---
title: "Hobbes–Wallis controversy"
chunk: 2/3
source: "https://en.wikipedia.org/wiki/Hobbes–Wallis_controversy"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T04:17:50.179873+00:00"
instance: "kb-cron"
---

=== The Elenchus ===
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. Hobbes had limited his interest to geometry, restricting the scope of mathematics.
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. Hobbes himself wrote to Samuel de Sorbière in the same year, saying the controversy was not merely scientific. He regarded the use of infinite quantities as the thin end of the wedge for a return of scholasticism, and behind Wallis he saw "all the Ecclesiastics of England". Sorbière visited Wallis in Oxford; but his analysis of Wallis as stereotypical pedant helped not at all in the quarrel.
Hobbes took care to remove some mistakes exposed by Wallis, before allowing an English translation of the De Corpore to appear in 1656. But he still attacked Wallis in a series of Six Lessons to the Professors of Mathematics, included with the De Corpore translation. Wallis defended himself, and re-confronted Hobbes with his mathematical inconsistencies. Hobbes responded with Marks of the Absurd Geometry, Rural Language, Scottish Church Politics, and Barbarisms of John Wallis, Professor of Geometry and Doctor of Divinity. It has been suggested that Hobbes was still trying to cultivate John Owen at this point: Owen was both the leading Independent theologian and Cromwell's choice as Vice-Chancellor of Oxford, and Hobbes softened his critical line on the universities while stoking up the quarrel with Wallis. 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. The various thrusts were parried by Wallis in a reply (Hobbiani puncti dispunctio, 1657).

=== Controversy over foundational matters ===
Wallis published a comprehensive treatise on the general principles of calculus (Mathesis universalis, 1657). Here he strongly advocated giving priority to the approach through arithmetic and algebra. This was quite contrary to the arguments of both Hobbes and Isaac Barrow. Hobbes set store on the "demonstrable" status of geometry, in the Six Lessons. Jon Parkin writes:

For Hobbes, his new form of geometrical demonstration was the finest example of what a nominalist science could achieve. It offered demonstrably certain knowledge. The creation and interaction of lines could clearly be conceived as a product of matter in motion, whose properties could be demonstrated with the highest level of certainty.[...] Wallis, by contrast was the foremost exponent of Cartesian analytical geometry.
Mathematicians sympathetic to Hobbes included François du Verdus and François Pelau, and some of his works were later translated into English for pedagogic use by Venterus Mandey; but he was not backed up by a "school". On the other side as critics were Claude Mylon, Laurence Rooke, Viscount Brouncker, John Pell, Christiaan Huyghens; much of the criticism Hobbes received was by private correspondence, or in the case of Pell direct contact. Henry Stubbe, later a vehement critic of the Royal Society, assured Hobbes in 1657 he had some (unnamed) supporters in Oxford.
Hobbes decided again to attack the new methods of mathematical analysis and by the spring of 1660, he had put his criticism and assertions into five dialogues under the title Examinatio et emendatio mathematicae hodiernae qualis explicatur in libris Johannis Wallisii, with a sixth dialogue so called, consisting almost entirely of seventy or more propositions on the circle and cycloid. Wallis, however, would not take the bait.

=== Hobbes and duplicating the cube ===
Hobbes then tried another tack, having solved, as he thought, another ancient problem, the duplication of the cube. He had his solution brought out anonymously in French, so as to put his critics off the scent. He slipped in algebraic terms in early efforts, by cubing √2 to the answer 2. While Hobbes would withdraw some arguments as erroneous, he distinguished between "errors of negligence" and "errors of principle", and found the latter much harder to admit. 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. Hobbes replied with an idiosyncratic appeal to a form of dimensional analysis, where algebraic quantities are non-dimensional. In general, his positions hardened after 1660.
Wallis publicly refuted the solution, but Hobbes claimed the credit of it. He republished it (in modified form), with his remarks, at the end of the 1661 Dialogus Physicus.