Squaring the Circle: Hobbes on Philosophy and Geometry

With great applause the algebrists then read Wallis his Algebra now published. A hundred years that geometrick pest Ago began, which did that age infest. The art of finding out the numbers sought, Which Diophantus once, and Geber taught: And then Vieta tells you that by this Each geometrick problem solved is. Savil the Oxford reader did supply Wallis with principles noble and high That infinite had end and finite should Have parts, but those without end allow’d; Both which opinions did enrage and scare All those who Geometricasters were. This was enough to set me writing ... ¹

I. In 1655 Thomas Hobbes published his Elementorum Philosophiae Sectio Prima de Corpore, ² in which he sets out the principles, methods, and ends of his philosophy, and his doctrines of the first part of that philosophy, which concerns body. Primary among these doctrines on body are his views on geometry. In among these Hobbes gave his solutions to several geometrical problems, including the quadrature of the circle, the ancient task of constructing with ruler and compass a line equal in length to the circumference of any given circle, a problem which was not then known to have no analytic solution.

Almost immediately there followed the refutation of Hobbes’s quadrature claim by the ex-roundhead cryptographer and then Savilian professor of geometry at Oxford, John Wallis. ³ These were the opening shots in a bitter dispute lasting eighteen years, in which Hobbes employed to the fullest degree his prodigious powers of invective and Wallis distinguished himself as the only mathematician to have taken Hobbes’s geometry seriously. Only three others—the Belgian philosopher Moranus, John Pell, sometime professor at Amsterdam and Breda, and Viscount Brounker—are known to have given it more than passing thought.

Wallis, a pioneering mathematician, was himself a proud, bitter, and unpopular man. Hobbes was certainly not one to shy away from a fight. But his geometry plays a more significant part in his thought than is generally recognized. Following Wallis at least in this, I accord some earnest thought to Hobbes’s geometry and hope to show its roots in his most general thought on the nature of philosophy. Being closer to these general principles than, for example, his political work, the geometry serves reciprocally to illuminate the premises of Hobbes’s philosophy.

I aim to show how an investigation of his geometry illuminates Hobbes’s views on metaphysics, logic, and philosophy of science. In common with his contemporaries Hobbes’s use of the term “philosophy” is broader than the use accorded today. In Hobbes’s case philosophy is, roughly speaking, the working out of the doctrines of logic and metaphysics to give knowledge of a particular sort. This knowledge may be divided according to its subject matter: body, man, and commonwealth. These “parts of philosophy” form a hierarchy extending, via moral philosophy (psychology), from the science of geometry to the science of politics, as the subject matter increases in size and complexity, from single bodies to complex commonwealths.

This hierarchy is epistemic. That is to say, there is a reductionism according to which the principles of sciences higher up in the hierarchy, may be deduced from those lower. As we shall see, scientific argument is demonstrative (deductive) and is couched in terms of cause and effect. Cause and effect themselves operate only through motion. These are the most general principles of Hobbes’s philosophy. Thus the foundation of science is to be found in the most universal treatment of the simplest bodies of which others are composed. This foundational science is, according to Hobbes, geometry.

Thus geometry is significant because (1) it lies at the foundation of Hobbes’s scientific hierarchy and (2) being foundational reflects more clearly than the other sciences Hobbes’s logical, epistemological, and metaphysical doctrines.

II. De Corpore, emphasizes Hobbes, deals with the fundamentals of philosophy, which he defines as “such knowledge of effects of appearances as we acquire by true ratiocination from the knowledge we have first of their causes or generation; And again of such causes or generations as may be from...