Theophrastus often called the father of botany. Aristotle and Theophrastus developed an extremely important type of science that is studied all over the world. Botany is so important because all the food that animals and people eat comes from plants, whether it be directly or indirectly.
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    Post by kittysnow Tue Jun 16, 2009 4:00 am


    • The Greeks did have inventions, but they were stronger in the world of ideas, and we owe much to them for this. They developed geometry and studied astronomy, geography, and mechanics. These studies formed the basis of much science that followed. Their philosophers developed speculative philosophy which is the foundation of much of our speculation and a good portion of our Mathematics. Their art and architecture were very influential and set styles that are still popular and highly copied today. Museums around the world have much material from ancient Greece which is often the most valuable part of their collection. A list of inventions follow:

      analog computer with clockwork mechanism
      cyclorama
      camera obscura
      steam driven jet engine
      Archimedian Screw
      astrolabe
      catapult
      hydraulic music organ
      coinage
      parchment
      trireme
      Ancient Greek Scientists:

      Alcmaeon of Croton
      Anaxagorus of Clazomenae
      Anaximander of Miletus
      Apollnius of Perga
      Archimedes of Syracuse
      Archytas of Tarentum
      Aristarchus of Samos
      Aristotle
      Callipus of Cyzicus
      Chalcidius
      Ctesibius of Alexandria




    Science in Ancient Greece! Chobot10


    Pythagorean Theorem or Pythagoras Theorem
    In mathematics, the Pythagorean theorem (American English) or Pythagoras' theorem (British English) is a relation in Euclidean geometry among the three sides of a right triangle (right-angled triangle in British English). It states:
    <BLOCKQUOTE>
    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).</BLOCKQUOTE>
    The theorem can be written as an equation:

    Science in Ancient Greece! 1455314a78f39a594485adbaf74d63f9
    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 Greek mathematician Pythagoras, who by tradition is credited with its discovery and proof, although it is often argued that knowledge of the theory predates him. (There is much evidence that Babylonian mathematicians understood the principle, if not the mathematical significance).

    If we let c be the length of the hypotenuse and a and b be the lengths of the other two sides, the theorem can be expressed as the equation:

    Science in Ancient Greece! 3ae71ab3eb71d3d182a3b9e437fba6ee
    or, solved for c:

    Science in Ancient Greece! 49a65217b7d35663efc6e558c0ffdba0
    If c is already given, and the length of one of the legs must be found, the following equations (which are corollaries of the first) can be used:

    Science in Ancient Greece! E04b8e3c08826db3752edb899f769796
    or

    Science in Ancient Greece! 872d95d9a5117061efdb25ac5c27587d
    This equation provides a simple relation among the three sides of a right triangle so that if the lengths of any two sides are known, the length of the third side can be found. 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 it reduces to the Pythagorean theorem.

    Proof using similar triangles

    Science in Ancient Greece! 180px-Proof-Pythagorean-Theorem.svg Science in Ancient Greece! Magnify-clip
    Proof using similar triangles



    Like most of the proofs of the Pythagorean theorem, this one is based on the proportionality of the sides of two similar triangles.
    Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. We draw the altitude from point C, and call H its intersection with the side AB. The new triangle ACH is similar to our triangle ABC, because they both have a right angle (by definition of the altitude), and they share the angle at A, meaning that the third angle will be the same in both triangles as well. By a similar reasoning, the triangle CBH is also similar to ABC. The similarities lead to the two ratios:

    Science in Ancient Greece! D572f81f304f85c73166417faae80f83
    These can be written as

    Science in Ancient Greece! 29b9421e06225c21345af7eec88891c8
    Summing these two equalities, we obtain

    Science in Ancient Greece! 70922f59b11b561621c245e11be0b61b
    In other words, the Pythagorean theorem:

    Science in Ancient Greece! 00922fdfae8fc68c7adc754af245d9d3

    Proof using similar triangles


    Like most of the proofs of the Pythagorean theorem, this one is based on the proportionality of the sides of two similar triangles.


    Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. We draw the altitude from point C, and call H its intersection with the side AB. The new triangle ACH is similar to our triangle ABC, because they both have a right angle (by definition of the altitude), and they share the angle at A, meaning that the third angle will be the same in both triangles as well. By a similar reasoning, the triangle CBH is also similar to ABC. The similarities lead to the two ratios:



    Science in Ancient Greece! D572f81f304f85c73166417faae80f83
    These can be written as



    Science in Ancient Greece! 29b9421e06225c21345af7eec88891c8
    Summing these two equalities, we obtain



    Science in Ancient Greece! 70922f59b11b561621c245e11be0b61b
    In other words, the Pythagorean theorem:



    Science in Ancient Greece! 00922fdfae8fc68c7adc754af245d9d3
    Science in Ancient Greece! 180px-Proof-Pythagorean-Theorem.svg



    Euclid's proof
    In Euclid's Elements, Proposition 47 of Book 1, the Pythagorean theorem is proved by an argument along the following lines. Let A, B, C be the vertices of a right triangle, with a right angle at A. Drop a perpendicular from A 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.
    For the formal proof, we require four elementary lemmata:

    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 Theorem)
    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 any square is equal to the product of two of its sides.
    4. The area of any rectangle is equal to the product of two adjacent sides (follows from Lemma 3).

    The intuitive idea behind this proof, which can make it easier to follow, is that the top squares are morphed into parallelograms with the same size, then turned and morphed into the left and right rectangles in the lower square, again at constant area.
    Science in Ancient Greece! 180px-Illustration_to_Euclid%27s_proof_of_the_Pythagorean_theorem.svg
    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.
    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 and BD are equal to FB and BC, respectively, triangle ABD must be equal to triangle FBC.
    8. Since A is collinear with K and L, rectangle BDLK must be twice in area to triangle ABD.
    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 = AB2.
    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 × BC
    14. Therefore AB2 + AC2 = BC2, since CBDE is a square.

    Garfield's proof
    James A. Garfield (later President of the United States) is credited with a novel algebraic proof.
    The whole trapezoid is half of an (a + b) by (a + b) square, so its area = (a + b)2/2 = a2/2 + b2/2 + ab.
    Triangle 1 and triangle 2 each have area ab/2.
    Triangle 3 has area c2/2, and it is half of the square on the hypotenuse.
    But the area of triangle 3 also = (area of trapezoid) − (sum of areas of triangles 1 and 2)

    = a2/2 + b2/2 + abab/2 − ab/2
    = a2/2 + b2/2
    = half the sum of the squares on the other two sides.
    Therefore the square on the hypotenuse = the sum of the squares on the other two sides.
    Science in Ancient Greece! Aa_garfield_pythag
    Proof by subtraction


    In this proof, the square on the hypotenuse plus four copies of the triangle can be asssembled into the same shape as the squares on the other two sides plus four copies of the triangle. This proof is recorded from China.
    Science in Ancient Greece! 180px-Pythagorean_proof_%281%29.svg

    [b]Similarity proof



    From the same diagram as that in Euclid's proof above, we can see three similar figures, each being "a square with a triangle on top". Since the large triangle is made of the two smaller triangles, its area is the sum of areas of the two smaller ones. By similarity, the three squares are in the same proportions relative to each other as the three triangles, and so likewise the area of the larger square is the sum of the areas of the two smaller squares.


    Proof by rearrangement

    Science in Ancient Greece! PythagScience in Ancient Greece! Pythagoras-2a

    A proof by rearrangement is given by the illustration and the animation. In the illustration, the area of each large square is (a + b)2. In both, the area of four identical triangles is removed. The remaining areas, a2 + b2 and c2, are equal. Q.E.D.Science in Ancient Greece! 180px-Pythagorean_graphic.svg


    This proof is indeed very simple, but it is not elementary, in the sense that it does not depend solely upon the most basic axioms and theorems of Euclidean geometry. In particular, while it is quite easy to give a formula for area of triangles and squares, it is not as easy to prove that the area of a square is the sum of areas of its pieces. In fact, proving the necessary properties is harder than proving the Pythagorean theorem itself (see Lebesgue measure and Banach-Tarski paradox). Actually, this difficulty affects all simple Euclidean proofs involving area; for instance, deriving the area of a right triangle involves the assumption that it is half the area of a rectangle with the same height and base. For this reason, axiomatic introductions to geometry usually employ another proof based on the similarity of triangles (see above).
    A third graphic illustration of the Pythagorean theorem (in yellow and blue to the right) fits parts of the sides' squares into the hypotenuse's square. A related proof would show that the repositioned parts are identical with the originals and, since the sum of equals are equal, that the corresponding areas are equal. To show that a square is the result one must show that the length of the new sides equals c. Note that for this proof to work, one must provide a way to handle cutting the small square in more and more slices as the corresponding side gets smaller and smaller.
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    Post by kittysnow Tue Jun 16, 2009 4:03 am

    (continued)

    Algebraic proof


    Science in Ancient Greece! 180px-Pythagproof.svg


    An algebraic variant of this proof is provided by the following reasoning. Looking at the illustration which is a large square with identical right triangles in its corners, the area of each of these four triangles is given by an angle corresponding with the side of length C.

    Science in Ancient Greece! B1081bfef2978a0dade324996d1647a2
    The A-side angle and B-side angle of each of these triangles are complementary angles, so each of the angles of the blue area in the middle is a right angle, making this area a square with side length C. The area of this square is C2. Thus the area of everything together is given by:

    Science in Ancient Greece! B8aee5ed10de05f6455590276e776044
    However, as the large square has sides of length A + B, we can also calculate its area as (A + B)2, which expands to A2 + 2AB + B2.

    Science in Ancient Greece! 16989f6501a46a61bae450f8b7acc469
    (Distribution of the 4) Science in Ancient Greece! Ad5cfec87e875a99a4bcce373ad6052f
    (Subtraction of 2AB) Science in Ancient Greece! 91102177e7c3d655c508328bcb572982

    Proof by differential equations


    One can arrive at the Pythagorean theorem by studying how changes in a side produce a change in the hypotenuse in the following diagram and employing a little calculus.
    Science in Ancient Greece! 180px-PythagoreanDerivation.svg

    As a result of a change da in side a,

    Science in Ancient Greece! Fe149ae472a81e724301b043eea90f04
    by similarity of triangles and for differential changes. So

    Science in Ancient Greece! 4642f5144cd83c155d3a6890c54cf40f
    upon separation of variables.
    which results from adding a second term for changes in side b.
    Integrating gives

    Science in Ancient Greece! Cf2fa53e0e06c962a7cbef052d366349
    When a = 0 then c = b, so the "constant" is b2. So

    Science in Ancient Greece! 9a16cee2305fb302ddb07c2fc80c0f71
    As can be seen, the squares are due to the particular proportion between the changes and the sides while the sum is a result of the independent contributions of the changes in the sides which is not evident from the geometric proofs. From the proportion given it can be shown that the changes in the sides are inversely proportional to the sides. The differential equation suggests that the theorem is due to relative changes and its derivation is nearly equivalent to computing a line integral.
    These quantities da and dc are respectively infinitely small changes in a and c. But we use instead real numbers Δa and Δc, then the limit of their ratio as their sizes approach zero is da/dc, the derivative, and also approaches c/a, the ratio of lengths of sides of triangles, and the differential equation results.[/b]
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    Post by wazzah!!! Tue Jun 16, 2009 5:11 am

    ok this took me a LONG time to type from somewhere that i found (i dont know how to copie and paste LOL)

    Public Water Works

    Public works were one of the greatest influences in Ancient Greece. They helped boost the economy, and acted as an art form, and they also led to a more sanitary life style. The system of planning the public works was invented by Hippodamus of Miletus, and was admired throughout the Hellenistic and Roman periods. Cities were built according to this scheme and old towns were reconstructed to fit this system. The Greeks were proud of the establishment of the public works and spent a lot of money on it.

    There were many ways to bring water into the city for people to use. Many great thinkers such as Archimedes, Hero, and Eupalinus discovered extraordinary ways to draw water more economically to the cities of Greece. Of all the many different inventions, there were three major inventions that made important contributions to the water supply of Greece.

    The three inventions are:

    # Archimedes' Screw - Archimedes, one of the greatest thinkers of ancient Greece, developed this invention. It was used to lift water from a lower elevation to a higher elevation by means of a tube that is internally threaded. The threads on the inside collect water and as the tube rotates, the water is brought up and put into a storage tank. This massive device was run by human power. The person running the screw, usually a slave, held onto a rail at the top and used his own muscle power to propel the water upward.

    # Aqueducts and Bridging - The Greeks also used techniques such as aqueducts and bridging valleys. They used these devices because the Greeks thought that the water could only be moved if it was moving downward or on a straight path. So in order to keep the water flowing they built aqueducts through mountains and built bridges over valleys. In the sixth century a Greek engineer by the name of Eupalinus of Megara built the aqueduct of Samos. This tunnel measured more than 3000 ft. long and it was started on opposite ends hoping to meet in the middle. When the two met, the tunnels were only fifteen ft. off from each other. On the average, aqueducts were about fourteen feet deep and they were completely lined with stone. The aqueducts were either single route or they branched off into many branches that supplied different areas with water. There was also a form of manhole covers that allowed the workers to access the aqueduct more easily if work needed to be done.

    # Siphon Principle - Hero, a Greek who lived after 150 B.C. was the first hydraulic engineer. He modernized the obtaining of water through a method known as the siphon principle. The siphon principle allows the pipes that carry the water to follow the terrain of the land and the aqueduct and bridging techniques were no longer used as often. For example, such a device was used for the citadel at Pergamon. The pipes that connected to the citadel had approximately 300 pounds of pressure per square inch and the pipes were most likely made of metal in order to withstand the pressure.

    Priests chosen to pray to Apollo had to drink from a secret spring at Colophon before praying. This water was thought to shorten the lives of the priests. The spring has very deep meaning because it was supposed to have formed from the tears of a prophetess. She had wept over the destruction of Thebes, her native city. There is also a punishment in Hell that uses water. People that were unmarried or uninitiated during their lives had the same punishment. The task was to fetch water from either a well or a stream and fill a broken, leaky wine vase for eternity.

    The slaves who had the responsibility of cleaning and repairing all of the public utilities. The more progressive cities had drains under the street that carried both fresh water and sewage. At times these slaves were used to watch over the fountains so that no one did their laundry or bathed in it. They also had to make sure that money thrown into the fountain for luck was not stolen by anyone.
    Science in Ancient Greece! Chobot11

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    Post by wazzah!!! Tue Jun 16, 2009 5:15 am

    LOL i forgot to put my name in my last post!
    my chobot name is flabbergasted and i happy lolz!
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    Science in Ancient Greece! Empty Science In Ancient Greece

    Post by Mimja Tue Jun 16, 2009 5:41 am

    Science is a great subject. Did you here of Science in Ancient Greece? Lets find out some information.

    Thales of Miletus is regarded by many as the father of science; he was the first Greek philosopher to seek to explain the physical world in terms of natural rather than supernatural causes.

    Science in Ancient Greece was based on logical thinking and mathematics. It was also based on technology and everyday life. The arts in Ancient Greece were sculptors and painters. The Greeks wanted to know more about the world, the heavens and themselves. People studied about the sky, sun, moon, and the planets. The Greeks found that the earth was round.

    Eratosthenes of Alexandria, who died about 194 BC, wrote on astronomy and geography, but his work is known mainly from later summaries. He is credited with being the first person to measure the Earth's circumference.

    Botany

    Greek influence on agriculture was the establishment of the science of botany. Botany is the study of all aspects of plant life, including where plants live and how they grow. The Greek philosopher Aristotle, who lived during the 300's BC, collected information about most of the plants known at that time in the world. He also studied other sciences and math.
    His student Theophrastus classified and named these plants. Theophrastus often called the father of botany. Aristotle and Theophrastus developed an extremely important type of science that is studied all over the world. Botany is so important because all the food that animals and people eat comes from plants, whether it be directly or indirectly.

    Earth Science

    Earth science is the study of the earth and its origin and development. It deals with the physical makeup and structure of the Earth. The most extensive fields of Earth science, geology, has an ancient history.
    Ancient Greek philosophers proposed many theories to account for the from and origin of the Earth. Eratosthenes, a scientist of ancient Greece, made the first accurate measurement of the Earth's diameter. The ancient Greek philosophers were amazed by volcanoes and earthquakes. They made many attempts to explain them, but most of these attempts to explain these phenomena sound very strange to most people today. For example, Aristotle, speculated that earthquakes resulted from winds within the Earth caused by the Earth's own heat and heat from the sun. Volcanoes, he thought, marked the points at which these winds finally escaped from inside the Earth into the atmosphere.
    Earth science allows us to locate metal and mineral deposits. Earth scientists study fossils. This helps provide information about evolution and the development of the earth. Earth science helps in locating fossil fuels, such as oil. These fuels compose a major part of the world economy. The Greeks came up with the idea of earth science, and most importantly laid the foundation for the scientists who lived hundreds of years after their time.

    Public Water Works

    Public works were one of the greatest influences in Ancient Greece. They helped boost the economy, and acted as an art form, and they also led to a more sanitary life style. The system of planning the public works was invented by Hippodamus of Miletus, and was admired throughout the Hellenistic and Roman periods. Cities were built according to this scheme and old towns were reconstructed to fit this system. The Greeks were proud of the establishment of the public works and spent a lot of money on it.
    There were many ways to bring water into the city for people to use. Many great thinkers such as Archimedes, Hero, and Eupalinus discovered extraordinary ways to draw water more economically to the cities of Greece. Of all the many different inventions, there were three major inventions that made important contributions to the water supply of Greece.

    The three inventions are:

    # Archimedes' Screw - Archimedes, one of the greatest thinkers of ancient Greece, developed this invention. It was used to lift water from a lower elevation to a higher elevation by means of a tube that is internally threaded. The threads on the inside collect water and as the tube rotates, the water is brought up and put into a storage tank. This massive device was run by human power. The person running the screw, usually a slave, held onto a rail at the top and used his own muscle power to propel the water upward.

    # Aqueducts and Bridging - The Greeks also used techniques such as aqueducts and bridging valleys. They used these devices because the Greeks thought that the water could only be moved if it was moving downward or on a straight path. So in order to keep the water flowing they built aqueducts through mountains and built bridges over valleys. In the sixth century a Greek engineer by the name of Eupalinus of Megara built the aqueduct of Samos. This tunnel measured more than 3000 ft. long and it was started on opposite ends hoping to meet in the middle. When the two met, the tunnels were only fifteen ft. off from each other. On the average, aqueducts were about fourteen feet deep and they were completely lined with stone. The aqueducts were either single route or they branched off into many branches that supplied different areas with water. There was also a form of manhole covers that allowed the workers to access the aqueduct more easily if work needed to be done.

    # Siphon Principle - Hero, a Greek who lived after 150 B.C. was the first hydraulic engineer. He modernized the obtaining of water through a method known as the siphon principle. The siphon principle allows the pipes that carry the water to follow the terrain of the land and the aqueduct and bridging techniques were no longer used as often. For example, such a device was used for the citadel at Pergamon. The pipes that connected to the citadel had approximately 300 pounds of pressure per square inch and the pipes were most likely made of metal in order to withstand the pressure.

    Priests chosen to pray to Apollo had to drink from a secret spring at Colophon before praying. This water was thought to shorten the lives of the priests. The spring has very deep meaning because it was supposed to have formed from the tears of a prophetess. She had wept over the destruction of Thebes, her native city. There is also a punishment in Hell that uses water. People that were unmarried or uninitiated during their lives had the same punishment. The task was to fetch water from either a well or a stream and fill a broken, leaky wine vase for eternity.
    The slaves who had the responsibility of cleaning and repairing all of the public utilities. The more progressive cities had drains under the street that carried both fresh water and sewage. At times these slaves were used to watch over the fountains so that no one did their laundry or bathed in it. They also had to make sure that money thrown into the fountain for luck was not stolen by anyone.
    Most of the public water-supply was used for public buildings, such as baths and street fountains. For example, in Alexandria, in Egypt, each house had a personal cistern for their own water for their own use. The slaves also had to clean these cisterns. These private owners of cisterns and users of water had to pay a water rate to the city. It is sort of like the first public utilities company.

    Biology

    Many important people contributed to Greek scientific thought and discoveries. Biology, a very vast and interesting topic, was studied by Hippocrates, Aristotle, Theophrastus, Dioscorides, Pliny, and Galen. These men were among the main researchers of Greek biology who contributed many ideas, theories, and discoveries to science. Some of their discoveries were observations, descriptions, and classifications of the various forms of plants and animal life. Other discussions in biology were natural selection and zoology.
    All living things were the basic concern of biology. Greek biologists were interested in how living things began, how they developed, how they functioned, and where they were found. These sorts of questions that ran through the biologists' minds are exactly how they began to discover the basics of life. At such an early time, about 300 B.C., science was just beginning to enter the minds of the Greeks. Aristotle, a Greek biologist, made contributions of his own to science. However, around 300 B.C. there was much more to be discovered, which enabled other scientists to add knowledge to the discoveries of Aristotle, during and after his time.
    Natural Selection is the manner in which species evolve to fit their environment - "survival of the fittest." Those individuals best suited to the local environment leave the most offspring, transmitting their genes in the process. This natural selection results in adaptation, the accumulation of the genetic variations that are favored by the environment.
    Many Greek scientists thought about natural selection and the origin of life. Anaximander believed that marine life was the first life on Earth and that changes happened to animals when they moved to dry land. Empedocles had the idea of chance combinations of organs arising and dying out because of their lack of adaptation. Aristotle, a Greek philosopher who contributed many works in the sciences, believed that there is purpose in the workings of nature, and mistakes are also made. He thought that nature working so perfectly is a necessity.
    Aristotle believed that nature is everything in the environment, like the sky rains, and the plants grow from the sun. Aristotle's theory fits very well with natural selection.
    Natural selection makes it necessary for animals and nature fit perfectly - 'survival of the fittest'. If they didn't, then that specific organism would die out, weeding out the characteristics that were unfit for that environment.
    That same organism's species might evolve over time and acquire adaptations suitable for the environment, so that newly evolved species can survive and flourish with offspring.
    Lucretius, who lived about 50 AD in Rome, believed that evolution was based on chance combinations; heredity and sexual reproduction entered only after earth itself had developed. Then with the organism developing characteristics that might make for survival in the environment, the organisms that don't have favorable characteristics are incapable of survival and disappear. These ideas from Greek scientists are all theories, of course, but the fossil evidence suggests that species evolved over time.

    Zoology

    Zoology is the study of animals, involves studying the different species of animals, the environment in which they live, and their organs. Aristotle was very persistent with his studies of the zoological sciences and made many contributions to how we study zoology today. He made observations on the anatomy of octopi, cuttlefish, crustaceans, and many other marine invertebrates that were remarkably accurate. These discoveries on the anatomy could have only been made by dissecting the animals. Through dissection, Greek zoologists studied the structures and functions of anatomies of various animals. Some structures that were studied were bones and membranes. However, to discover and learn about the diversity of animals, Greek zoologists had to narrow their areas of study by attempting to classify the organisms.

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