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Famous Mathematician: John Wallis

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    Introduction

    The paper focuses on the life and works of English Mathematician John Wallis, who is considered to be the most important and influential mathematician before Sir Isaac Newton (http://www.britannica.com/eb/article-9075990/John-Wallis). In addition to being a mathematician, he was a considered as an eminent scientist in many field such as divinity, due to his adopted profession as a priest; cryptography, which was also one of his professions in the English Civil War; mechanics; music; philology etc (Scott 1).

    Background

    John Wallis was born on 23rd November 1616 at Ashford in East Kent. His father was Reverend John Wallis who was a rector in Ashford. John Wallis started his schooling in Ashford, but had to leave three years later due to the outbreak of Plague (Scott 2). He then attended the James Movat’s grammar school in Tenterden, Kent, and was later sent to Martin Holbeach’s school in Felsted, Essex (O’Connor & Robertson 6). In the former he became proficient in grammar, while in the later he became very good in languages like Greek, Hebrew and Latin. At the age of fifteen, when he returned home for his Christmas holidays, he came upon a book of arithmetic and was immediately interested in it. He mastered the book within a fortnight with the help of his brother starting his genius streak. But in those days mathematics was not considered to be a scholarly pursuit, so later he was sent to Emmanuel College, Cambridge to study medicine, as his family wished him to be a doctor (Rouse Ball 1).

    Though his interest in the field of medicine was minimal, he continued to study anatomy and became the first person to publicly defend the theory of circulation of blood which was proposed by his teacher Francis Glisson. He took a range of topics in the college such as ethics, metaphysics, geography, astronomy, medicine and anatomy; and graduated with a degree in arts in the year 1637, since there was no one in the college to guide him in mathematics, and continued his studies till he received the Master’s Degree in the year 1640 (Scott 4). In this year he was ordained by the bishop of Winchester and appointed chaplain to Sir Richard Darley at Butterworth in Yorkshire. Between 1642 and 1644 he was chaplain at Hedingham, Essex and in London and was later accepted as a fellow of the Queen’s College, Cambridge (O’Connor & Robertson 7). Here, Wallis discovered his skills with cryptography and used them to decode the letters of the Royalists for Parliamentarians. This was the time of the Civil war between both these parties. In the year 1643, he was given charge of the church of St Gabriel in Fenchurch Street, London, for his services to the Parliamentarians. In 1945 he married Susanna Glyde, vacated his fellowship and returned to London where he attended scientific meetings weekly with a group of scientists.

    This group later evolved into the establishment of the Royal Society of London (O’Connor & Robertson 8). In 1948 he earned the anger of Parliamentarians when he opposed to the execution of Charles 1. Inspite of this, he was appointed Savilian Professor of Geometry at Oxford, where is took the degree of D.D in 1649. He held the Savilian chair for over 50 years until his death on 28th October 1703 (Rouse Ball 2). Besides his scientific pursuits, he was also wrote papers on theology, logic, and philosophy and even invented the first system for teaching deaf-mutes.

    Contributions in Mathematics

    During his tenure in Oxford, John Wallis became known as a leading mathematician of his times. He worked extensively in the field of geometry and laid the foundation for calculus (Taylor 2). He first made a study of the previous works by many scientists like Kepler, Cavalieri, Roberval, Toricelli, Descartes etc., and later wrote his ideas on the field of calculus.

    The first point of starting was due to the subject of Indivisibles, which was a popular subject at those times. He went through the works of Torricelli that used Cavareli’s methods and found that he could find a means by which the quadrature of the circle would be possible, and started working on the problem. This laid the foundation of his most famous work of Arithmetica Infinitorium which was published in 1656. The Arithmetica Infinitorium rs related to the quadrature of curves. Wallis used the principles of Analytical Geometry by Descartes to perform the operations (Scott 14). He established the formula

    π/2 = (2.2.4.4.6.6.8.8.10..) / (1.3.3.5.5.7.7.9.9…)                  (Rouse Ball 27)

    The above result was discovered, when Wallis was Wallis discovered this result when he was attempting to compute the integral of (1 – x2)1/2 from 0 to 1 and hence to find the area of a circle of unit radius. He solved this by using the Kepler’s Law of Continuity to evaluate the integral, even including fractional and negative indexes. These were not possible using Cavalier’s method of indivisibles. For this purpose, he introduced the word interpolation in this work, which was used to deal with the fractional indexes.

    This method for evaluating integrals was used by Newton in his work on the Binomial theorem (Rouse Ball 28). Many scientists like Huygens, Hobbes, Fermat etc., who had initially read the book Arithmetic Infinitorium challenged Wallis’s argument especially his usage of indivisibles and induction and his assumption of a range of continuous and definable values between numbers of a sequence. Nevertheless this method of indivisibles or using infinitely small quantities was increasingly being used amongst his contemporaries. This was indeed the basis of Integral Calculus (Stedall 2004).

    A lesser known but equally important work of his was the Tract on Conic Sections, published in the year 1655, where Wallis described the curves which could be obtained as  cross sections by cutting a cone with a plane as properties of algebraic coordinates. This was where he had first used the concepts of Descartes on the Analytic Geometry (Rouse Ball 30).

    In 1659 Wallis published a tract containing the solution of the problems on the cycloid which had been proposed by Pascal. Here he explained all the principles which were laid down in his earlier book, and he also explained how this could be used for the rectification of algebraic curves. He gave a solution of the problem to rectify the semi-cubical parabola x³ = ay² which had been discovered in 1657 by his pupil William Neil (Rouse Ball 33)

    In addition to all this, Wallis is also considered to be a very important historian of mathematics which probably was due to his immense interest in the subject. He used his knowledge of Greek to translate ancient Greek Texts such as Ptolmey’s Harmonics, aristarchus’ On the Magnitudes and distances of the Sun and Archimedes’s Sand-Reckoner. His Treatise on Algebra gave a wealth of historical material and was also very important because the book is the first which accepts the existence of accepts negative roots and complex roots. He showed that a3- 7a = 6 has exactly three roots and that they are all real (O’Connor & Robertson 18)

                                                               References

    Books

    Scott J F, “The Mathematical Work of John Wallis, D.D., F.R.S., (1616-1703)”, 1981,

    American Mathematical Society, ISBN 0828403147

    Wallis J, Stedall J A (Translator), “The Arithmetic of Infinitesimals”, 2004, Springer

    Websites

    http://www.britannica.com/eb/article-9075990/John-Wallis

    O’Connor J J, Robertson E F, “John Willis”, February 2002,

    http://www-history.mcs.st-andrews.ac.uk/Biographies/Wallis.html

    Rouse Ball W W. “John Willis 1616 -1703”, Short Account of the History of

    Mathematics’ (4th edition, 1908), http://www.maths.tcd.ie/pub/HistMath/People/Wallis/RouseBall/RB_Wallis.html

    Taylor P, “John Wallis and the Roof of the Sheldonian Theatre – Structural Engineering

                in Oxford in the 17th Century”, 2005, Society of Oxford University Engineers,

                http://www.soue.org.uk/souenews/issue4/wallis.html

     

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