Socially, the formation of the group of Royal Society members, and the status of the publication Philosophical Transactions, was brought to a point as the quarrel proceeded, with Hobbes playing the outsider versus the self-selecting guild.
Quentin Skinner writes: "There is no doubt that at the personal level Wallis behaved badly (as was widely conceded at the time).
"[3] The fact that Wallis was a Presbyterian, a university man, and an anti-Royalist during the civil war made him "three times an enemy to Hobbes", as Anthony Gottlieb points out in The Dream of Enlightenment.
[4] Part of the significance of the controversy is that Hobbes felt that, in the later stages, the Royal Society was in some way complicit in the attacks from Wallis, despite the fact that he had many friends as Fellows in it.
Hobbes in Leviathan (1651) joined others in attacks on the existing Oxbridge academic system, essentially a monopoly in England of university teaching.
The issue of the universities was heavily loaded at the time, and the orthodox Presbyterian minister Thomas Hall lined up with Vindiciae literarum (1654).
Wilkins wrote a preface to Vindiciae academiarum; the main text by Ward mentioned Hobbes, who was the particular target of an appendix.
But the emergence of the full scope of the philosophy of Hobbes in Leviathan lost him allies who may have shared somewhat in his starting assumptions, but who felt a need to distance themselves from his conclusions, as Ward did in his Philosophicall Essay of 1652.
[9] Ward went on to make a full-dress attack on Hobbes the philosopher, the In Thomae Hobbii philosophiam exercitatio epistolica of 1656, dedicated to Wilkins.
[10] Errors in De Corpore, in the mathematical sections, opened Hobbes to criticism also from John Wallis, Savilian Professor of Geometry.
Wallis's Elenchus geometriae Hobbianae, published in 1655, contained an elaborate criticism of Hobbes's attempt to put the foundations of mathematical science in its place within knowledge.
The book was dedicated to John Owen, and in prefatory remarks Wallis (a Presbyterian) avows that his differences with Hobbes are largely rooted in theology.
[13] Hobbes took care to remove some mistakes exposed by Wallis, before allowing an English translation of the De Corpore to appear in 1656.
Further, the religious dimension (Scottish Church Politics refers to the Presbyterianism of Wallis, not shared by Owen) has been seen as a presage of later analysis of Behemoth, the book Hobbes wrote in 1668 as a post-mortem on the English Revolution.
He was led to argue that the doctrine of nth roots in algebra (one contribution of Wallis) did not adequately model the geometric notions based on area and volume.
René François Walter de Sluse walked through Hobbes's proof in one version, clearing the radicals to come down to a numerical assertion it implied (97,336 = 97,556), which could only be accepted as an approximation.
He had reasoned out his own conclusions years before from speculative principles, and he warned them that if they were not content to begin where he had left off, their work would come to naught.
It included the accusation that Hobbes used purely verbal tactics, preferring his own semantics of a term such as "air", to cast doubt on the existence of a vacuum.
Three years later he brought his three mathematical achievements together in Quadratura circuli, Cubatio sphaerae, Duplicitio cubii, and as soon as they were once more refuted by Wallis, reprinted them with an answer to the objections.