It was then notated that if triangle ABC is a right triangle, with a right angle at C, then c2 = a2 + b2. Earlier, the converse of this theorem appears to have been used. This became proposition number 47 from Book I of Euclid's Elements ("Pythagorean," 2007).
Although this theorem is traditionally associated with Pythagoras, it is actually much older.
More than a millennium before the birth of Pythagoras, four Babylonian tablets were created demonstrating some knowledge of this theorem, circa 1900-1600 B.C.. At the very least, these works represent the knowledge of at least special integers known as Pythagorean triples that satisfy it.
In addition, the hind papyrus, created around 1650 B.C., shows that Egyptians had knowledge of the theorem as well. However, the first proof of this theorem is still credited to Pythagoras, despite the fact that some scholars believe it was independently discovered in several different cultures ("Pythagorean,"…...
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References
Meserve, B.E. (2007). Pythagoras, theorem of. Grolier Multimedia Encyclopedia. Retrieved December 6, 2007, from Grolier Online.
Mourelatos, a.P.D. (2007). Pythagoras and Pythagoreanism. Encyclopedia Americana. Retrieved December 6, 2007, from Grolier Online.
Pythagoras. (2007). Encyclopedia Britannica. Retrieved December 5, 2007, from Encyclopedia Britannica Online.
Pythagorean theorem. (2007). Encyclopedia Britannica. Retrieved December 5, 2007, from Encyclopedia Britannica Online.
Pythagoras was born in 569 BC in Samos, to Mnesarchus of Tyre and Pythias of Samos. Mnesarchus was a merchant and so Pythagoras had the opportunity to visit many lands as a child traveling with his father. Besides these facts there is very little that is known of the childhood of Pythagoras. There is no doubt that he would have been properly educated and there is evidence to show that Pythagoras was influenced by three philosophers the most important of which was Pherkydes. Another person to influence his life was Thales and it is believed that it was Thales that stoked the interest of mathematics in Pythagoras. The pupil of Thales Anaximander is believed to have contributed many ideas in geometry and cosmology that were to influence the views of Pythagoras. Pythagoras as a young man moved to Egypt and remained there till the Persians invaded Egypt and…...
This idea was accepted by most of the philosophical schools of the time, including the Atomists.
Plato took quite a different approach and found that ideas, as noted, and saw idas as existing outside of human consciousness. Plato's doctrine of recollection holds that learning is the remembering of a wisdom that the soul enjoyed prior to its incarnation, another aspect of the idea that there are ideal forms "remembered" by the soul in this world, and this is actually a mythical statement of this view that neither reason nor the intelligible order that it reveals is alien to the human soul. The soul is seen as existing before life here on earth and as remembering the ideals it knew before birth. Protagoras would not have engaged in this sort of argument, jus as he avoided arguments about the existence of the gods as being outside of sensory experience.
The twenty-one pieces of the work, minus the Overture, are divided into two acts, 8 pieces in the First Act, and 13 in the Second. 8 to 13 is an example of the Golden Ratio. There are also 49 entrances in The Magic Flute, divided up as 19 in Act I and 30 in Act II. This too is an example of the Golden Ratio. Furthermore, the Overture contains a division between 81 bars and 130, yet another Golden Ratio.
Golden Ratio is thus contained within Golden Ratio, an image of the endless repetition of the primordial forms. Each individual part of the Creation is complete unto itself. One can take apart the Cosmos and find perfect miniature "worlds" that can be put back together to form a coherent whole. According to the Classical canon of art, the human body is built upon the Golden Ratio. By drawing lines through…...
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Works Cited
Benstock, Seymour L., ed. Johann Sebastian: A Tercentenary Celebration. Westport, CT: Greenwood Press, 1992.
Boyd, Malcolm, and John Butt, eds. J.S. Bach. Oxford: Oxford University Press, 1999.
Chua, Daniel K.L. Absolute Music and the Construction of Meaning. Cambridge, England: Cambridge University Press, 1999.
Faulkner, Quentin. Wiser Than Despair: The Evolution of Ideas in the Relationship of Music and the Christian Church. Westport, CT: Greenwood Press, 1996..
Philosophy and Psychology of the Mind and Body
Throughout human history, philosophers, doctors, and most recently, psychologists, have attempted to understand the relationship between the mind and body and how it results in human beings' awareness and perception of reality. At least since the golden age of Greek philosophy, thinkers have been aware of an ostensible distinction between the mind and body, a distinction that nonetheless allows for some intermingling such that physical issues affect the mental state just as mental issues may result in physical symptoms. Thus, if one desires to truly understand how contemporary estern psychologists and philosophers consider the nature of consciousness via the interaction between mind and body, one must trace the history of these concepts starting with the Greek philosophers, moving through the Middle Ages and the Renaissance and on to the eighteenth and nineteenth centuries, when psychology first began to develop as a formal discipline.…...
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Works Cited
Bunge, M. (2010). The mind-body problem. Matter and Mind, 287(2), 143-157.
Hergenhahn, B.R. (Ed.). (2009). An introduction to the history of psychology. Belmont, CA:
Wadsworth.
Kendell, R.E. (2001). The distinction between mental and physical illness. British Journal of Psychiatry,178, 490-493.
Parmenides is one of Plato's most important dialogues, according to both ancient and modern scholars, and focuses on the critique of the theory of forms, based on the influence of pre-Socratic thinkers such as Pythagoras, Parmenides, and Heraclitus. The theory of Forms is founded on the assumption that a higher, spiritual realm of Forms, or Ideas, exists beyond the world of physical things.
The realm of Forms has a hierarchical order, the highest level being that of the Form of Good. The physical world, as perceived by the senses is in constant flux, therefore making knowledge derived from it variable and restricted. The realm of Forms, however, is only apprehensible by the mind and is eternal and changeless. Each Form is actually a pattern of a certain category of things in the physical world -- things which are only an imperfect copy of the perfect Forms.
Although in the Phaedo dialogue, Socrates…...
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References:
1. Allen R.E, "Plato's Parmenides," Minneapolis, University of Minneapolis Press, 1983
2. The Cambridge Companion to Plato, edited by Richard Kraut, New York, Cambridge University Press, 1992
3. Encyclopaedia Britannica -- articles on Plato and the history of philosophy, 1997 edition
Joyce's Ulysses
Claude Rawson is best known as a scholar of Jonathan Swift and the eighteenth century, but Rawson's has also used the savage irony of Swift's modest proposal for a series of essays which consider Swift's invocation of cannibalism in light of a longer tradition (in Anglo-Irish relations) of imputing cannibalism literally to the native Irish as a way of demonizing their "savagery" or else to implying a metaphorical cannibalism to describe the British Imperial exploitation of those native Irish. Rawson reapproaches these Swiftian subjects in a more recent essay entitled "Killing the Poor: An Anglo-Irish Theme" which examines what Rawson calls the "velleities of extermination" in a text like Swift's "Modest Proposal" (Rawson, 300). Rawson examines how Swift's ironic solution of what to do with the poor of Ireland (eat them as food) undergoes, in various later iterations by Anglo-Irish writers including Shaw and ilde, transformation into a rhetorically…...
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Works Cited
Burgess, Anthony. ReJoyce. New York: W.W. Norton, 1965.
Ellmann, Richard. Ulysses on the Liffey. New York and London: Oxford University Press, 1972.
Henke, Suzette. James Joyce and the Politics of Desire. New York and London: Routledge, 1990.
Joyce, James. Ulysses. Ed. Hans Walter Gabler. New York: Vintage, 1986. Print.
He looks at thee methods: histoy (melding infomation about the divese geogaphical oigins of algeba with the poblems themselves), multiple epesentations (using notation, naative, geometic, gaphical, and othe epesentations togethe to build undestanding), and the object concept of function (teaching functions without genealizing about how taits of an individual elate to taits of a goup). The aticle seves to offe some inventive solutions to a common poblem in math education: How to make mateial elevant and compelling to a beadth of students.
Matinez, a.A. (2010). Tiangle sacifice to the gods. 1-11.
The aticle looks at Pythagoas, paticulaly the mythology suounding his life and his most famous discovey, the Pythagoean theoem. It calls into question the histoical evidence on which mathematics teaches base thei teaching of this theoy. The autho points out how vey little is known about Pythagoas and how he has been canonized by the math discipline because his theoy…...
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references the impact that Newton's work had on mechanical applications. Lastly, the piece points out how Newton used the thought patterns associated with calculus in what appears to the modern reader as a work of geometry (with respect to his book "The Principia"). In this way, the article functions as a reminder of how scientific discoveries are created, which is by building upon the theories of others and by giving weight to the importance to mathematical principles.
Nature Closer to the Ancient than the Renaissance View?
In his book, The Idea of Nature, Collingwood analyzes the principle characteristics of three periods of cosmological thinking in the history of European thought: Greek, Renaissance, and the Modern. By taking such an approach, Collingwood makes it possible for his readers to distinguish the similarities as well as fundamental differences between the modern view of Nature and that of Greek and Renaissance cosmology. But, perhaps Collingwood's work is more valuable because it demonstrates how both Greek and Renaissance schools of thought have made the modern view of nature possible. In other words, the modern view of nature has evolved from both Greek and Renaissance cosmology, with each period laying the foundation for the next to build on. To that extent, an assertion that the modern view of Nature more closely resemblances one period rather than another cannot, strictly speaking, be made…...
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Works Cited
Collingwood, R.G. "The Idea of Nature." Oxford: Clarendon Press, 1945.
He died four years before Peurbach's matriculation, leaving the University without an astronomy lecturer. However, his library and instruments were probably accessible to Peurbach.
While it is known that Peurbach travelled throughout Europe between the years 1448 and 1453, there is no record of the precise dates. At the time, he also had an international reputation as an astronomer of note, despite the fact that he had not publications at the time. He did however lecture in Germany, France and Italy.
After lecturing at Bologna and Padua, these universities offered him permanent appointments as lecturer, but Peurbach turned these down. During his travels he also met the leading Italian astronomer of the time, Giovanni Bianchini, in Ferrara. Bianchini also offered Peurbach a post at an Italian university. Peurbach however remained unwilling to be tied to any specific institution of learning and turned down the offer. In 1453, Peurbach returned to Vienna…...
One of the most enduring legacies of the Pre-Socratic concern with the source of all sentient life is atomism, advocated by Democritus. The definition of atoms as things "that cannot be cut up or actively divided up and split," seems more like a precursor to modern physics rather than philosophy (80) Sophists such as Critas asked questions, not about the natural world, but the moral and political life of citizens and acted as "independent, often itinerants teachers of wisdom," and above all conveyers of "political skills" that were of vital importance in a free-for-all democracy like ancient Athens, where the ability to "speak well" and persuasively was a vital survival skill (6;96). The Sophists were rivals to Socrates because they claimed that morality was based in convention, and did not exist outside of social institutions like language.
The diversity of the Pre-Socratic philosophers highlights the complexity of what constituted philosophy in…...
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Works Cited
Pre-Socratic Reader: Selected Fragments and Testimonial.
Edited by Patricia Curd. Translations by Richard D. McKirahan. Jr. New York: Hackett Publishing Company, 1996.
Pythagorean Triple is a term derived specifically from the theorem that a right triangle displays the equation. This theorem, named aptly the Pythagorean Theorem, states that "in any right-angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides" (Mack, 2010). How do the two relate? A Pythagorean Triple is simply the term derived for a, b, and c, provided that 1) a, b, and c are positive integers and 2) a, b, and c fulfill the equation. Silverman's Number Theory chapter on "Pythagorean Triples" has highlighted the relevant proof for the equation of the Pythagorean Theorem. What further equations can be used to find subsequent Pythagorean Triples?
The primitive Pythagorean triple begins with set T (3,4,5). Already, set T. fulfills both the requirements 1) and 2) as stated above; that the numbers in the set are positive integers and, with the use of…...
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Resources
Mack, J., & Czernezkyj, V. (2010). The tree in Pythagoras' garden. Australian Senior Mathematics Journal, 24(2), 58. Retrieved from EBSCOhost.
Silverman, Joseph H. "Pythagorean Triples." A Friendly Introduction to Number Theory. Third ed. Pearson Prentice Hall, 2006. Web.
Turner, P. (2006). Making Pythagoras count. Australian Senior Mathematics Journal, 20(1), 48. Retrieved from EBSCOhost.
Weisstein, Eric W. "Pythagorean Triple." From MathWorld -- A Wolfram Web Resource.
Ethnography
There is no such thing as a time machine. Ancient history can only be understood by modern peoples through the cultural documentation that was left behind. ritings from the period of the New Testament exist but they do not provide information into every aspect of everyday life. Consequently, historians and scholars must analyze the documents that are in existence in order to gain a greater understanding into the world's past. One technique that makes it possible for current populations to understand ancient texts is the use of literary ethnography. This procedure is the endeavor to use qualitative means to learn about and to better understand various cultural documentation and ideology which mirror that culture's society. Particularly of importance to ethnography is the ways and means of knowledge acquisition of a culture and also the system of meanings and which dictate that culture, such as language and the roles of…...
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Works Cited:
Aphthonius of Antioch. "Progymnasmata."
Diogenes and Crates. "Principal Representatives of Cynic Philosophy."
Goethe and Marlowe, Faust
The Faust myth provides a writer with a chance to explore religious issues through the theme of damnation, while also allowing the writer to identify with the damned protagonist through a shared sense of ambition. This is palpable in both Marlowe's and Goethe's different versions of the Faust legend -- in both cases, it seems like the ambitious "striving" (to use a crucial Goethean word for Faust's essential nature) of the main character is mirrored by the author's ambition to present broad swathes of human and indeed divine experience on stage or into the reader's imagination. A comparison of the endings of these two different handlings of the Faust legend will, I think, illustrate crucial differences between not only Goethe's and Marlowe's differing literary ambitions, but also their different religious or spiritual worldviews.
In reality Marlowe's Faustus seems like Marlowe himself -- someone who is interested in gaining…...
" James a.S. McPeek
further blames Jonson for this corruption: "No one can read this dainty song to Celia without feeling that Jonson is indecorous in putting it in the mouth of such a thoroughgoing scoundrel as Volpone."
Shelburne
asserts that the usual view of Jonson's use of the Catullan poem is distorted by an insufficient understanding of Catullus' carmina, which comes from critics' willingness to adhere to a conventional -- yet incorrect and incomplete -- reading of the love poem. hen Jonson created his adaptation of carmina 5, there was only one other complete translation in English of a poem by Catullus. That translation is believed to have been Sir Philip Sidney's rendering of poem 70 in Certain Sonnets, however, it was not published until 1598.
This means that Jonson's knowledge of the poem must have come from the Latin text printed in C. Val. Catulli, Albii, Tibulli, Sex. Aur. Propertii Opera omnia…...
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Works Cited
Alghieri, Dante Inferno. 1982. Trans. Allen Mandelbaum. New York: Bantam Dell, 2004.
Print.
Allen, Graham. Intertextuality. Routledge; First Edition, 2000. Print.
Baker, Christopher. & Harp, Richard. "Jonson' Volpone and Dante." Comparative
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