He received his early education in Shrewsbury School under a Mr Hotchkin and was admitted as a sizar at Magdalene College, Cambridge, on 24 March 1753, being also Millington exhibitioner.

In fact Waring was very young and did not hold the MA, necessary for qualifying for the Lucasian chair, but this was granted him in 1760 by royal mandate.

The Meditationes Algebraicae (1770), where many of the results published in Miscellanea Analytica were reworked and expanded, was described by Joseph-Louis Lagrange as 'a work full of excellent researches'.

In this work Waring published many theorems concerning the solution of algebraic equations which attracted the attention of continental mathematicians, but his best results are in number theory.

He also published a theorem, due to his friend John Wilson, concerning prime numbers; it was later proven rigorously by Lagrange.

He devoted himself to the classification of higher plane curves, improving results obtained by Isaac Newton, James Stirling, Leonhard Euler, and Gabriel Cramer.

In 1794 he published a few copies of a philosophical work entitled An Essay on the Principles of Human Knowledge, which were circulated among his friends.

It is indicative that he was one of the subscribers of John Landen's Residual Analysis (1764), one of the works in which the tradition of the Newtonian fluxional calculus was more severely criticised.

In the preface of Meditationes Analyticae Waring showed a good knowledge of continental mathematicians such as Alexis Clairaut, Jean le Rond d'Alembert, and Euler.

The theory of convergence of series (the object of which is to establish when the summation of an infinite number of terms can be said to have a finite 'sum') was not much advanced in the eighteenth century.