The word literally means "rope-stretchers" or "rope-fasteners. A famous Egyptian piece of papyrus gives us another ancient estimation for pi. Chronologically, the next approximation of pi is found in the Old Testament. A fairly well known verse, 1 Kings , says: "Also he made a molten sea of ten cubits from brim to brim, round in compass, and five cubits the height thereof; and a line of thirty cubits did compass it round about" Blatner, Debates have raged on for centuries about this verse.
According to some it was just a simple approximation, while others say that " However, most mathematicians and scientists neglect a far more accurate approximation for pi that lies deep within the mathematical "code" of the Hebrew language. In Hebrew, each letter equals a certain number, and a word's "value" is equal to the sum of its letters. Interestingly enough, in 1 Kings , the word "line" is written Kuf Vov Heh, but the Heh does not need to be there, and is not pronounced. With the extra letter , the word has a value of , but without it, the value is The ratio of pi to 3 is very close to the ratio of to Tsaban, This figure is far more accurate than any other value that had been calculated up to that point, and would hold the record for the greatest number of correct digits for several hundred years afterwards.
When the Greeks took up the problem, they took two revolutionary steps to find pi. Antiphon and Bryson of Heraclea came up with the innovative idea of inscribing a polygon inside a circle, finding its area, and doubling the sides over and over. Later, Bryson also calculated the area of polygons circumscribing the circle. This was most likely the first time that a mathem atical result was determined through the use of upper and lower bounds. Unfortunately, the work boiled down to finding the areas of hundreds of tiny triangles, which was very complicated, so their work only resulted in a few digits.
Blatner, 16 At ap proximately the same time, Anaxagoras of Clazomenae started working on a problem that would not be conclusively solved for over years. After imprisonment for unlawful preaching, Anaxagoras passed his time attempting to square the circle. Cajori wri tes: "This is the first time, in the history of mathematics, that we find mention of the famous problem of the quadrature of the circle, the rock that upon which so many reputations have been destroyed Anaxagoras did not offer any solution of it, and seems to have luckily escaped paralogisms" Cajori Since that time, dozens of mathematicians would rack their brains trying to find a way to draw a square with equal area to a given circle; some would maintain that they had found methods to solve the problem, while others would argue that it was impossible.
The problem was finally laid to rest in the nineteenth century. The first man to really make an impact in the calculation of pi was the Greek, Archimedes of Syracuse. Where Antiphon and Bryson left off with their inscribed and circumscribed polygons, Archimedes took up the challenge. However, he used a slightly dif ferent method than they used. Archimedes focused on the polygons' perimeters as opposed to their areas, so that he approximated the circle's circumference instead of the area.
He started with an inscribed and a circumscribed hexagon, then doubled the si des four times to finish with two sided polygons. Archimedes, 92 His method was as follows This results in a decreasing sequence a1, a2, a Gradually, "the lead The earliest value of pi used in China was 3. Near the end of the 5th century, Tsu Ch'ung-chih and his son Tsu Keng-chih came up with astonishing results, when they calculated 3. His final result was that 3.
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Unfortunately, this equation is not too useful in calculating because it requires too many iterations before convergence, and the square roots become quite complicated. He did not even use his own formula in his calculation of pi. Beckmann, 92 Still, it was an innovative discovery that would open many doors in the future. In , Adrianus Romanus used a circumscribed polygon with sides to compute pi to 17 digits after the decimal, of which 15 were correct. O'Connor, 3 Just three years later, a German named Ludolph Van Ceulen presented 20 digits, using the Archimede an method with polygons with over million sides.
Van Ceulen spent a great part of his life hunting for pi, and by the time he died in , he had accurately found 35 digits. His accomplishments were considered so extraordinary that the digits were cut into his tombstone in St. Peter's Churchyard in Leyden. Still today, Germans refer to pi as the Ludolphian Number to honor the man who had such great perseverance. Cajori, It should be noted that up to this point, there was no symbol to signify the ratio of a circle's circumference to its diameter.
It was not immediately embraced, until , when Leonhard Euler began using the symbol pi; then it was quickly accepted. Cajori, In , John Wallis used a very complicated method to find another formula for pi. One source describes his method as "extremely difficult and complicated" Berggren, while another source says it is "remarkable" Cajori, Cajori, In , James Gregory wrote about a formula that can be used to calculate the angle given the tangent for angles up to 45 pi.
This elegant formula is one of the simplest ever discovered to calculate pi, but it is also fairly useless; terms of the series are required to get only 2 decimal places, and 10, terms are required for 4 decimal places. O'Connor, 3 To compute digits, "you would have to calculate more terms than there are particles in the univ erse" Blatner, However, this formula set the stage for a handful of other formulas that would be more effective. Sur ely, the 17th-century mathematicians were onto something.
It was just a matter of time until they discovered a formula that was even better. The world didn't have to wait too long, after all, before another formula was discovered. Blatner, 43 In fact, Machin took the initiative to calculate p i with his new formula, and computed places by hand.
Cajori, Over the next years, several men used the exact same formula to find more and more digits. In , an Englishman named William Shanks used the formula to calculate places of pi. Many years later, it was discovered that somewhere along the line, Shanks had omitted two terms, with the result that only the first digits were correct. Berggren, All these efforts, however, had not contributed to the solution of the ancient problem of the quadrature of the circle" Struik, The first step was taken by the Swiss mathematician Johann Heinrich Lambert when he proved the irrationality of pi first in and then in more detail in Cajori, Some people felt that his proof was not rigorous enough, but in , Adrien Marie Legendre gave ano ther proof that satisfied everyone.
Not long after this Jones became tutor to Philip Yorke, later 1st Earl of Hardwicke , who became lord chancellor and provided an invaluable source of introductions for his tutor. It was probably around that Jones first came to Isaac Newton's attention when he published Synopsis, in which he explained Newton's methods for calculus as well as other mathematical innovations. In Jones was able to acquire Collins's extensive library and archive, which contained several of Newton's letters and papers written in the s.
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These would prove of great interest to Jones and useful to his reputation. Born half a century apart, Collins and Jones never met, yet history will forever link them because of the library and mathematical archive that Collins started and Jones continued, arising from their shared passion for collecting books. The son of an impoverished minister, Collins was apprenticed to a bookseller. Essentially self-taught like Jones, he had also gone to sea and learned navigation. On his return to London he had earned his living as a teacher and an accountant. He held several increasingly lucrative posts and was adept at disentangling intricate accounts.
Collins's modest ambition had been to open a bookshop, but he was unable to accumulate enough capital.
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In , however, he was elected to the Royal Society of which he became an indispensable member, assisting the official secretary Henry Oldenburg on mathematical subjects. Collins corresponded with Newton and with many of the leading English and foreign mathematicians of the day, drafting mathematical notes on behalf of the Society. In spite of these he was turned down. However Jones's former pupil, Philip Yorke, had by now embarked on his legal career and introduced his tutor to Sir Thomas Parker , a successful lawyer who was on his way to becoming the next lord chief justice in the following year.
Jones joined his household and became tutor to his only son, George c. This was the start of his life-long connection with the Parker family. Around the time that Jones bought Collins's library and archive, Newton and the German mathematician Gottfried Leibniz were in dispute over who invented calculus first. In Collins's mathematical papers, Jones had found a transcript of one of Newton's earliest treatments of calculus, De Analyst , which in he arranged to have published. It had previously been circulated only privately.
President of the Royal Society since , Newton was reluctant to have his work published and jealously guarded his intellectual property. However, he recognised an ally in Jones. In Jones joined the committee set up by the Royal Society to determine priority for the invention of calculus. Jones made the Collins papers with Newton's correspondence on calculus available to the committee and the resulting report on the dispute, published later that year, Commercium Epistolicum , was based largely upon them.
Though anonymous, Commercium Epistolicum was edited by Newton himself and could hardly be viewed as impartial. Unsurprisingly it came down on Newton's side. Today it is considered that both Newton and Leibniz discovered calculus independently though Leibniz's notation is superior to Newton's and is the one now in common use.
By Jones was firmly positioned among the mathematical establishment. In his patron Sir Thomas Parker was made lord chancellor and in was ennobled as Earl of Macclesfield. Shirburn castle became a home too for Jones who was, by then, almost a family member.
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Besides the law, Parker had a scholarly interest in many subjects including science and mathematics and was a generous patron of the arts as well as the sciences. He was influential in the appointment of Halley as astronomer royal in But there was an obverse side to the first earl's character. It seems that together with his great abilities and ambition there was also a dangerous lust for wealth. He was accused of selling chancery masterships to the highest bidder and of allowing suitors' funds held in trust to be misused.
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Parker resigned as lord chancellor in but he was nevertheless impeached. Some of his assets were sold and his name was struck from the roll of privy councillors but he did not have to forfeit Shirburn which remains in the Macclesfield family to this day. Some dignity was restored when in he was one of the pallbearers at Newton's funeral. Thomas's son, George Parker, became an MP for Wallingford in and spent much of his time at Shirburn where, with Jones's guidance, he added to the library and archive that Jones had brought with him.
Pi Day: How did they first calculate pi?
George Parker developed an interest in astronomy and with the help of a friend, the astronomer James Bradley who became the third Astronomer Royal in on the death of Halley , he built an astronomical observatory at Shirburn. Among the many influential mathematicians, astronomers and natural philosophers he corresponded with was Roger Cotes , the first Plumian Professor of Astronomy at Cambridge and considered by many to be the most talented British mathematician of his generation after Newton.
He had been entrusted with the revisions for the publication of the second edition of Newton's Principia.