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

Pope Francis Quotes on Racism, Xenophobia, Immigration

Pope Francis Quotes on Racism, Xenophobia, Immigration Pope Francis has received praise for his forward-thinking views since 2013 when he became the first pontiff from Latin America. While the Catholic Church leader has not backed same-sex marriage or reproductive rights, he’s suggested that gay people and women who’ve had abortions deserve empathy and forgiveness, a departure from previous pontiffs. Given his views on these issues, progressives wondered what the pope might have to say about race relations when he made his first visit to the United States in September 2015. At that time, racial tensions continued to run high in the nation, with police killings and allegations of police brutality routinely making the news and trending on social media networks. Prior to his U.S. visit, Pope Francis had not specifically commented on the Black Lives Matter movement, but he had weighed in on racism, xenophobia, stereotypes, and diversity around the world. Familiarize yourself with the pope’s views on race relations with the following quotes. All Forms of Intolerance Should Be Fought Pope Francis  came down hard on intolerance while speaking to a group from the Simon Wiesenthal Center in Rome in October 2013. He highlighted the center’s goal â€Å"to combat every form of racism, intolerance, and anti-Semitism† and noted that he’d recently reaffirmed the Catholic Church’s condemnation of anti-Semitism. â€Å"Today I wish to emphasize that the problem of intolerance must be confronted in all its forms: wherever any minority is persecuted and marginalized because of its religious convictions or ethnic identity, the well-being of society as a whole is endangered and each one of us must feel affected,† he said. â€Å"With particular sadness I think of the sufferings, the marginalization and the very real persecutions which not a few Christians are undergoing in various countries. Let us combine our efforts in promoting a culture of encounter, respect, understanding, and mutual forgiveness.† Although the pope could have limited his discussion of religious intolerance, he included intolerance based on ethnic identity in his speech as well, an indication that he’s concerned about the treatment of all minority groups. The World Cup as an Instrument of Peace When the FIFA World Cup kicked off in June 2014, many sports fans focused exclusively on whether their favorite teams would advance in the soccer (football) tournament, but Pope Francis offered a different viewpoint on the games. Before the opening match between Brazil and Croatia, Francis said the World Cup could teach the public a great deal about solidarity, teamwork, and honoring opponents. â€Å"To win, we must overcome individualism, selfishness, all forms of racism, intolerance, and manipulation of people,† he said. One cannot be a self-centered player and experience success, he said. â€Å"Let nobody turn their back on society and feel excluded!† he said. â€Å"No to segregation! No to racism!† Francis is reportedly a lifelong fan of the Buenos Aires soccer team San Lorenzo and hoped the World Cup served as a â€Å"festival of solidarity between peoples.† â€Å"Sport is not only a form of entertainment but also - and above all I would say - a tool to communicate values that promote the good that is in humans and help build a more peaceful and fraternal society,† he said. End Racism Against U.S.-Bound Migrants A year before real estate mogul-turned-President Donald Trump branded some undocumented immigrants from Mexico as rapists and drug traffickers, Pope Francis called on the United States to adopt a humanitarian approach to the migrants crossing the border, especially children. â€Å"Many people forced to emigrate suffer, and often die tragically,† the pope stated on July 15, 2014, in a message addressing a global conference in Mexico. â€Å"Many of their rights are violated, they are obliged to separate from their families and, unfortunately, continue to be the subject of racist and xenophobic attitudes.† Francis could have framed the situation on the U.S.-Mexico border as a humanitarian crisis without invoking racism and xenophobia, but he made a point to recognize how attitudes about â€Å"the other† influence immigration policy. The pope has a history of advocating for refugees, remarking on an Italian island in 2013 that the public was indifferent to the dire circumstances in which North African and Middle Eastern migrants find themselves. Stereotypes and the Criminal Justice System On Oct. 23, 2014, Pope Francis addressed a delegation from the International Association of Penal Law. Speaking to the group, Francis discussed the widespread idea that public punishment is the solution to difficult social problems. He expressed his disagreement with this view and questioned the motives of public punishment. â€Å"Scapegoats are not only sought to pay, with their freedom and with their life, for all social ills such as was typical in primitive societies, but over and beyond this, there is at times a tendency to deliberately fabricate enemies: stereotyped figures who represent all the characteristics that society perceives or interprets as threatening,† he said. â€Å"The mechanisms that form these images are the same that allowed the spread of racist ideas in their time.† This is the closest Francis came to addressing the Black Lives Matter movement before his visit to the U.S. in September 2015. Like many activists in the movement, Francis suggests that racial scapegoating factors into why society favors taking freedom away from some groups and placing them behind bars for years rather than remedy the social ills that keep prisons overflowing. Embracing Differences While discussing tensions between Catholics and Muslims in January 2015, Pope Francis once again emphasized the need to accept differences. He told a delegation affiliated with the Pontifical Institute of Arab and Islamist Studies that â€Å"patience and humility† are musts in the Islamic-Christian dialogue to avoid fueling â€Å"stereotypes and preconceptions.† â€Å"The most effective antidote to every form of violence is education about discovering and accepting difference as richness and fertileness,† Francis said. As his other remarks on diversity indicate, accepting differences can apply to religious faith, ethnicity, race and much more. The lesson to be learned, according to the pope, is that people don’t divide themselves and lash out against others based on differences.

Portfolio Essay Example | Topics and Well Written Essays - 1000 words - 1

Portfolio - Essay Example The topic on the selling of human organs is of importance due to its emerging and contentious nature. Several questions have arisen from the point of view that organs are being sold, raising both heath and ethical concerns. The American history is also of major concern considering the need for our generation to understand where our country has come from and get to understand the differences and similarities between then and now. The three essays I wrote revolves around these two major topics, here I will explain in detail my findings and the positive and negative remarks that I have established as I review them. Following an article on the ‘Capitalism Magazine’ by Holcberg about the issue of human organs being predestined for sale, I developed a deep interest to review the topic so as to present Holcberg’s thoughts systematically. I found the report raising very serious concerns about legalizing of selling of human organs. I clearly articulated his concerns while presenting the facts that he had considered. My introduction was very captivating and captured the concerns at a first glance. The reasoning behind the article was not based on my own thoughts but that of the author. The topic limited me to the thoughts of the author; an aspect that I believe brought some biasness in the article and made the topic narrow instead of taking the broader perspective of capturing the opinions of many. However, the brevity and the systematic articulation of my points made the essay to read and comprehend. Several improvements to the article could have made it more interesting and more informative. I only focused on an individual have thought that was a challenge; this could have been enhanced by considering not only the opinion of others but also could have reviewed some literature on the topic. Another challenge was in the way I presented my work, all the points I presented them in one long paragraph. This could have been improved by articulating my poin ts in individual short and precise paragraphs. It is apparent that the paper limited use of my own thoughts an aspect that made me as the writer to be a reviewer and had no opportunity to put across a personal opinion, especially because the topic had been of concern to me. Grammatical errors and poor sentence structure also prevailed in my work; an aspect contributed by the fact that I did not take time to edit my work, I learnt a lesson and has decided that next time I will be keen to recheck. My second essay was geared towards giving a summary of the Krugman’s article that was published in the New York Times focusing on the similarities and differences between the America we had at independence and today. I presented the reasons for his argument as presented in the media. The thoughts were analyzed taking into concern the key points that outstand when comparing the state of the country at the two different times. All the information was presented in one paragraph since the y were direct and self-explanatory. However, the narrow presentation was evidenced by the regard of Krugman’s thoughts only, without having a point of view from other individuals. My opinion of the situation influenced the summary, such that I was prompted to think against the arguments of the author. However, to restrain myself from his facts and opinions I had to break down the arguments into points that he had mentioned. This facilitated the development of a summary in a great

The Reformation Essay Example | Topics and Well Written Essays - 500 words

The Reformation - Essay Example Martin Luther with the help of his colleagues introduced a new order of prayers at their Electoral Saxony of 1527. He completed many religious literary pieces including his German Mass which was his simpler form of his earlier Latin Mass. The idea was to make religious teachings and scriptures easier for common people. He believed that Catholic Church was not letting people understand the religion, instead they were coming up with their interpretations of the teachings of Christ as they found fitting. What started as a question or a challenge to Catholic teachings turned into a massively popular protestant movement in Christianity. The Scientific Revolution Galileo Galilei (1564-1642) was one of the greatest scientists of the era of Scientific Revolution. He was an Italian physicist, mathematician, philosopher and astronomer. Galileo constructed his famous theory of heliocentrism which apparently was contradictory to Biblical Scriptures. According to this theory, Galileo correctly demonstrated the orbit of the solar system by stating that the sun is the center around which the earth and other planets revolve. This didn’t sit well with the Catholic Church who blamed Galileo of committing heresy as his theory was against the reading of the scriptures; â€Å"The world is firmly established, it cannot be moved† (1 Chronicles 16:30). Galileo faced a trial on suspicion of heresy. Today he is recognized as the iconic challenger in the face of Church’s Authority. In 1633, Galileo faced his trial which found him guilty of heresy. He was imprisoned and his famous work on solar system orbits, Dialogue was banned.

Reasons for Travel Essay Example for Free

Reasons for Travel Essay More and more people all over the world prefer to spend their holidays travelling. Rich or poor, old or young, they strive to leave the place where they live or work, and move to another spot of our planet — at least for two to four weeks a year. They travel to cities and towns, mountains and lakes, across oceans and seas. There are several reasons why the popularity of travel and tourism is growing. The first one is educational value of travelling. Man has always moved from one place to another in search of knowledge. Even now, in the age of technology and global communication, travelling to a different region or country can help discover new ideas, technologies and inventions. The best way of studying geography is travelling, the best way to help you master a foreign language is travelling, too. Moreover, a special kind of tourism has developed, called educational tourism when people travel to study a foreign language or to take up a course in one or several other subjects. The second reason why people travel is entertainment and rest. To see great buildings and natural wonders, listen to national music, get some knowledge about traditions and ways of different countries, taste new cuisine gives us new emotions, helps drive away the stress. Thirdly, there are special reasons. A lot of new types of travelling have appeared like ecotourism, educational tourism, sports tourism. A lot of people travel on business. Now, with the growth of international trade people have a lot of chances to do business with foreign partners, and they do it willingly. Some people have additional reasons to like travelling. When travelling, they do sports, or take care of their health, or get new contacts.

A Little Piece of Heaven Essay -- Descriptive Essay Examples

A Little Piece of Heaven Teacher's Comment: This essay, which received an "A," is strong because the vivid and careful descriptions enliven a commonplace topic: a simple walk in the woods. The writer creates an effective persona though a flashback to "a corner of captured memories." The writer recreates the child's persona, questioning and then understanding the motives of the counselor who is leading the group. Whenever I am troubled or confused, I always plunder through my mind to a corner of captured memories in my childhood. Here in this corner, tucked safely away from all of the mundane facts and figures, is a place I once visited as a small child. This spot has never failed to create a wealth of wonder and serenity for me. Join me now as I take a journey back through time. The day was an unbelievably hot one for the usual cool summers that North Carolina is so popular for. The dusty, faded-blue van in which my journey began was tightly packed with many eager and sweaty children, all of whom were anxious with anticipation of our day's fieldtrip. We were on our way to a particular stream that is well known in the small town o f Chesswood, which is located deep in the heart of the Carolina mountains. Upon finally reaching our destination, we unglued ourselves from the hot, vinyl seats and tumbled out of the van. I had become extremely hot and my skin was sticky from the sweat and dirt of afternoon play. The camp counselor called for all fourteen of us to line up in single-file and follow her lead. I did not understand why we had to do this, but as we travele d down the path that led to the stream, I began to understand her reasoning. The descending path was very steep and narrow. The sand and un... ... in the air. Among this kaleidoscope of brilliant hues appeared tangy-oranges, fiery-reds, crystal-clear blues, and fresh-greens. As the day came close to an end, the setting sun also displayed a work of art through this tree. The light, that bravely beamed through the V, created a heavenly aura. I felt as if angels were gracefully dancing on the colored droplets of water. The colors had changed somewhat and consisted mostly of a transparent-white mixed with brilliant oranges and yellows. Soon afterward, my little piece of Heaven was broken up by the shouts of our counselor telling us it was time to return to camp. I know I had to physically leave that stream back then, but it has never left my memory. Even though you did not visit that place, it is now in your memory also. I hope you have enjoyed experiencing this little piece of Heaven with me.

Cleverness: Woman and Virtue Essay

he association of intelligence or cleverness with virtuous or admirable women had begun as early in the querelle as 1558 4, with Marguerite de Navarre’s collection of short stories now called the Heptameron. Many of the stories, such as Story 2, Day 1 (â€Å"The Mule-Driver’s Wife†) and Story 2, Day 3 (â€Å"Sister Marie and the Prior†) are simple examples of feminine virtue as chastity, but several stories present this virtue in combination with wit. At this point it is still quite early in the querelle, too early for any reshaping of the feminine ideal. Accordingly, Mme. de Navarre seems to view intelligence as more of an asset than a virtue, providing a woman with two advantages: the ability to defend her own chastity, and to avenge herself of wrongs done her. Story 5, Day 1 is an example of the first: a poor boatwoman â€Å"as virtuous as she was clever† outwits two lecherous friars, leaving them stranded on different small islands in the middle of the river. They entreat her not to thus put them to shame. â€Å"I should be doubly foolish if, after escaping out of your hands, I were to put myself into them again,† she replies. â€Å"Wait now, sirs, till the angel of God comes to console you; for you shall have nought that could please you from me to-day. â€Å"5 Her cleverness, which the male narrator-character makes certain to point out as one of her good qualities, allows her to singlehandedly preserve her chastity. This conventional view of feminine virtue is made progressive by the heroine’s intelligence. Most virtuous female characters would have either had to rely on the protection of men, or have suffered the fate of the mule-driver’s wife: death rather than disgrace. The boatwoman’s intelligence gives her power. The second advantage of intelligence has a much more shocking example: that of Story 8, Day 1. In this tale, the â€Å"upright and virtuous wife† of a man named Bornet learns that her husband and his friend are sharing a plot to sleep with her maid-servant. Rather than simply prevent this from happening, the wife substitutes herself for the servant, â€Å"not in the manner of a wife, but after the fashion of a frightened maid. This she did so well that her husband suspected nothing. â€Å"6 When the friend’s turn comes, he takes her ring as a keepsake, and the ring alerts the husband to the deception. The cuckoldry is considered just recompense for his wrongs, while the (adulterous! ) wife is still considered virtuous enough to give her husband a lofty lecture on marital faithfulness. Here, the wife’s cleverness gives her astonishing power over her husband, and because she uses it to avenge his straying eye, it renders her own actions blameless. Surely, without the supplementary virtue of intelligence, this wife would not have any virtue left at all according to the standards of the time. Intelligence may not yet be enough of a virtue to create a virtuous woman in the world of the Heptameron, but like a virtue it already covers a multitude of sins. The fact that these tales and those like them are intermixed with less pro-woman stories of vulgar or stupid women does not render them any less of a defense of women’s virtue and intelligence; rather, it is only a consequence of Mme. de Navarre’s intention to create a wholly realistic environment. In order to depict the diverse group storytellers recounting their various views on â€Å"the ill-turns which have been done by Women to Men and by Men to Women,† she had to give both sides equal voice, as would only be natural in a two-sided exchange. The presence of the stories praising intelligence in women remains significant and a mark of progress. But although these heroines’ intelligence is undoubtedly a benefit to them and an estimable attribute of their characters, it does not afford them the full dignity of a true virtue. It enables escape by deception and revenge by trickery; it serves as a plot device; it provides humor. It is wit or cleverness, rather than wisdom or genius; and unlike chastity, it is not sacred or worth dying for. Intelligence as a genuine virtue would have to wait for later works of the querelle des femmes.