John Wallis: Mathematical Biography

Considered to be the most influential mathematician prior to Sir Isaac Newton, John Wallis was born in 1616 in Ashford, Kent, England, the son of Reverend John Wallis. At first locally educated, Wallis was first introduced to Math while at school in Felsted, Essex. At the time, however, Math was not considered a primary academic study. Enrolling in Emmanuel College, Cambridge, Wallis' plan was to become a Doctor but, after graduating, he went on to obtain his Masters degree in 1640, before becoming a Priest. Wallis was elected to a fellowship at Queen’s College, Cambridge in 1644 but resigned the following year after his marriage to Susanna Glyde.

Wallis became interested in cryptography and once refused to teach the principles to German students due to his recognizing the potential of such techniques for his country. Although a Royalist, Wallis still served the government after the Civil War that ousted King Charles I in 1649, and on the restoration of the Monarchy in 1660, became Chaplain to King Charles II.

In London, about 1643 Wallis joined a group of scientists later known as the Royal Society and 6 years later was appointed as chair of Geometry at Oxford University where he remained until his death in 1703. A renowned insomniac, Wallis often did mental calculations while lying in bed at night. Reputedly, he once calculated the square root of a 56 digit number and later correctly recited the answer from memory to 27 digits accuracy. Throughout his life, Wallis made great contributions in the fields of Calculus, Geometry, and Trigonometry and was an adherent of the practice of using arithmetic concepts even for geometric concepts. Other fields in which he made significant contributions include English grammar, Theology, Logic, and Philosophy.

Published in 1656, Wallis begins by proving the law of indices which shows:

He then goes on to find the area under the curve y=x^m and proves that it is proportional to the area of a parallelogram, of same base and height as the curve, to a ratio of 1:1+m.

From this he goes on to derive his formula for π which is

The more terms that are added to the numerator and denominator, the close the result gets to π. This formula was found while attempting the integral

which is the same as computing the area of the unit circle.

Accepting both negative and complex roots, Wallis showed, in his 1685 work, that a³-7a=6 has exactly three real roots and stated that the rule for finding the number of positive and negative roots of a polynomial only works for equations whose roots are all real. Descartes, who established the rule, had not considered that they must all be real.

Wallis designed a structure that could span a square space while only being supported at the edges. It was made up of a pattern of short and identical interlocking pieces, each of which is attached to another piece at each end and supporting two other pieces in-between.

(See picture here)

Another clever aspect of this design is that the pieces support each other, imposing only vertical forces, so that the joints don’t need any screwing, nailing, or gluing.  While designing this structure, Wallis did what we now know as structural analysis, solving a set of 25 simultaneous equations by hand.

Wallis did some very interesting thinking in this area. Following Euclid’s proposition that, A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the interior angles on the same side equal to two right angles. Wallis argued that concentric circles must then also be parallel lines because they possess the same properties. Furthermore, if they are parallel, then the angles made by a line intersecting both circles will be equal. If a polygon inscribed in a circle makes a certain angle with a line bisecting the polygon, then this angle will become smaller as the number of sides increases. Assuming the number of sides equals infinity, not only will that angle become zero, that the polygon becomes a circle. The conclusion is that there is no angle made from a circle and its tangent line.

Wikipedia

A History of Mathematics by Carl B. Boyer

maths.tcd.ie

gap.dcs.st-and.ac.uk

A Brief History of Algebraic Notation by Lynn Stallings, School Science and Mathematics 100.5 (May 2000): p230

Leibniz on the Foundations of the Calculus: The Question of the Reality of Infinitesimal Magnitudes.(17th-century German mathematician Gottfried Wilhelm Leibniz), Douglas M. Jesseph.
Perspectives on Science (Spring-Summer 1998): p6(1)

soue.org.uk