A History of Pi by Petr Beckmann (PDF)

Description

The history of pi, says the author, though a small part of the history of mathematics, is nevertheless a mirror of the history of man. Petr Beckmann holds up this mirror, giving the background of the times when pi made progress — and also when it did not, because science was being stifled by militarism or religious fanaticism.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐It seems that by 2000 BCE, the significance of PI was appreciated, and a rough approximation was known. It appears that the Babylonians and the Egyptians were aware of its existence and significance. The symbol for PI was not generally used until the 18th century CE. Incidentally, the equal sign did not come into general use until the 16th century. Early on, it was the priests in Babylon, Egypt, and Maya societies that were the learned ones – the astronomers, calendar makers, and timekeepers. Interestingly, the Mayans “could out-calculate the Egyptians, the Babylonians, the Greeks, and all Europeans up to the Renaissance,” with their vigesimal, positional notation system.An entire chapter is spent on the early Greeks and their contributions. The philosopher Antiphon enunciated the “principle of exhaustion,” which had a profound influence on the quest to determine a more accurate value of PI. We also learn of the contributions of Hippocrates and his use of reductio ad absurdum (reduction to the absurd). There is also the contributions of Hippias of Elis, Eratosthenes, and Euclid. Following the Greeks were the Romans, whom the author characterizes as thugs. Their contribution to science, according to the author, was “mostly limited to butchering antiquity’s greatest mathematician, burning the Library of Alexandria, and slowly stifling the sciences that flourished in the colonies of their Empire.” It was a Roman soldier that killed the great thinker Archimedes. How sad.Born in 287 BCE, is the great Archimedes of Syracuse. He was the first to provide a method of calculating PI to any desired degree of accuracy. His polygonal method remained unsurpassed until 19 centuries later. I wanted to mention here, that the author provides a bit of mathematical detail throughout this chapter and the rest of the book, which one might find difficult to follow. Fortunately, it is not necessary to understand the details in order to appreciate the developments of these great thinkers.We segue now into the middle ages. Unfortunately, we see no significant progress in the method of determining PI until Viète (1540-1603). Other notables that contributed to the value of PI were Fibonacci (1180-1250), and Christiaan Huygens (1629-1693). Unfortunately, throughout history we have too many instances of the destruction of scientific thought. The torch of Alexandria was extinguished by militaristic Rome, the intellectual life in Babylon was wiped out by the militaristic Assyrians, and the golden age of Muslim science was stifled by the militaristic Turks, according to the author. We have similar stories from India and China.By the time of the Renaissance, we see ever increasing accuracy of the numerical value of this constant. Trigonometric functions and logarithms have come to the rescue. At this time, Viète was the first to represent PI by an analytical expression of an infinite sequence of algebraic operations. In time, we see people trying to calculate PI to an ever-increasing number of decimal places. This is interesting since the digits beyond the first few decimal places are of no practical scientific value. We learn of further developments revolving around PI with Leibniz, Newton, and the development of calculus. Interestingly, it was in Newton’s lifetime that the circle ratio was first denoted by PI. Later, Euler developed formulas for PI by the truckload. The author presents a bit of math in these chapters showing us just how some of these formulas were conceived, but don’t worry, you can understand the text without comprehending all the derivations. In the 1700’s, the irrationality of PI was established by Johann Lambert and Adrien-Marie Legendre. By 1840, the existence of transcendental numbers is proven, and we find out that PI is just such a number, this being proven by Lindemann in 1882. This had important implications as it showed that the possibility of squaring the circle by Euclidean construction was impossible – not that that stopped people from trying.The author concludes saying, “The history of PI is only a small part of the history of mathematics, which itself is but a mirror of the history of man. That history is full of patterns and tendencies whose frequency and similarity are too striking to be dismissed as accidental. Like the laws of quantum mechanics, and in the final analysis, of all nature, the laws of history are evidently statistical in character.” The conclusion to an excellent book, I thought.

⭐A review of the Book: (a history of) Pi by Gerald T. WestbrookBackground. The story of Pi or II is a remarkable story. I have written, privately, about Pi and used it in the teaching of two Hispanic youths. I expect to mentor additional youths in the future. As I mentor, in each case it has been explained that this material may not be an immediate fit to their current mathematical situation. It is not designed for that. Rather it is aimed at depicting some of the more interesting and intriguing aspects of this field. It is aimed at illustrating that there can indeed be a “Joy of Mathematics” in a field where one might think there can never be any joy. It is also aimed at stimulating ones interest in this subject, that might lead to an interest in such fields as engineering, science, business, insurance, statistics and economics. Several subjects will be covered* Infinite Series. Two types are noted. (1) Diverging (2) Converging* Arithmetic Convention – The use of repeated dots, such as …. at the end of a number, means that the digits continue on and on randomly. Example: ‘ = 3.141592….to six decimal places. The front cover of this book shows it out to over 100 places. And a table after the index shows it for 10,000 decimal places. It noted it was calculated in July of 1961 via the formula (page 184, 185):Pi = 24*arctan(1/8) + 8*arctan(1/57) + 4*arctan(!/239).Now this equation looks rather interesting, but I will not dig into it’s derivation, but it does give a history of the calculation of Pi to more and moredecimal places.* July 1961 to 100,265 places.* February 1966 to 250,000 places.* February 1967 to 500,000 places.This book was printed in 1971, so any additional milestones are not shown. However a Google search indicates, as of January 6, 22010 at 2.7* 1012 , orDerivation of Pi Definition: The ratio of the circumference (C) of a circle to it’s diameter (D) is a constant. Hence for any circle, no matter how big,C/D = a constant = 3.14159…..At one time politicians in Indiana tried to work up a law to state thatC/D = 3.0. They failed. I would suggest that if one asked many high school students why Pi is the value it is, about 99% would not be able to answer. One might ask how does one prove this law. Rather than using the high level of mathematics covered in the above book, I prefer the following method. The answer can be seen in a series of circles that just enclose geometrical objects called polygons, with n sides. Four example four polygons follow:n = 3 Trianglen = 4 Squaren = 5 Pentagonn = 6 Hexagon As n gets larger and larger, the sum of the sides of these polygons become an approximation of the circumference. Trigonometry provides an equation for the Length (L) of one side of a polygon of order n, circumscribed by a circle of diameter (D), namely:D = L / Sin(180/n) or L = D * Sin(180/n) or L/D = Sin(180/n).The sum (S) of all sides of the polygon, of order n, becomes S = n * LAnd C is the limit, as n goes to infinity, of S = n * LSince D = L / Sin(180/n or L = D*Sin(180/n)or S = n * D * Sin(180/n)and S/D = n * Sin(180/n)Hence C is the Limit, as n goes to infinity, of n * Lor of n * D * Sin(180/n)Examine the table belown 180/n L/D = Sin(180/n) S/D = sum of all sides3 60 0.874 3 * 0.874 = 2.6224 45 0.707 4 * 0.707 = 2.8285 36 0.588 5 * 0.588 = 2.9406 30 0.500 6 * 0.500 = 3.0009 20 0.342 9 * 0.342 = 3.07818 10 0.174 18 * 0.174 = 3.132180 1 0.0175 180 * 0.01745 = 3.141Infinity 0 In the limit S/D = C/D = 3.14159…. = Pi

⭐Enjoyed reading. Interesting history.

⭐Petr Beckmann presents the history of Pi in an interesting and very detailed manner. He does (intentionally) veer off the subject, but only to help the story. He sometimes presents non-Pi related details about the several key people who were involved in creating better ways to calculate Pi, but this extra information about these people only enhances the story.The math used to explain some of the breakthroughs related to calculating Pi is often very heavy and not for everyone. However, as the author explains, readers can simply skip through the math and focus on the bigger story. For this reason, the high-level math does not distract from the book.The book was written before the advent of personal computers (and well before cell phones and tablets) so the final chapters, which get into the use of computers to calculate Pi, are dated. But, the rest of the history presented in the book is quite interesting.

⭐This is a fascinating insight into the history of this strange number. Starting with how the ancients worked out how to use Pi to computers calculating Pi to huge numbers of decimal places covering the many ways developed to calculate Pi down the centuries all explained in an easy to understand way. The book is not right up to date as it was written many years ago, but that does not detract from it’s value.The author also has a sense of humour that shows itself in the many digs he has at some of he famous names of the past who he feels were perhaps not as clever as we thought!

⭐Item as described and on promised time

⭐Everyone with interests in math and history should read this book.The author has a sharp vision on things and I like that.It was fun to read, it even taught me some things about Pi I did not know.Great!

⭐Very good.

⭐The thing I really liked about the book was the quite personal style of the author. From the Early Greeks to the computer age he gives a good historical example of how PI has been derived.

Keywords

Free Download A History of Pi in PDF format
A History of Pi PDF Free Download
Download A History of Pi 2015 PDF Free
A History of Pi 2015 PDF Free Download
Download A History of Pi PDF
Free Download Ebook A History of Pi