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3.1415926535897932384626433832795028841971693993

William Shanks, a thirtyfive year old amateur mathematician from Corsenside, came to settle in HoughtonleSpring in 1847 with his new wife Jane Elizabeth, at about the same time that Houghton got its new Rector, the Hon & Rev John Grey.
William became the Master of a private boarding school in Nesham Place, and would routinely spend hours on a morning calculating and expanding the value of Pi, before checking his calculations on an afternoon.
Pi is a mathematical constant whose value is the ratio of any circle's circumference to its diameter, and many of us will only ever have used this during Maths lessons when at school (area of a circle = Pi r squared). Article Copyright © Books of the North 2009.
William continued with his calculations and managed to expand the calculation of the decimal of Pi up to 607 places. In the early months of 1853 William published a book entitled ‘Contributions to mathematics, comprising chiefly the rectification of the circle to 607 decimals etc’. The book contained a list of many subscribers, three of which were local: Rector John Grey and M.A.C of HoughtonleSpring subscribed a copy each, as did Nathaniel Ellison Esq of Morton House. Article Copyright © Books of the North 2009.
William’s true fame arose in 1873 when he calculated Pi to 707 places. This was to be the longest expansion of Pi for over a hundred years, and was only bettered by a computer in the 1970s.
William Shanks died at the age of 70 years, and was buried at Houghton Hillside Cemetery on June 17th 1882. His wife joined him there on October 26th 1904. The impressive headstone is thought to have been bulldozed during the cemetery clearance of 1973/4. Article Copyright © Books of the North 2009.
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1812 – William Shanks was born at Corsenside, Northumberland, on January 25th January 1812. Article Copyright © Books of the North 2009.
1846 – William married Jane Elizabeth Pringle in the second quarter of 1846 in the City of London. Elizabeth was born in Westminster, London in 1815.
1847 – William moved to HoughtonleSpring.
1851 – William, aged 38 years, lived at Quality Hill, HoughtonleSpring. His occupation was Classical & Mathematical Teacher. Also living with William was his wife, Jane Elizabeth, aged 36 years, and his widowed motherinlaw, Sarah Pringle, aged 65 years. A servant, Jane Calbreath, was also at the premises.
1853 –William published a book entitled ‘Contributions to mathematics, comprising chiefly the rectification of the circle to 607 decimals etc’. He had managed to expand the calculation of the decimal of Pi up to 607 places.
1861 – William lived at Nesham Place and was School Master of a Private Boarding School. Also living in the premises were: His wife, Jane Elizabeth, aged 46 years; Sarah Pringle, mother in law, aged 75 years; William Routledge, aged 18 years, Assistant Master (English); Ann Oliver, Domestic Cook; Alice Oliver, Housemaid; and eleven pupils with the age range of 7 – 18 years. William’s twelveyearold nephew, John Shanks, was a pupil at this time.
1871  William and Elizabeth continued to live at Nesham Place as a School Master of the boarding school. Jane Monk was their domestic cook and Catherine Potts was their domestic servant. The school had eleven pupils with the age range of 9 – 14 years. These pupils were from all over the northeast including Yorkshire, Durham and Northumberland. Article Copyright © Books of the North 2009.
1873 – William calculated Pi to 707 places (later identified as being only correct to 527 places in 1944).
1881  69 year old William still lived at Nesham Place with his wife Jane Elizabeth, and their domestic servant Jane Shaw. Their nextdoor neighbours were George & Josephine Wheatley, the Houghton confectioners.
1882 – William died aged 70 years, and was buried at Houghton Hillside Cemetery on June 17th 1882.
1891 – Widowed Jane Elizabeth, aged 76, lived at the Manor House, Nesham Place. Living with her was her niece, Margaret E Pole, aged 44 years, who was born in Southampton. Both ladies were living off their own means, and could afford their domestic servant, Mary Robinson. Jane Elizabeth’s neighbours at this time were physicians, surgeons and general medical practitioners.
1901  Jane Elizabeth, aged 86 years, lived at 12 Nesham Place, with her niece Margaret (interestingly the new occupant of the Manor House is Dr James G Houseman, doctor of medicine). Their servant was Mary E Petch. Jane Elizabeth’s neighbours were now coal miners, hewers and agricultural labourers, quite a contrast to the life she had been used to among the well heeled. Article Copyright © Books of the North 2009.
1904  Jane Elizabeth died aged 89 years and was buried at Houghton Hillside Cemetery on October 26th 1904.
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This article is dedicated to the memory of William Shanks. With thanks to Linda Bromfield for help with genealogical research.
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PAGE UPDATED: 28/08/2012
William Shanks moved to HoughtonleSpring, a small town in County Durham, in 1847. In this coal mining area near Durham, England, he ran a boarding school but he used his leisure hours working on mathematics, particularly on calculating the decimal expansion of p. He was influenced to undertake this task by William Rutherford from Edinburgh. In 1853 Shanks published a book entitled Contributions to mathematics, comprising chiefly the rectification of the circle. In the same year William Rutherford gave 440 decimal places in the expansion of p and, later in the same year, Shanks, in a collaboration with Rutherford, gave 530 places. This was a busy year for Shanks, for also in 1853 he gave 607 decimal places in the expansion of p which had been independently checked as correct to the first 500 of those places. At this point Shanks rested in his calculations of the decimal expansion of p, but he continued to write mathematical works. Between 1854 and 1874 Shanks published nine mathematical memoirs in the Proceedings of the Royal Society of London. Shanks is famed for his calculation of p to 707 places in 1873, which, unfortunately, was only correct for the first 527 places. Despite these errors he did manage to correct some errors in the expansion of to 607 places which he had given twenty years earlier. The method Shanks used in his calculation was based on the formula p/4 = 4 tan1(1/5)  tan1(1/239) which had been discovered by Machin in 1706 and used by him to correctly calculate to 100 decimal places. Shanks also calculated e and Euler's constant ? to a great many decimal places. He published a table of primes up to 60,000, found the natural logarithms of 2, 3, 5 and 10 to 137 places, and the values of 212m+1 for m = 1, 2, 3, ..., 60. In 1944 D F Ferguson calculated p using the formula p/4 = 3 tan1(1/4) + tan1(1/20) + tan1(1/1985). He found that his value disagreed with that of Shanks in the 528th place. Ferguson discovered that Shanks had omitted two terms which caused his error. Of course calculating p to 707 places is now a trivial matter using a computer algebra package such as Maple. Shanks spent many long tedious days calculating; he would calculate new digits all morning and then he would spend all afternoon checking his morning's work. Now at a press of a button we get 707 places: 3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706 7982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381 9644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412 7372458700660631558817488152092096282925409171536436789259036001133053054882046652138414695194151160 9433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949 1298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051 3200056812714526356082778577134275778960917363717872146844090122495343014654958537105079227968925892 354201996 Article by: J J O'Connor and E F RobertsonFrom: http://www.absoluteastronomy.com/topics/William_Shanks William Shanks (January 25, 1812 – summer 1882, HoughtonleSpringHoughtonleSpring HoughtonleSpring is a town in the county of Tyne and Wear, North East England that has its recorded origins in Norman times. It is situated almost equidistant between the cathedral city of Durham 7 miles to the southwest and the City of Sunderland about 6 miles to the northeast.... , DurhamDurham Durham is a city in North East England. It lies at the heart of the City of Durham local government district. It is the county town of County Durham.... , EnglandEngland native_name =conventional_long_name = Englandcommon_name = Englandimage_flag = Flag of England.svgimage_coat = England COA.svgsymbol_type = Royal Coat of Arms... ) was a BritishUnited Kingdom of Great Britain and Ireland The United Kingdom of Great Britain and Ireland was the formal name and the state form of the United Kingdom from 1 January 1801 until 12 April 1927.... amateurAmateur An amateur is generally considered a person attached to a particular pursuit, study, or science, without formal training or pay. Conversely, an expert is generally considered a person with extensive knowledge, Aptitude, and/or training in a particular area of study, while a professional is someone who also makes a living from it.... mathematicianMathematician A mathematician is a person whose primary area of study and/or research is the field of mathematics.... . Shanks is famous for his calculation of pPi Pi or p is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean geometry; this is the same value as the ratio of a circle's area to the square of its radius.... to 607 places, accomplished in the year 1873, which, however, was only correct up to the first 527 places. This error was highlighted in 1944 by FergusonFerguson Ferguson may refer to:PeoplePlacesIn the United States:* Ferguson, Iowa* Ferguson, Kentucky* Ferguson, Missouri, a suburb of St.... (using a mechanical desk calculator). Shanks earned his living by owning a boarding schoolBoarding school A boarding school is a school where some or all pupils not only study, but also live during term time, with their fellow students and possibly teachers.... at HoughtonleSpringHoughtonleSpring HoughtonleSpring is a town in the county of Tyne and Wear, North East England that has its recorded origins in Norman times. It is situated almost equidistant between the cathedral city of Durham 7 miles to the southwest and the City of Sunderland about 6 miles to the northeast.... , which left him enough time to spend on his hobby of calculating mathematical constants. His routine was as follows: he would calculate new digits all morning; and then he would spend all afternoon checking his morning's work. To calculate p, Shanks used Machin's formulaJohn Machin John Machin, , a professor of astronomy at Gresham College, London, is best known for developing a quickly converging series for Pi in 1706 and using it to compute Pi to 100 decimal places.... : Shank's approximation was the longest expansion of p until the advent of the electronic digital computerComputer A computer is a machine that manipulates Data according to a list of Code .The first devices that resemble modern computers date to the mid20th century , although the computer concept and various machines similar to computers existed earlier.... about one century later. Shanks also calculated e and the EulerMascheroni constantEulerMascheroni constant The Euler?Mascheroni constant is a mathematical constant recurring in mathematical analysis and number theory, usually denoted by the lowercase Greek letter .... ? to many decimal places. He published a table of primePrime number In mathematics, a prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC.... s up to 60 000 and found the natural logarithmNatural logarithm The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e , where e is an irrational number constant approximately equal to 2.718281828.... s of 2, 3, 5 and 10 to 137 places. Shanks died in HoughtonleSpring in summer 1882, aged 70, and is buried in the local Hillside Cemetery.