John Wallis

John Wallis was a central figure in the early modern mathematical revolution. He was an English clergyman and a highly influential mathematician. He is credited with helping to develop infinitesimal calculus. This branch of mathematics allows us to study continuous change. Wallis was a contemporary of Isaac Newton. His work helped bridge the gap between older geometric methods and modern analysis.

Wallis used a unique way to think about shapes and areas. He extended the method of indivisibles developed by Bonaventura Cavalieri. He imagined that a flat plane was made of an infinite number of parallel lines. He also viewed it as an infinite number of tiny parallelograms. To describe these tiny parts, he used the term infinitesimal. He represented this concept using the notation 1/∞. To represent the idea of something that goes on forever, he introduced the symbol ∞.

His mathematical work covered several distinct areas. In 1655, he published a treatise on conic sections. This was the first book to define these curves analytically as curves of the second degree. This work made the ideas of René Descartes much easier to understand. In 1656, he published his most important work, *Arithmetica Infinitorum*. This book focused on the analysis of infinite series and integration. He also contributed to trigonometry and the study of continued fractions.

Wallis's life was shaped by the political and social changes of 17th-century England. He was born in Ashford, Kent, as the third of five children. He moved to Tenterden in 1625 due to an outbreak of the plague. He studied at Emmanuel College, Cambridge, earning his B.A. in 1637 and his M.A. in 1640. During the English Civil War, he served as a chief cryptographer. He helped Parliament decipher Royalist dispatches between 1643 and 1649. He believed that ciphers using a variable key were much more secure than older methods.

In 1649, Wallis was appointed to the Savilian Chair of Geometry at Oxford University. He held this prestigious position for 54 years. This long tenure allowed him to become one of the greatest intellectuals of his era. His appointment was partly due to political reasons, but his work quickly justified his position. He was also part of a group of scientists that would eventually become the Royal Society. Beyond math, he wrote on theology, logic, and English grammar. He even worked on a system to help a deaf boy learn to speak.

One of his most complex achievements involved the area under a curve. He developed a method to find the area enclosed by a curve like y = x^m. He proved that the ratio of this area to a specific parallelogram was 1/(m + 1). He also showed how to find the area of curves by expanding them into powers of x. This allowed him to calculate areas for various mathematical functions. He used a technique called interpolation to handle difficult curves like the circle. This led to what is now known as the Wallis product.

Wallis also made breakthroughs in finding the length of curved lines. This process is called rectification. In 1659, he provided a solution to problems regarding the cycloid. He also used his methods to rectify the semicubical parabola. This was a significant discovery because many people thought such curves could not be rectified. His work helped expand the boundaries of what mathematicians thought was possible. By connecting algebra with geometry, Wallis helped lay the foundation for much of modern mathematical analysis.