John Wallis

By: Jake Arends

# To Infinity and Beyond!

### Credited with the invention and introduction of the infinity symbol (∞), as well as several other formulas John Wallis is given "partial credit for the development of modern claculus."

(wikipedia)

John Wallis, born in Ashford Kent on November 23, 1616, is said to have been “the most influential English mathematician before Isaac Newton.” (Britannica) It was not until later in his life, however, that he pursued the scholarly interests of math. Wallis’ education originally directed him towards a career as a doctor. Wallis switched interests after beginning his studies at Emmanuel College in Cambridge, and obtained his Bachelor of Arts in 1637. In 1640 he obtained his Masters.

Throughout his education Wallis remained in touch with his mathematical interests. In 1640 he was ordained as a priest, but exhibited great mathematical skill by deciphering cryptic royalist dispatches intercepted by the parliamentarians. In 1645 Wallis married and was forced to resign form his position as a priest. Wallis then moved to England where in “1647 his serious interest in mathematics began when he read William Oughtred's Clavis Mathematicae (“The Keys to Mathematics”).” (Britannica). In 1649 Wallis was appointed Savilian Professor of Geometry at the University of Oxford. He held this post until his death on October 28, 1703.

It was during his time at Oxford that Wallis worked on a majority of his mathematical theories and formulas. Wallis not only brought a new approach to the theory of quadratures in his Arithmetica Infinitorum (1655) (Arithmetic of Infinities). He also developed the symbol for infinity, and was the first to demonstrate the uses of exponents, by use of his negative and fractional exponents. In 1657 Wallis published a second work, the Mathesis Universalis (“Universal Mathematics”), on algebra, arithmetic, and geometry, in which he further developed notation. (Brittanica). Wallis also published papers the papers Tractatus de Sectionibus Conicis (1659; “Tract on Conic Sections”), and Treatise on Algebra, in which he studies equations which he applied to the properties of conoids. It was in this work that he anticipated the concept of complex numbers. His most recognized theory, however, is what is known as the “Wallis Product” Wallis expressed the area under a curve as the sum of an infinite series and used clever and unrigorous inductions to determine its value. To calculate the area under the parabola,

(1)
\begin{align} \int^1_0 x^2\,dx \end{align}

Wallis considered the successive sums

(2)
\begin{align} \frac{0 + 1}{1 + 1} = \frac{1}{3} + \frac{1}{6} \end{align}
(3)
\begin{align} \frac{0 + 1 + 4}{4 + 4 + 4} = \frac{1}{3} + \frac{1}{2} \end{align}
(4)
\begin{align} \frac{0 + 1 + 4 +9}{9 + 9 + 9 + 9} = \frac{1}{3} + \frac{1}{18} \end{align}

and infered by "induction" the general relation

(5)
\begin{align} \frac{0^2 + 1^2 + 2^2 + ... + n^2}{n^2 + n^2 + n^2 + ... + n^2} = \frac{1}{3} + \frac{1}{6n} \end{align}

By letting the number of terms be infinite, he obtained 1/3 as the limiting value of the expression. With more complicated curves he achieved very impressive results, including the infinite expression now known as Wallis' product:

(6)
\begin{align} \frac{4}{\pi} = \frac{3}{2} * \frac{3}{4} * \frac{5}{4} * \frac{5}{6} * \frac{7}{6} * ... \end{align}

Wallis' many findings have led to the development of modern calculus. Even Isaac Newton learned from and developed Wallis' formulae. Newton credits his work on binomials and calculus to his research of Wallis' paper the Arithmetica Infinitorum.

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