The History Of Pi Essays - Pi, Approximations Of,

Pi The History of Pi A little known verse in the bible reads ?And he made a molten sea, ten cubits from the one brim to the other; it was round all about, and his height was five cubits; and a line of thirty cubits did compass it about(I Kings 7, 23).? This passage from the bible demonstrates the ancient nature of the irrational number pi. Pi in fact is mentioned in a number of verses throughout the bible. In II Chronicles 4,2, in the passage describing the building of the great temple of Solomon which was built around 950BC, pi is given as equal to three. This value is not very accurate at all and should not even be considered accurate for it's time, however it should be noted that precision was not needed for the task that was being performed and we should let the general concept of pi that the biblical characters posses impress us. Present knowledge suggests that the concept of pi first developed in 2000 BC in two separate cultures. The Babylonians used pi at a value of 25/8 while an entirely different culture, the ancient Egyptians used pi at a value of 256/81. While the biblical calculation of pi=3 most likely came from crude measurement, there is strong reason to believe, because of the relative accuracy of the values, that the Babylonians and Egyptians found pi by means of mathematical equations. In the Egyptian Rhind Papyrus, which is dated around 1650 BC, there is strong evidence supporting that the Egyptians used 4(8/9)2 =3.16 for their value of pi. At that point in history, and for the majority of modern history, pi was not seen as an irrational number as it is today. The next culture that investigated pi was the ancient Greeks. Starting in 434 BC Greeks were unraveling the mysteries of pi. The mathematician Anaxagoras made an unsuccessful attempt at finding pi, which he called squaring the circle and in 414 BC, 20 years after Anaxagoras failed in his attempt to square the circle, Aristophanes refers to the work of Anaxagoras in his comedy ?The Birds?. It took over 100 years for the Greeks to finally find a value for pi. In 240 BC Archimedes of Syracuse showed that 223/71*pi*22/7. Archimedes knew, what so many people today do not, that pi does not equal 22/7 and he made no claim to have discovered the exact value of pi. However if we take the average of his two bounds we obtain pi=3.1418, which was an error of about 0.0002. Archimedes found the most accurate value of pi up to that time and his value would be used exclusively until the next discovery in the world of pi. The next major finding concerning pi did not occur in the western world, but in China by Tsu Chung-chi'h who approximated pi at 355/113 in 480 AD. Next to nothing except for this work is known about Tsu Chung-chi'h's life but it is very unlikely that he had any awareness of Archimedes work. We shall now notice how during the dark ages of Europe, the lead in the research of pi is passed to the East. Aryabhata, working on his own in Persia without any outside information in 515 AD was able to approximate pi to 3 decimal places. A mathematician from Baghdad named Al'Khwarizimi worked with pi however the most accurate finding of pi to date was found even more east in Samarkand by Al-Khashi. In 1430 AD he approximated pi to 16 decimal places, the most to date. His work however, would be the last of note from the east as the European Renaissance brought about a whole new mathematical world. The first notable discovery in the approximation of pi from the European Renaissance was by Viete in 1593 AD. He expressed pi as an infinite product by using only 2's and square roots. In 1610 Ludolph van Ceulen demonstrated the new thought coming out of the Renaissance by calculating pi to 35 decimal places. Around the same time, Snell refined Archimedes's method of calculating pi, and Snell's work was used by Grienberger to calculate pi to 39 decimal places in 1630. In 1655 Wallis showed that pi/2=2/1*2/3*4/3*4/5*6/5*6/7*8/7*8/9..... The 18th centuary brought about great

The political debate Essay Example

The political debate Essay Example The political debate Essay The political debate Essay The political debate on whether it is best to have a strong or weak presidency has been going on since the dawning of the American Republic, but such a question usually crops up at a time when the security of the American people is in peril, or there is a clear and definite state interest for the American government to limit certain civil liberties of the people for the benefit of the general population. History has been replete on this political discourse especially since the United States convincingly expanded its sphere of influence across the entire world starting with the subjugation of the Philippine Islands to American colonial rule in 1899. As American influence, even imperialism and hegemony, spread across all parts of the world until the present time, the attacks on the democratic life of the American people and the existence of the Republic has been unceasing such as when Japanese militarists bombed Pearl Harbor at the start of World War II and the decades-long communist t hreat of the Soviet Union and allied socialist republics. In pursuit of the protection of the American Republic, the American presidency has always sought to explore ways and means on how to best defend the nation, militarily and diplomatically, without resorting to the curtailment of the civil liberties of the American people. However, for the most part, the American government with the US president at its head, has usually challenged the limits of such civil liberties. No best example can ever be had than the McCarthyist communist witch-hunts of the 1950s where suspected American communists from all sectors of American society were invited to the US Congress only to be vilified and falsely accused of their participation in the Communist Party of the USA. While there was no participation of the American president then in such congressional proceedings, it is clear that he acquiesced in the entire affair led by Sen. McCarthy, notwithstanding the complicity of security agencies of the federal government which are directly under the American President. On the other hand, as perceived curtailments of civil liberties happened throughout American contemporary history, there have been campaigns and struggles from American interest groups to call for the relegation of the American presidency as simply a watchtower of other government branches as the presidency has far too awesome executive powers that can curtail not only on civil-political rights but may e ven infringe on socio-economic and property rights as well. However, the resolution of the debate has always been in favor of strengthening the American presidency as it is clearly within the interest of the state and even the mandate of the American president to ensure the welfare and protection of the American people from threats from within and without. However, in this era of global terrorism and the rise of Islamic fundamentalism the question of a strong presidency emerged once again. The debate is in the light of the recent discovery by the American public that President George W. Bush secretly authorized security forces in 2002 to gather intelligence through wiretapping of telephones and mobile phones, and eavesdropping on emails and other telecommunications services to swoop down on terrorist correspondence within the United States and abroad, without the fulfilling the statutory requisite of a warrant from a surveillance court. Moreover, the controversy erupted in the wake of the approval of the reauthorization of the US Patriot Act, a controversial law that granted very broad powers to security forces to swoop down on perceived enemies of the state, particularly suspected Islamic fundamentalists operating inside the continental United States. Nonetheless, the law is deemed by liberals and progressives as a direct attack on co nstitutional rights of the American people, especially when previously strict surveillance measures are now relaxed, notwithstanding assertions by the security community that the operations in pursuit of the US Patriot Act and the war on terror is not within the purview of the court. If this be so, this would also be tantamount to a curtailment of the judicial power and supremacy of the US Supreme Court to review all acts and omissions of the other branches and government, even if in the guise of national security. The controversy has also elicited further tensions between an American public that was already weary from the ravages of the Iraq War as many feared that even American homes and offices are no longer spared from the operations of the American government in its fight against terror. Nonetheless, the Bush administration has maintained that the president has inherent war powers which may not be expressed but implied in the Section on Executive power in the American constitut ion. Truth to tell, the secret authorization given by President Bush to spy on the American people may find legal basis in the inherent war powers afforded the president, unless the statutory provisions ordering the securing of a warrant prior to surveillance are more controlling. Assuming without conceding that the Constitution grants such residual powers to the American president, the propriety of the authorization is most questionable, as it comes at the heels of growing American discontent over the war in Iraq, especially when the basis for invading Iraq (i.e. direct link with Al-Qaida and production of weapons of mass destruction) was not convincingly and truthfully conveyed by the president to the American people. Nonetheless, the act is well within the purview of the executive’s powers despite the public outrage that ensued upon discovery of the authorization.