Rather remarkably Raphson was made a member of the Royal Society in 1689, several years before he graduated. He was proposed for election by Edmond Halley who had himself been elected at a very early age. Raphson's election to that Society was on the strength of work which was published in his book Analysis aequationum universalis in 1690. This contained the Newton method for approximating the roots of an equation.
In Method of Fluxions Newton describes the same method and, as an example, finds the root of x3 - 2x - 5 = 0 lying between 2 and 3. Although written in 1671 it was not published until 1736, so Raphson published the result nearly 50 years before Newton.
Raphson's relation to Newton is important but not particularly well understood. In  Copenhaver writes:-
Raphson was one of the few people whom Newton allowed to see his mathematical papers. As early as 1691, he and Edmund Halley were involved in plans to publish Newton's work of the early 1670's on quadrature of curves, a project fulfilled only in 1704, and then in a much different form. In 1711, Roger Cotes and Willian Jones arranged for Raphson to see some of Newton's papers '... pertinent to his design of writing an History of the Method of Fluxions'.Raphson did indeed write his History of Fluxions which did not appear until 1715 after Raphson had died. It is unclear how pleased Newton was with this work despite its clear position in favour of Newton's claims over those of Leibniz. Certain letters which had passed between Newton and Leibniz appeared as an appendix to a reprint of Raphson's book in 1716-1718. Immediately a row broke out and Johann Bernoulli showed his anger. An attempt was made by Newton to calm things down when he wrote to Johann Bernoulli saying:-
I stopt [Raphson's History of Fluxions] coming abroad for three or four years.However, Newton admitted in a letter to Varignon that he was responsible for the letter being added to Raphson's book:-
When I heard that Mr Leibnitz was dead I caused what had passed between him and me to be printed at the end of Raphson's book because copies thereof had been dispersed by Mr Leibnitz.This was not Raphson's only publication relating to Newton's work. He translated Newton's algebraic work from Latin to English. Newton's Arithmetica universalis was translated by Raphson and appeared as Universal arithmetick in 1720 after Raphson's death.
Early in his career Raphson published a mathematical dictionary. In 1691, the year Raphson was elected to the Royal Society, Ozanam published Dictionnaire mathématique. Raphson produced his shorter version A mathematical dictionary in 1702 which is:-
A mathematical dictionary or a compendious explication of all mathematical terms, abridg'd from Monsieur Ozanam and others ... written by J Raphson FRS.Raphson published a second edition of his analysis book and, at the same time, De spatio reali which is an application of mathematical reasoning to theological issues. Raphson wrote a second theological work Demonstratio de deo in 1710.
De spatio reali discusses space and in it Raphson talks of 'real space' which he thinks of as being independent of the mind that perceives it. He discusses the infinite, distinguishing between the potentially infinite and the actual infinite. In discussing motion he argues that space is infinite but the collection of moving objects in it is finite.
Raphson's ideas of space and philosophy were based on Cabalist ideas. The Cabala was a Jewish mysticism which was influential from the 12th century on. It was an oral tradition and initiation into its doctrines and practices was passed on. Cabala developed several basic doctrines which were strong influences on Raphson's philosophical thinking. The doctrines included the withdrawal of the divine light, thereby creating primordial space, the sinking of luminous particles into matter and a "cosmic restoration" that is achieved by Jews through living a mystical life.
In these two works by Raphson De spatio reali and Demonstratio de deo, cosmology, natural philosophy, mathematics and his Cabalist beliefs combine. Of course his religious beliefs greatly influenced all his thinking. Newton's views of space were strongly influenced by Christian beliefs, and possible just slightly by his interaction with Raphson.
Article by: J J O'Connor and E F Robertson