The vociferous public quarrel between the philosopher Thomas Hobbes (1588-1679) and the mathematician John Wallis (1616-1703) was a strange and unpleasant episode in the history of mathematics. In his *De Corpore* of 1655, Hobbes claimed quadrature of the circle, rectification of curvilinear arcs, and solutions of other outstanding geometric problems. Wallis replied in the same year with *Elenchus Geometriae Hobbianae*, in which he pointed out the many errors and generally deplorable state of Hobbes' geometry. This unleashed a continuing exchange of rebuttal and invective lasting until the end of Hobbes' life. Some of the flavor of the dispute can be gleaned from such titles as Hobbes' *Six Lessons to the Professors of the Mathematiques* (the other Oxford professor in the fray was Seth Ward, Savilian Professor of Astronomy), Wallis' *Due Correction for Mr. Hobbes ... for not saying his Lessons right*, or Hobbes' *Markes of the Absurd Geometry, Rural Language, Scottish Church-Politicks, and Barbarismes of John Wallis*.

Hobbes, who first encountered (and became enamored of) Euclid's *Elements* at the age of forty, was no mathematical match for Wallis (Savilian Professor of Geometry at Oxford), and this lopsided contest is usually mentioned (e.g., by J. F. Scott in his 1938 book on Wallis) with some puzzlement that Wallis should bother with someone so obviously his mathematical inferior. Jesseph, who teaches philosophy at North Carolina State University, and who is also clearly mathematically competent and historically knowledgeable, intends " ...not to rehabilitate Hobbes's mathematical reputation, but rather to focus on his mathematical work as a way of improving our understanding of his philosophy and the context in which it developed." He believes that we need to consider Hobbes' philosophy as a whole, and the opposed political views of Wallis and Hobbes, to gain a full comprehension of the battle.

Chapter One, "The Mathematical Career of the Monster of Malmesbury," begins with the story of an alleged quadrature of the circle by the aging Longomontanus (Christian Severin Longborg) and its refutation by John Pell. When Longomontanus rejected a trigonometric identity used in the refutation, Pell appealed to several mathematicians for alternative proofs of the identity, among them Descartes, Cavalieri, Roberval, and -- ironically -- Thomas Hobbes. The account of this incident is followed by a survey of the mathematical and quasi-mathematical work of Hobbes throughout his long career, and an overview of the dispute between Hobbes and Wallis that lists titles and summarizes contents of the works that each wrote in criticism of the other. Jesseph then reviews classical conceptions of mathematics, focusing mainly on the *Elements*, and explicates the classical problems of quadrature, trisection, and duplication. We then have a report on the state of mathematics in the 17th century, a time of transformation that included disputes over the primacy of arithmetic and algebra (Wallis, Descartes) as opposed to the primacy of geometry (Hobbes, Barrow). The chapter concludes with a description of Cavalieri's method of indivisibles and contemporary reactions to it.

Chapter Two, "The Reform of Mathematics and of the Universities," provides an outline of how Hobbes' notion of the proper role of a university in a commonwealth was in conflict with the views of Wallis and Seth Ward (and a multitude of others). The 1651 publication of *Leviathan*, with its radical materialism and denial of the soul's immortality, offended many, and its message that universities should be subservient to the state offended the academic world in particular. Jesseph sees the deep philosophical, theological, and political differences between Hobbes on the one hand, and Wallis and Ward on the other, as significant factors in the ensuing Hobbes/Wallis battle. Ward was actually the first of the two Oxford professors to publicly criticize Hobbes, in an appendix to his 1654 rebuttal of views on university reform propounded by the Cromwell ally John Webster. It was during the next year that Hobbes published *De Corpore* as a philosophical system so good (he thought) as to guarantee the solution of any problem, mathematical or otherwise. In this spirit, he added at the last minute a chapter containing a circle quadrature ".... whose ineptitude probably surpassed Ward's fondest wishes" according to Jesseph. Ward and Wallis then seemingly split the task of demolishing Hobbes into two parts, with Wallis attempting to refute every mathematical bit of *De Corpore* and Ward doing the same for the philosophy and political theory of the work.

Chapter Three, "*De Corpore* and the Mathematics of Materialism)," examines Hobbes' philosophy of mathematics, especially his departures from the traditional views of the nature of mathematical objects. Hobbes was troubled by the Euclidean definitions of point (that which has no parts) and line (breadthless length), and substituted his own ideas that a point is a body without magnitude while a line is the path by which a point travels, ideas that were sarcastically attacked by Wallis. Hobbes similarly replaced Euclid's treatment of ratio and proportion with his own version cast in terms of body and motion, and tried to improve on Euclid's obscure definition of straight line. He further attempted to incorporate work of Galileo, Cavalieri, and Roberval into his materialistic conception of mathematics, with his versions sometimes so close to the originals that Wallis remarked " ... when something true is included among these things of yours, it is not really your own, but taken from somewhere else." It is worth mentioning that Wallis himself had been accused of the same behavior, as for example in his work (reflecting Roberval's) on the rectification of the Archimedean spiral. The chapter concludes with more detail on the circumstances surrounding the *De Corpore* circle quadrature.

Chapter Four, "Disputed Foundations," contrasts the views of Wallis and Hobbes on the nature of mathematics, especially the question of whether mathematics does depend on the structure of the material world (Hobbes) or does not (Wallis). There is another extended discussion of their differing views of ratio and proportion, and then a more interesting review of their ideas on the "angle of contact" between a circle and its tangent, a so-called "horned angle" (one side straight, one curved) that if considered an actual magnitude seems to violate the axiom of Archimedes. Wallis sided with Peletier in not considering this kind of angle a genuine magnitude, while Hobbes somewhat agreed with Clavius that it is an infinitely small magnitude, but also found his way out of the dilemma by declaring that rectilinear, curvilinear, and mixtilinear angles are three different kinds of magnitudes that cannot be directly compared. Finally there is a section on the combatants' differing views of infinite processes. Wallis used infinitesimals in a much freer way than did Cavalieri, considering them a "notational variant" of the method of exhaustion, according to Jesseph. That casual approach invited criticism from Hobbes, of course, who with more justification than usual saw the entirety of Wallis' *Arithmetica Infinitorum* as without solid foundation.

Chapter Five, "The *Modern Analytics* and the Nature of Demonstration," begins with a discussion of Hobbes' conception of genuine knowledge, involving both an Aristotelian concern that causes be taken into account and a nominalist position that reasoning involves the manipulation of names (as opposed to the Cartesian view that reasoning is an intellectual process independent of language). Hobbes' ideas on the demonstrability of reasoning that begins with definitions (not suppositions, as in natural science) is seen as the source of his hope that any geometrical problem should be solvable -- a hope, Jesseph says, that " ... led him to underestimate the difficulty of the classical problems such as the quadrature of the circle." A middle section looks at the philosophical notions of analysis and synthesis, with analysis, considered as a method of discovery, usually seen as preliminary to synthesis (a method of demonstration). It is noted that Descartes sometimes seemed to reverse that order, and that Wallis enthused about analysis, which he essentially identified with algebra. In his *Treatise of Algebra* (1685), Wallis went so far as to suggest that algebra " ... was in use among the Grecians [Euclid, Archimedes, Apollonius, Pappus, etc.], we need not doubt; but studiously concealed (by them) as a great Secret." Hobbes' analysis/synthesis distinction, based on order of reasoning as related to cause and effect, is then discussed at some length. A final brief section notes Hobbes' rejection of analytic geometry, and especially its excessive use of symbols. Hobbes felt that the new algebraic methods did not meet his criteria for rigorous demonstration, but Jesseph suggests that Hobbes' lack of understanding of those methods also played a part in the rejection.

Chapter Six, "The Demise of Hobbesian Geometry," reviews two main points about Hobbes' philosophy of mathematics: " ... (1) mathematics is a generalized science of body and (2) the first principles of mathematics must express the causes by which mathematical objects are generated." It is noted that Hobbes had formerly admitted technical errors in his mathematical work (while insisting his general ideas were still sound), but eventually rejected any criticism of his mathematics and even recanted his earlier admissions of mistakes. In the 1650s, for example, Hobbes accepted the prevailing standards of the European mathematical community, and publicly acknowledged the validity of criticism on technical points (being careful to attribute such errors to negligence rather than ignorance), but by 1661 he rejected algebraic analysis that seemed to conflict with his geometrical results. Finding no mathematicians who would agree with him that arithmetic and algebra cannot validly be applied to geometry, Hobbes in a letter of 1664 to Sorbière expressed doubt about the Pythagorean Theorem, and in 1666 claimed that any result depending on that theorem should be taken as not yet proven. And in a 1669 attempted quadrature, he claimed a result that would assign to Pi the value 3.2 exactly. The chapter also analyzes personal, ideological, and political aspects of Hobbes' exclusion from the Royal Society (of which Wallis and Ward were founding members), and its effect on Hobbes' public controversies with Wallis and others.

Chapter Seven, "The Religion, Rhetoric, and Politics of Mr. Hobbes and Dr. Wallis," is devoted to extramathematical features of the Hobbes/Wallis dispute, with a reminder that the attacks of Wallis and Ward on Hobbesian mathematics were at least partially motivated by their view of Hobbes as expounding views that were inimical to religion and to the universities. Hobbes was obviously outside the mainstream of 17th century European thought, while Wallis " ... fits neatly into ... the Presbyterian strain of English Puritanism." The public roles and stances of Wallis and Hobbes are traced through the period of the English civil war, the Commonwealth, and the Restoration. Jesseph skillfully sorts out the competing religious strains during this period of changing authority, and provides an extended discussion of allegations of atheism against Hobbes by Wallis, Ward, and many others. A middle section discusses the rhetorical style of the time, showing that (e.g.) the charges of poor Latin exchanged by Hobbes and Wallis were not peculiar to the debates of those two. A final section presents accusations of various kinds of disloyalty (regarding Commonwealth and later Restoration) between the two adversaries.

The final chapter, "Persistence in Error," subtitled "Why Was Hobbes So Resolutely Wrong?", summarizes the preceding material in a succinct first paragraph: Hobbes erroneously thought his materialistic version of geometry could solve all problems; Wallis persisted in refuting Hobbes' quasi-mathematical results because he considered Hobbes' religious and political views dangerous; and the lengthy public debate included a multitude of issues in addition to questions of circle-squaring. Jesseph then outlines a sociological view of scientific knowledge that sees one's science or mathematics as depending on one's political or social views, focussing especially on Shapin and Schaffer's *Leviathan and the Air-Pump*, only to rebut that view as inadequate in a subsequent section -- inadequate to explain the Hobbes/Wallis dispute, but also in general.

The body of the book is followed by an entertaining appendix of samples of Hobbesian mathematics, with commentary by the author. We are given Hobbes' contribution to the joint quadrature refutation organized by Pell (mentioned in Chapter One). This is followed by: two defective attempts from *De Corpore* to establish a result equivalent to (correctly) integrating x^{n} from 0 to a; two attempted circle quadratures (also from *De Corpore*); a rectification of the Archimedean spiral in the style of Roberval; an attempted cube duplication; and the 1669 quadrature leading to 3.2 as the value of Pi. These efforts are marked by a prolix writing style that eschews algebra in favor of Euclidean synthetic demonstration, elaborate geometrical diagrams with many extraneous and unused features, and of course critical steps in an argument that simply do not follow from what comes before.

*Squaring the Circle* is a welcome addition to the literature on the history of mathematics. It is quite well-written, and thoroughly explores all aspects of the conflict between Hobbes and Wallis. It also makes a good case that the picture of Hobbes as a mathematical buffoon needs softening. By taking pains to examine Hobbes' ideas, it shows that at least some of his mathematical efforts were not completely ridiculous. The book also carries with it a heavy load of scholarly apparatus. There are 371 footnotes in its 383 pages, and 25 pages of references at the end indicate the enormous scope of Jesseph's investigation. The book carefully considers the foundations of a kind of thinking that can lead to erroneous results. Readers who think that an important object of study will find the entire book valuable, while those interested mainly in the mathematical aspects of the dispute might consider a strategy consisting of careful reading mainly of Chapters One, Two, and Six and the Appendix. At any rate, this book should certainly be on the shelves of any decent college or university library.

David Graves (

dgraves@elmira.edu) is Associate Professor of Mathematics at Elmira College, where he is active as a pianist, and has taught courses in cryptology, opera, and history of astronomy as well as the usual run of mathematics courses.