John Wallis: a Man before his Time

Stipple engraving of John Wallis by R. Cooper via Wikimedia Commons

John Wallis was born to Joanna Chapman and Reverend John Wallis in Ashford, Kent in 1616. He was the 3rd of 5 children but was primarily raised by his mother, because his father died when he was 6 years old. Later he moved to Tenterden, Kent with his mother after there was an outbreak of the plague in his home town. In Tenterden he attended the James Movat’s grammar school and here he showed his true potential. When he turned 13 he felt that he was ready for the university and within a year he was attending the Martin Holbeach’s school located in Felsted. He took many classes and learned several languages yet mathematics became his passion. Hs family’s plan was for him to become a doctor, but one night he found an arithmetic book, and over a few weeks he mastered it with his brother’s help. This was his first step towards becoming one of the most valuable mathematicians of the time, and along the path of discovering many things so revolutionary they feel as if they are before their time.

Later in his life Wallis attended Emmanuel College Cambridge where finally no one could disrupt his mathematics studies. He studied many other subjects including astronomy, medicine and cryptography. Unlike his family’s plan he never wished to become a doctor, but he did spend time studying medicine. It is worth noting that he was the first person to defend Francis Glisson’s theory on blood circulation. After graduating with a Bachelor of Arts degree in 1637 he followed with his masters in 1640. He was later given a fellowship at Queens’ College, Cambridge, and here he took orders and assisted in deciphering royalist messages. He was against the execution of Charles I as he felt that it was lasting hatred from the Independents. During the civil war he was a great aid for the Parliamentarians, and because of this the Parliamentarians placed him in charge of the Church of St Gabriel in Fenchurch Street. During the same year his mother died and he was given the family’s estate. Not only being a genius but also applying this to aid his country captivates my interests as he was a dedicated man. Being a man of brilliance Wallis frequently did large mental calculations. This is due to his insomnia which kept him awake for many long hours. He occupied them by calculating numbers while lying in his bed, and not only honed his skill, but also his memory as he would recite them the next morning. One of his greatest mental math problems was when he not only calculated the square root of a 53 digit number. I personally find this to be fascination, as he not only revolutionized many fields but he also was so dedicated that all of his time was spent on mathematics. His feats of mathematics alone bring him above his peers but this dedication and knowledge furthers his status as being beyond his time.

He later married Susanna Glyde in March of 1645, but in doing so he forfeited his fellowship at Queen’s College. From here he traveled back to London and started small group meetings of scientists. This would later grow into the Royal Society of London. His true passion for mathematics was revealed during the Society meetings. This was sparked when Oughtred’s Clavis Mathematicae was read out loud. Wallis mastered the book in a matter of weeks and went on to write his own book. His book, Treatise of Angular Sections, went unpublished for 40 years. He also developed solutions to 4th degree equations. These solutions were similar to those of Thomas Harriot’s, another mathematician of the time who graduated from Oxford. Wallis was appointed to be the chair of Geometry at Oxford in 1649 where he remained for the remainder of his life.

The first page of the Arithmetica Infinitorum, 1656, via Wikimedia Commons.

The first page of the Arithmetica Infinitorum, 1656, via Wikimedia Commons.

During his lifetime he produced many important mathematical works. These include his Arithmetica Infinitorum, a book, and the method of indivisibles. Arithmetica Infinitorum was published in 1656 and extends the methods of analysis used by Descartes and Cavalieri. It soon became the standard book being used to teach due to how it expanded the field, and is recognized as his most important work. It focuses on conic sections and states that all planes made from them can be represented with algebraic coordinates as well as featuring Wallis’ work on integral calculus, solving for several integrals such as x-1 and xn. The book also gives an accurate number for π, and it does this by representing π by a series and then solving. He was also the first to use the principle of interpolation. This principle constructs new points of data inside of the range of discrete known points in order to achieve a solution. This method was used through the 17th century by mathematicians and is still used in engineering today. It was in this book that the symbol ∞ was used for infinity was first seen. Wallis selected this because its curvature can be traced out infinitely many times. ∞ is a symbol that was derived from the Ouroboros, or the snake biting its own tail, which represented endlessness. Wallis’s greatest work impacted his future works as well as others. In 1659 Wallis used its principles in his work on cycloids. This idea was previously proposed by Pascal but Wallis was the one to reach a solution. Consequently in doing this he also applied his principles to algebraic curves. This allowed for a solution to a semi-cubical parabola, something that had troubled previous mathematicians, this solution was found by William Neil. The Arithmetica Infinitorum alone is a work of greatness, as it allowed for several of the most troubling problems to be solved. There are few others to compare it to that had such an impact. It revolutionized the field as it was truly before its time.

In his lifetime Wallis also published another book titled Treatise on Algebra. Here he demonstrates that negative roots are complex numbers and that it is possible to factor a polynomial into roots featuring complex numbers. It also challenges Descartes’ rule of signs, and because of this Wallis received much criticism from the mathematical community. Wallis’s view on negative numbers was different than what we except today, as he believed that they were less than nothing. This did not hamper his works as he demonstrated complex knowledge of them.

Wallis published one final book in his life, called Algebra. Well known to be ahead of its time, it features the systematic use of formulae. Wallis uses this to analyze the space that a particle moving at a constant velocity is at any time. He used the ratio of space to the length rather than what many previous mathematicians, and this new revolutionary idea opened up many possibilities for solving problems. This book also featured a second edition titled Opera that was notably expanded from the first, as it featured many more examples. The amount of work that went into all of the books that Wallis published along with the concepts inside bewilders me, and shows his dedication to mathematics.

Wallis also did work in the field of physics. When the theory of collision of bodies was proposed to the Royal Society, Wallis alongside other mathematicians was tasked in sending feedback and similar solutions in order to support the theory. This theory went on to become what is called conservation of momentum. Wallis was the only one to consider a situation other than just perfect elastic bodies, doing work with imperfect ones as well. Later on he made other contributions to physics such as his work with center of gravity and dynamics. Wallis was a fantastic mathematician and a brilliant mind. Wallis is remembered for his many contributions in mathematics and physics and rightfully deserves to be remembered.