## Abstract

The dispute that raged between Thomas Hobbes and John Wallis from 1655 until Hobbes's death in 1679 was one of the most intense of the ‘battles of the books’ in seventeenth-century intellectual life. The dispute was principally centered on geometric questions (most notably Hobbes’s many failed attempts to square the circle), but it also involved questions of religion and politics. This paper investigates the origins of the dispute and argues that Wallis’s primary motivation was not so much to refute Hobbes’s geometry as to demolish his reputation as an authority in political, philosophical, and religious matters. It also highlights the very different conceptions of geometrical methodology employed by the two disputants. In the end, I argue that, although Wallis was successful in showing the inadequacies of Hobbes’s geometric endeavours, he failed in his quest to discredit the Hobbesian philosophy *in toto*.

The dispute that raged between Thomas Hobbes and John Wallis from 1655 until Hobbes's death in 1679 was one of the most intense of the ‘battles of the books’ in seventeenth-century intellectual life. The quarrel began over geometric matters, when Wallis published a lengthy refutation of the mathematics in Hobbes's 1655 treatise *De corpore* under the title *Elenchus geometriae hobbianae*.^{1} A large part of the significance of this dispute comes from the fact that it involved two of the most influential figures in the English Republic of Letters. On one side was Hobbes, whose 1651 masterwork *Leviathan* had solidified his reputation as a political thinker and controversial proponent of the ‘new philosophy’, but who saw his once-considerable scientific and mathematical credentials demolished in the course of the dispute. On the other was Wallis, whose status as Oxford's Savilian Professor of Geometry and founding member of the Royal Society put him very much at the centre of the English scientific scene. Yet his participation in the quarrel did not measurably enhance his intellectual standing.

Hobbes claimed that his principles had enabled him to solve the great unsolved problems of geometry—notably the quadrature of the circle, the arbitrary section of an angle and the duplication of the cube.^{2} In support of his claims for the superiority of his geometry Hobbes included a circle quadrature and various other geometric efforts in his *De corpore*; when Wallis promptly refuted these in his *Elenchus*, Hobbes replied with revised efforts and mounted a vigorous counterattack against Wallis's own geometric principles. The ensuing exchange of polemics acquired a life of its own, and Hobbes found his geometric efforts almost universally scorned. In a futile attempt to preserve his imagined results from refutation, Hobbes rejected the basic principles on which his opponents' arguments were based, but his geometric programme collapsed into incoherence as he found himself forced to question such basics as the Pythagorean Theorem, the validity of the traditional trigonometric tables or the Archimedean bounds for the value of π.

Viewing the wreckage to which Wallis had reduced Hobbes's geometric argumentation, Christiaan Huygens wondered why he had even bothered to expose the errors and incoherence in Hobbes's geometry. Writing to Wallis in 1656, Huygens declared himself ‘amazed that you judged [Hobbes] worthy of such a lengthy refutation, although I read your learned and rather sharp *Elenchus* with some pleasure’.^{3} This was, in all respects, a very public debate, and it focused on issues where one might not initially expect a controversy, namely technical questions of mathematics and the adequacy of proposed demonstrations. Public disputes and debates typically have a political aspect, and this was no exception as it involved matters of political obligation and church governance in addition to questions of mathematics. Yet it would be a serious misinterpretation to think that the controversy was entirely political, or entirely religious, or entirely mathematical. The episode illustrates the interplay among a variety of factors, and it cannot be reduced to questions of pure geometry or pure politics.

My purpose here is to address two questions. First, what was ultimately at stake in this dispute? Second, what consequences did the dispute have both for Hobbes and Wallis? My remarks are grouped into three sections: the first takes up the question of what motivated the controversy, and the second examines issues concerning the criteria for rigorous demonstration that played a central role throughout the dispute. The final section examines the consequences of the controversy.

## The dispute in its context

The first thing that becomes evident in surveying the Hobbes–Wallis dispute is the fact that it was not a dispassionate search for the truth. This much is obvious from some of the harsh language employed by the disputants, with Wallis denouncing Hobbes's *De corpore* as a ‘shitten piece’^{4} and Hobbes complaining of the ‘Levity and Scurrility’ employed by Wallis while invoking ‘Vespasians law’, or the principle that good manners forbid one from initiating the use of harsh language, but there is no harm in replying in kind.^{5} This level of invective shows that the participants in the dispute held quite passionately to their views, and it suggests that there was far more in play than the success or failure of a few geometric demonstrations. Indeed, issues in religion, church government and politics appear, along with lengthy exchanges over subtle points of grammar and philology.

One might be tempted to think that the mathematical aspects of the dispute were unimportant, so that at the most fundamental level the *real* issues were those concerning politics and religion. Viewed in this light, the controversy would have involved geometric matters almost by accident, and geometry would have mattered only to the extent that it provided the disputants a convenient arena in which to trade polemics. As I hope to show, there is little to be said in favour of such a reading of the controversy, for there were also matters of considerable mathematical significance at stake. Nevertheless, there is one respect in which this sort of approach gets things right: there were certainly social factors at work in the Hobbes–Wallis dispute, and we cannot hope to understand the controversy without attending to them.

In trying to determine what was ultimately at stake in this controversy, we can usefully begin by taking Huygens' question seriously: why, indeed, did Wallis go to the trouble of publishing lengthy and detailed refutations of Hobbes's geometric efforts? One might think that Wallis's decision to attack Hobbes's geometry could be explained entirely in terms of his quarrelsome temper. Wallis was, as Richard Westfall put it, ‘a bellicose character engaged in endless quarrels and controversies’.^{6} In fact, the catalogue of Wallis's polemical writings is quite extensive, covering all manner of natural-philosophical, mathematical and theological subjects.^{7} There can be little question that Wallis relished the role of polemicist, but one should not think that his interest in the refutation of Hobbesian geometry was exclusively the consequence of his combative temper. It is noteworthy that the English theologian and natural philosopher Thomas White published a circle quadrature in his *Exercitatio geometrica. De geometria indivisibilium, & proportione spiralis ad circulum*.^{8} The level of argument in this piece was no better than anything to be found in Hobbes, but Wallis did not bother to attack it.^{9} We are left to conclude that there was something significant in Hobbes's philosophy that motivated Wallis to engage in the lengthy and vitriolic denunciation of all things Hobbesian.

In point of fact, Wallis made no great secret of his motivations for attacking Hobbes's geometry, and the presence of theological and political motives is well attested in a 1659 letter to Huygens. He wrote:But regarding the very harsh diatribe against Hobbes, the necessity of the case, and not my manners, led to it. For you see, as I believe, from other of my writings how peacefully I can differ with others and bear those with whom I differ. But this was provoked by our Leviathan (as can be easily gathered from his other writings, principally those in English), when he attacks with all his might and destroys our universities (and not only ours, but all, both old and new), and especially the clergy and all institutions and all religion. As if the Christian world knew nothing sound or nothing that was not ridiculous in philosophy or religion; and as if it has not understood religion because it does not understand philosophy, nor philosophy because it does not understand mathematics. And so it seemed necessary that now some mathematician, proceeding in the opposite direction, should show how little he understand this mathematics (from which he takes his courage). Nor should we be deterred from this by his arrogance, which we know will vomit poison and filth against us.

^{10}

Decades later Wallis offered a similar assessment of his motivations in the Preface to the first volume of his *Opera mathematica*: ‘Certain works written long ago against *Thomas Hobbes* (pseudo-geometer) will not be found here, for I would not want to seem to triumph over one now dead. Nevertheless, as things then stood, it was something that had to be done, when he had set himself up in the guise of a great geometer and dared to offer false suggestions to our unsuspecting youth in matters of religion.’^{11}

We can thus distinguish two issues that Wallis claimed as principally motivating his campaign against Hobbes: the role of the English universities and the status of the clergy (or, indeed, religion itself). Beyond these two concerns, Wallis saw the refutation of Hobbes's geometry as a way to discredit his philosophy generally. Because Hobbes insisted that his philosophy was grounded in fundamental truths that offered secure foundations for geometry, physics and politics, exposing his geometric failures would show the inadequacy of his system *in toto*.

Hobbes's political theory requires that the power of the civil sovereign be absolute and undivided. As a consequence, he held that universities must submit to the dictates of the sovereign in all matters. This extends, ironically enough, to geometry, since Hobbes notoriously claimed in *Leviathan* that the sovereign could ban the teaching of the subject and order ‘the burning of all books of Geometry’ if he should judge geometric principles ‘a thing contrary to [his] right of dominion, or to the interest of men that have dominion’.^{12} Hobbes saw universities as critically important in inculcating the virtue of obedience to sovereign authority and he took it to be ‘manifest, that the Instruction of the people, dependeth wholly, upon the right teaching of Youth in the Universities’.^{13} In Hobbes's estimation, the Civil War was in large measure a university-inspired conflict, and the preservation of order in the realm required that the universities at Oxford and Cambridge be reformed so as to purge them of false doctrines that promoted instability.

This critique of the universities was part of a broader debate over the status of the English university system in the aftermath of the Civil War. Puritan critics of the universities sought to align the educational system with the principles of true government and religion.^{14} Like Hobbes, the Puritan reformers complained that the university curriculum was overly dependent on Scholastic and Aristotelian teachings that should be cast aside in favour of the ‘new philosophy’. These reformers also shared Hobbes's view that traditional theology (‘school divinity’) was tainted by its association with Aristotelian philosophy. Matters had gone so far that in 1653 the Barebones Parliament entertained a proposal to abolish the universities altogether.^{15} Although Hobbes was surely no radical Puritan, he and the reformers agreed that the universities could not be trusted to change their ways, so that state power must be employed to enforce a new educational order.^{16}

Wallis's colleague Seth Ward (Savilian Professor of Astronomy) took up the cause of defending the universities in 1654 with his *Vindiciae academiarum*. He published the tract anonymously, principally in reply to the *Academiarum examen* of the Puritan reformer John Webster, but he also included two appendices, one directed at the writings of the reformer William Dell and the other responding to Hobbes. The appearance of Ward's *Vindiciae academiarum* precipitated Hobbes's dispute with Wallis, although Wallis himself had no part in its publication. Evidently seeking to goad Hobbes into making public his supposed circle quadratures and suspecting the lamentable state of his geometric efforts, Ward dared the philosopher from Malmesbury to present his geometric results to the learned world.^{17}

In the course of events, Ward proved prescient. When Hobbes's *De corpore* appeared, its twentieth chapter contained a supposed circle quadrature that had been hastily revised multiple times even as the book was being printed, as well as several intemperate passages directed against Ward, under the name ‘Vindex’ (in allusion to his authorship of the *Vindiciae academiarum*). The level of geometric argument in this chapter was as poor and riddled with paralogisms as Ward had anticipated when he predicted that he would be ‘able to find a great number in the University, who will understand as much or more of [Hobbes's geometrical pieces] then he desires they should, indeed too much to keep up in them that Admiration of him which only will content him’.^{18}

Ward and Wallis both defended the institutional autonomy of the universities, arguing that the reformers' portrayal of them was inaccurate and uninformed. They expressed particular horror at the thought that the civil sovereign could order ‘the Publicke reading of [Hobbes's] *Leviathan* in the Universities’.^{19} Both men held that the universities were fully capable of exercising their proper educational function without excessive external interference from the state: no subversive doctrines were taught, the authority of Aristotle was hardly absolute and the new science was an object of intense study. This project nevertheless required a sort of conceptual balancing act because both Savilian professors had been intruded into their chairs by the Parliamentary visitation in 1649. They consequently owed their positions to an exercise of sovereign power that (in their assessment) was no longer warranted in the 1650s. To whatever extent Hobbes's critique of the universities might seem persuasive to a contemporary reader, Wallis was motivated to show that the Hobbesian programme was an amalgam of error and fallacy. Hobbes's geometry was therefore a tempting target for Wallis's attack.

Hobbes's conception of sovereignty has consequences for church government that parallel those for the universities. English debates over church polity in the 1650s were dominated by the conflict between two models: the Presbyterian and the Independent. Presbyterian ecclesiology places the church under the authority of a body of ministers and elders—the presbytery—who are granted their authority directly from God and do not answer to the civil sovereign. In contrast, the Independents held that the freely-gathered congregation was the true model of the New Testament church, and they opposed the imposition of a national ecclesiastical structure, whether in Presbyterian or Anglican form. By placing the church and its organization under the control of the civil sovereign, Hobbes decisively rejected the central tenet of Presbyterianism. The issue had particular significance at Oxford, where John Owen (a leading Independent) had been appointed Vice-Chancellor of the university in 1652. The issue of church governance was far from settled in the 1650s, and Wallis was an active participant in debates concerning it, defending Presbyterian ecclesiology and attacking the scriptural basis for Independency.^{20} As a Presbyterian minister and Doctor of Divinity, Wallis therefore found abundant reason to reject Hobbes's political theory.

Hobbes's metaphysics provided still more to provoke Wallis's antipathy. Hobbes embraced a thoroughgoing materialism that was difficult to reconcile with any traditional notion of the Deity, and he was widely suspected of endorsing principles that implied atheism, even though he frequently insisted that his views were in conformity with the tenets of Protestant Christianity.^{21} Wallis and the proponents of traditional theology held that God is to be understood as an essentially immaterial spirit, radically distinct from the world of material bodies. In *Leviathan* Hobbes notoriously declared that the expression *immaterial substance* is insignificant because it combines ‘two Names, whose significations are contradictory and inconsistent’,^{22} while also insisting ‘that which is not Body, is no part of the Universe: And because the Universe is All, that which is no part of it, is *Nothing*; and consequently *no where*’.^{23} To Wallis and other religious traditionalists, this could only mean ‘that you deny (and not just in words) not only angels, and immortal souls, but also the great and good God Himself’.^{24}

Hobbes reserved some of his harshest words for the proponents of ‘school divinity’, or professors, like Wallis, who worked out subtle theories on the nature of God and the relationship between God and the world. School divines, Hobbes held, are agents of the ‘Kingdome of Darknesse’ whose doctrines are an amalgam of false and pernicious philosophy that rest on ‘canting and fraud’, the intent of which is ‘to entangle shallow wits’ and mislead them.^{25} If Hobbes had his way, Doctors of Divinity such as Wallis would be cast out of the universities and their teachings would be outlawed.

In sum, Wallis saw Hobbes and his philosophy as a threat, not merely to his livelihood but to his entire vision of a well-ordered society. If Hobbes's doctrines were allowed to go unchallenged, the consequences for religious belief, political stability and his own position were potentially catastrophic. Yet, Wallis also found himself in something of a delicate position. He was, after all, a proponent of the ‘new philosophy’ that sought to supplant the traditional Aristotelian-Scholastic approach to nature. As a visible exponent of the mathematics, mechanics and astronomy associated with the ‘moderns’ against the traditionalists,^{26} Wallis needed to show that the benefits of the new philosophy could be had without danger to piety and true religion. As his associate Joseph Glanvill would put the matter in 1665, it was alarming that ‘divers of the brisker *Geniusses*, who desire rather to be accounted *Witts*, then endeavour to *be so*, have been willing to accept *Mechanism* upon *Hobbian* conditions, and many others were in danger of following them into the *precipice*’*.*^{27} The corrective to such a danger lay in showing that the mechanistic principles of the new philosophy could be harmonized with traditional Christian teaching, so that ‘the meanest intellects may perceive, that *Mechanick Philosophy* yields no security to *irreligion*, and that those that would be *gentilely* learned, and ingenious, need not purchase it, at the *dear* rate of being *Atheists*’.^{28} Thinkers such as Wallis and Glanvill opted to confine mechanism to the material world, leaving the non-physical realm of God and the soul beyond the reach of the ‘mechanical philosophy’. Hobbes, in contrast, held that the material world is the only world, so he opted for a (highly controversial) reading of the Scriptures in which neither God nor the soul is an immaterial substance.

Sensitive to the difficulty of showing that embracing the mechanistic new philosophy did not require a commitment to thoroughgoing materialism of the Hobbesian sort, Wallis undertook the refutation of Hobbes's geometry as a way of showing that the philosopher from Malmesbury lacked the intellectual standing to hold forth on weighty matters of religion or politics. One could phrase this by saying that the struggle over questions in geometry was in many respects a struggle for intellectual authority. By refuting Hobbes's geometry, Wallis hoped to show that the Hobbesian philosophy was rotten to the core, and that ‘whoever stumbles so horribly in geometry, where demonstrative proofs have a place, can hardly be thought to walk more securely in other matters’.^{29} In the letter of dedication to the *Elenchus*, he made a similar point, claiming that once Hobbes's geometry had been refuted, ‘that man, so full of airy talk, might be quite deflated and others, less skilled in geometry, may know that there is nothing more to be feared from this Leviathan’.^{30}

## Hobbes and Wallis on the foundations of mathematics

Although the dispute between Hobbes and Wallis was ignited by conflicts involving the universities and questions of church governance, it persisted long after these had been largely resolved with the restoration of the Stuart monarchy. When Charles II assumed the throne in 1660, the radicals' threat to the universities had passed.^{31} Likewise, the struggles over church organization were essentially settled with the 1662 Act of Uniformity. The Restoration did not, however, put an end to the conflict between Hobbes and Wallis, as their exchange of polemics continued well in to the 1670s.^{32} Much of this debate concerned foundational issues in mathematics, most notably the criteria for rigorous demonstration. In the course of the dispute it became clear that Hobbes and Wallis had widely diverging conceptions of mathematics, and it can help to illuminate their dispute by examining their competing accounts of the metaphysics and method of mathematics.

From Greek Antiquity onward, mathematics was traditionally seen as a science whose objects are fundamentally distinct from those of ordinary experience. According to this view, material bodies may approximate the lines, figures and solids of geometry, but the truths of mathematics concern abstract entities different in kind from material bodies. In the Aristotelian tradition, the objects of mathematical investigation are held to be abstracted or mentally separated from material things. The first two definitions in Euclid's *Elements* exemplify this theme.^{33} The line, defined as ‘Length without breadth’, was typically understood as an abstraction in which the concept of length is mentally separated from breadth; likewise, the point ‘That which has no part’ is an abstraction in which the concept of a part is stripped away from that of an object. As the sixteenth-century Jesuit geometer Christopher Clavius explained: ‘[n]o example [of a point] can be found in material things, unless you mean that the extremity of the sharpest needle expresses some similitude to a point; which nevertheless is wholly untrue, since this extremity can be divided and cut to infinity, but a point must be supposed altogether indivisible.’^{34} Wallis accepted this general account, claiming in the first chapter of his *Mathesis Universalis* that ‘We call those parts of mathematics pure in which quantity is considered absolutely, so far as it is abstracted from matter’.^{35}

Hobbes based his approach to mathematics on his distinctively materialist principles and identified the objects of mathematics with material bodies. Where Euclid and the tradition had understood points, lines and surfaces as abstracted from the realm of matter, Hobbes declared:Though there be no Body which has not some Magnitude, yet if when any Body is moved, the Magnitude of it be not at all considered, the way it makes is called a LINE, or one single Dimension; & the Space through which it passeth is called LENGTH; and the Body itself a POINT; in which sense the Earth is Called a

*Point*, and the Way of its yearly Revolution, the *Ecliptick Line*.^{36}

Thus, a Hobbesian point is a material body sufficiently small that its magnitude can be neglected in a demonstration; the path traced by a moving point is a line or curve, and a surface is traced by a moving line. Such points have magnitude and are divisible, although their size and divisibility are disregarded in the course of a demonstration.

Wallis poured scorn on this doctrine, asking: ‘Who ever, before you, defined a point to be a body? Who ever seriously asserted that points have any magnitude?’^{37} In Wallis's view, Hobbes's materialistic foundation for geometry (and, indeed, mathematics generally) fails to account for the abstract, immaterial nature of mathematical objects, which excludes ‘corporeity’ or a dependence upon the structure and contents of the material world.^{38} A recurring theme in Wallis's polemics against Hobbes is that his system has no place for immaterial objects such as God, angels or the soul. Further, he held that this dogmatic commitment to materialism has made Hobbes incapable of understanding the nature of mathematics and its demonstrations, which are not confined to the realm of material objects. Hobbes retorted that Wallis's approach amounted to the doctrine that ‘a point is nothing’, which he took to imply that, on Wallis's principles, there could be no proper science of geometry.^{39}

These differences over the nature of mathematical objects also led to a fundamental difference of opinion on the criteria for rigorous mathematical demonstration. According to a tradition that traces back to the Aristotelian *Posterior Analytics*, proper demonstrations must be based on premises that are true, better known than the conclusion and expressing the causes that bring about the conclusion. This doctrine raised some questions in the case of mathematical demonstrations, since it is not immediately clear how such demonstrations might employ causes. As a result, there was a significant discussion in the sixteenth and seventeenth century of whether mathematics could satisfy the criteria for Aristotelian science.^{40} Wallis defended the causality of mathematical demonstrations, arguing in his *Mathesis universialis* that mathematical definitions specify the essences of the things defined, so that conclusions follow ‘immediately as from a true and proximate cause’.^{41} This kind of causality was understood as ‘formal causality’, in which the form or essence of the defined object functions as the cause of the derived conclusions. As Isaac Barrow (the Lucasian Professor of Mathematics at Cambridge) put it in the sixth of his 1664 *Lectiones mathematicæ*:In truth, such, and no otherwise, is the causality and mutual dependence of the terms of a mathematical demonstration, namely a most close and intimate connection of them with one another; which may yet be called a

*formal causality*, since the remaining affections result from that one property first assumed, as from a form.^{42}

Hobbes was emphatic in his support for the thesis that mathematical demonstrations must be causal, but he rejected the notion of ‘formal causality’ as an incoherent bit of Aristotelian metaphysics that should be abandoned. In his scheme, only mechanical causes (i.e. matter and motion) are admissible. Because he took geometric objects to be material bodies, and motion to be the only cause, Hobbes concluded that proper demonstrations must proceed from definitions that specify the motions that generate figures, angles or solids.^{43} In the twelfth chapter of the tract *De principiis et ratiocinatione geometrarum* he outlined his reasoning:All demonstrations are flawed, unless they produce knowledge, and unless they proceed from causes, they do not produce knowledge. Further, demonstrations are flawed unless their conclusions are demonstrated by construction, that is, by the description of figures, that is, by the drawing of lines. For every drawing of a line is motion: and so every demonstration is flawed, whose first principles do not contain the definitions of motions by which figures are described.

^{44}

This doctrine found no favour with Wallis, who frequently complained that Hobbes's principles are ‘plainly physical’ and far removed from the true understanding of mathematics.^{45}

Hobbes took this conception of demonstration to the extreme of insisting that algebraic or arithmetical calculation are irrelevant to the status of a geometric result. In his view, the process of ‘drawing lines into lines’ that generates a surface is an essentially geometric operation that cannot be identified with algebraic or arithmetical multiplication. When others refuted his supposed geometric results through calculations showing that, for instance, his results conflicted with the established Archimedean bounds for π, Hobbes responded by declaring that such bounds were calculated by an illegitimate intrusion of algebra into geometry. Thus, reliance on trigonometric tables could never impugn a geometric demonstration. Discussing the objections that Wallis and others had raised to one of his supposed circle quadratures, Hobbes declared:But this was opposed by the said professors, in part from the tables of sines, tangents, and secants, and in part from the authority of Archimedes. Yet because those tables were constructed by the multiplication of lines by numbers (whose product they falsely computed as a number of squares), and by the extraction of roots from those squares (which roots they falsely computed as a number of lines), the argument taken from those tables has no refutative force. And since Archimedes himself demonstrates his dimension of the circle by the extraction of roots, his authority need not weigh in this matter.

^{46}

His obstinate insistence on this conception of geometric demonstration as essentially non-algebraic made Hobbes impervious to the shortcomings in his own geometric work and was catastrophic for his entire mathematical programme. Indeed, as previously mentioned, he found himself forced to reject the Pythagorean Theorem in order (as he imagined) to defend his geometric efforts against refutation.^{47}

The divergent conceptions of mathematical demonstration that separated Hobbes and Wallis can be further illustrated in their reaction to the ‘method of indivisibles’ introduced by the Italian mathematician Bonaventura Cavalieri in his 1637 *Geometria inidivisibilibus continuorum nova quadam ratione promota*. Both Hobbes and Wallis accepted the method and employed it, but they held very different views about its foundation and ultimate justification.

The method is rooted in the intuition that we can reason about the relative areas of two figures by considering ‘all the lines’ contained in them. If we take the circle ABCD and the oblong figure EFGH of equal height (figure 1), we can consider the common tangent LM and let it pass through the two figures to arrive in the position IK. In Cavalieri's parlance, the passage of the line LM (which he termed the ‘*regula*’) produces ‘all the lines’ of ‘the indivisibles’ of the figures.^{48} Cavalieri's procedure takes indivisibles to comprise a new kind of magnitude that could be treated in accordance with the traditional doctrine of ratios.^{49} To determine the ratio of the areas of two curvilinear figures, he introduced postulates concerning the relations between indivisibles of figures. Thus, he postulated that the indivisibles of congruent figures are equal; that if figure F_{1} is a proper part of F_{2}, then the indivisibles of F_{1} are less than those of F_{2}; and if figure F_{1} can be decomposed into figures F_{2} and F_{3}, the indivisibles of F_{1} are equal to the sum of the indivisibles of F_{2} and F_{3}. Cavalieri's basic procedure was to establish a ratio between the indivisibles of two figures, and then to conclude that the areas of the figures stand to one another in the same ratio as their indivisibles. In his words:It is clear from this that when we want to find what ratio two plane figures or two solids have to one another, it is sufficient for us to find what ratio all the lines of the figure stand in (and in the case of solids, what ratio holds between all of the planes), relative to a given

*regula*, which I lay as the great foundation of my new geometry.^{50}

This is the basis for the theorem that has become known as ‘Cavalieri's principle’: when two figures have equal altitudes and sections made by lines parallel to their bases at equal distances stand in a given ratio, the areas of the figures also stand in that ratio.

Cavalieri insisted that his method did not treat figures as literally composed from lines, and thus did not encounter the notorious difficulties of composing the continuum from indivisible parts. His successors, however, were far less cautious on this point. Wallis, in particular, opened his 1656 *De Sectionibus conicis nova methodo expositis Tractatus* by declaring: ‘I suppose, to begin with, (according to the Geometry of Indivisibles of Bonaventura Cavalieri) any plane to be made up (so to speak) out of an infinity of parallel lines; or (which I prefer) from an infinity of parallelograms of the same altitude. Let the altitude of any one of them be 1/∞ of the whole … and the altitude of all together being equal to the altitude of the figure.’^{51} In his *Arithmetica infinitorum* (also published in 1656) Wallis approached geometric problems by using infinite series summations to determine the areas of figures, taking the figure to be an infinite sum of infinitesimal elements. To solve the problem of determining the area enclosed by the cubic parabola, he proceeded by comparing ratios of sums of cubic numbers, beginning with the observation that

From these initial cases, Wallis concluded ‘by induction’ that in the infinite case the ratio of an infinite sum of increasing cubic numbers to an infinite sum of cubes equalling the greatest of those cubes must be exactly 1:4. Then, taking the cubic parabola as an infinite sum of lines that increase as the cubic power (i.e. as *x*^{3}) on the interval AT (figure 2), and comparing it to a rectangle composed of an infinite sum of lines equal to the final value TO, he concluded:Therefore the whole figure AOT (Consisting of the infinity of straight lines TO, TO &c. in triplicate ratio of the arithmetically proportional straight lines AT, AT, &c.) will be to the parallelogram TD (consisting of just as many lines all equal to TO) as one to four, Which was to be shown. And consequently, the semiparabola AOD (the residuum of the parallelogram) is to the parallelogram itself as one to four.

^{52}

This approach is difficult to reconcile with traditional standards of rigorous demonstration, both because it relies on a problematic ‘induction’ to the infinite case from a few initial instances and because it takes finite geometric magnitudes to be infinite sums of indivisible, infinitesimal elements. Wallis was content to claim that, properly understood, the method was nothing more than a notational variant of classical techniques,^{53} but this pretence is difficult to take seriously.

Hobbes's attitude toward the method of indivisibles was more complex. On the one hand, he denounced Wallis's use of it, as when he dismissed the Savilian professor's inductions: ‘Egregious Logicians and Geometricians, that think an *Induction* without a *Numeration* of all the particulars sufficient to infer a Conclusion universal, and fit to be received for a Geometricall Demonstration!’^{54} He also argued that Wallis's doctrines destroyed the only possible foundation of the method of indivisibles:you think it will pass for current, without proof, that a Point is nothing. Which if it do, Geometry also shall pass for nothing, as having no ground nor beginning but in nothing. But I have already in a former Lesson sufficiently shew'd you the consequence of that opinion. To which I may add, that it destroys the method of Indivisibles, invented by Bonaventura; and upon which, not well understood, you have grounded all your scurvy book of

*Arithmetica Infinitorum;* where your Indivisibles have nothing to do, but as they are supposed to have Quantity, that is to say, to be Divisibles^{55}

Hobbes insisted that there could be no proper reasoning based on the notion of an infinitely large or infinitely small quantity, and therefore rejected Wallis's methods as misguided and fallacious.^{56}

In the face of such criticisms, one might conclude that Hobbes opposed the method of indivisibles *tout court*. However, a closer look at the evidence shows that Hobbes thought the method was rigorously demonstrative when interpreted against the background of his materialistic ontology for mathematics. Moreover, he actually used the method of indivisibles in the attempt to determine the areas of curvilinear figures.

Hobbes held that the key to the method of indivisibles was to adopt his understanding of points, lines and planes. Taking points as extended bodies whose magnitude can be neglected allows a line or curve to be literally composed of (finitely extended) points. Likewise, taking a straight line to be a parallelogram of determinate length but negligible (yet still extended) breadth enables the composition of plane figures from collections of lines. As he explained in the second dialogue of his 1660 *Examinatio et emendatio mathematicæ hodiernæ*:Those things that can exceed one another when multiplied are homogenous, and these are measureable by a measure of the same kind, as lines are measurable by lines, surfaces by surfaces, and solids by solids. However, things heterogeneous are measured by different kinds of measures. Nevertheless, if lines are considered as the most minute parallelograms, as they are considered by those who use the method of demonstration that Bonaventura Cavalieri calls the doctrine of

*Indivisibles*, then there will be a ratio between a *straight line* and a *plane surface*. And indeed such lines, when multiplied, can exceed any given finite plane surface.^{57}

Thus, Hobbes held that taking points to be extensionless ‘nothings’ or lines to be breadthless lengths makes it impossible to use the method; but he saw his own materialistic foundations for geometry as putting indivisibles on a secure foundation that avoids the use of infinitesimals and delivers correct results.

In the seventeenth chapter of *De corpore* Hobbes adopted the method of indivisibles and used it to find the areas of figures that he termed ‘deficient’.^{58} In his terminology, the deficient figure *ABEFC* is made by the motion of the line *AB* as it moves toward *CD*, diminishing continually until it vanishes at point *C.* The complement of the deficient figure is *BDCFE*, and the sum of the deficient figure and its complement will be the rectangle *ABDC*. The area of the figure is determined by the rate of diminution: if the rate is completely uniform, the diagonal *BC* results and the ratio of deficient figure to its complement will be 1:1. Where the rate of diminution is not uniform, a curvilinear figure will result. Hobbes's task in the seventeenth chapter is to establish a general rule for computing the ratio of the area of a deficient figure to its complement, given a specified rate of diminution. The central theorem is stated in the second article of chapter 17:A Deficient Figure which is made by a Quantity continually decreasing to nothing by proportions every where proportionall and commensurable, is to its Complement, as the proportion of the whole altitude, to an altitude diminished in any time, is to the proportion of the whole Quantity which describes the Figure, to the same Quantity diminished in the same time.

^{59}

Thus, if the line *AB* (figure 3) decreases as the square of the diminished altitude, the area of the deficient figure will be twice that of the complement. And, more generally, if the line *AB* decreases as the *n*th power, the ratio of the deficient figure to its complement will be *n*:1*.* The critical step in Hobbes's argument is to take the deficient figure *ABEFC* to be composed of all the lines parallel to AB (such as *HF* and *GE*) while its complement is composed of all the lines parallel to BD (such as *QF* and *OE*).^{60}

What emerges from this investigation is that the Hobbes–Wallis controversy involved significant differences over mathematical method and the criteria for proper demonstration. Thus, the conflict went beyond certain technical errors in Hobbes's supposed geometric results, nor was it simply a quarrel about the status of English universities or the limits of sovereign power in matters of religion. It should be noted that Hobbes's mathematical programme had certain strengths. His rejection of Wallis's use of infinitesimal magnitudes and his problematic ‘inductions’ to the infinite case comport well with traditional standards of rigorous demonstration. On the other hand, his attempted demonstrations of important results almost invariably involve fallacious reasoning, and his rejection of algebraic methods condemned his mathematics to failure.

## Consequences of the dispute

The most salient result of this long-running dispute was the demolition of Hobbes's reputation as a geometer. He had clearly hoped to establish himself as one of the great mathematicians of Europe by delivering long-sought solutions to central problems, yet his efforts on this front ensnared him in a mass of self-contradiction and incoherence. This is best evidenced by the fact that his various circle quadratures yield quite different results—so that if the reasoning behind any one of them were (*per impossible*) true, the rest would be destroyed;^{61} yet Hobbes insisted that they were all correct and properly demonstrated.^{62} Beyond this, Hobbes found himself jettisoning established mathematical results such as the Pythagorean Theorem in the vain attempt to shield his supposed circle quadratures from refutation.

Hobbes also failed to win any adherents to his thoroughly materialistic conception of geometry. The programme of *De corpore*, which understands geometric objects as literal physical bodies endowed with magnitude, was disregarded by philosophers and philosophically-minded geometers in the seventeenth century. This is despite the fact that some of Hobbes's criticisms of Wallis's use of indivisibles were generally on the mark. Wallis's introduction of infinitesimal elements and his practice of relying on shaky ‘inductions’ from a few initial cases are impossible to reconcile with traditional criteria for rigorous demonstration. Yet these methods were never seriously called into question, while Hobbes's mathematics was almost universally ignored. Thus, if we think of the dispute in purely mathematical terms, Wallis emerges as the clear winner.

Nevertheless, there is an odd irony here. For all that Wallis could triumph over Hobbes's mathematical efforts, he did not succeed in his principal goal of destroying Hobbes's reputation as a philosopher and political theorist. Hobbes is still read widely today, and his importance for political philosophy is undeniable. Wallis, although hardly an obscure figure, never attained the degree of influence that Hobbes had over subsequent generations of theorists. As we have seen, Wallis was convinced that the refutation of Hobbesian geometry was the way to refute Hobbes's entire philosophy, but in this he was very much disappointed. Thus, if we keep score by asking which of our two combatants had the greater impact on the development of European intellectual history, it seems that Hobbes came out the winner.

Further, Hobbes himself saw his political philosophy as intimately connected to his theory of demonstration, and he accepted Wallis's contention that a refutation of his geometry would be catastrophic for his political theory. He notoriously claimed that both civil philosophy and geometry are demonstrable sciences, since we know the causes that bring about both civil institutions and mathematical objects.^{63} Indeed, this is precisely why Hobbes insisted on maintaining the adequacy of his false quadratures and other failed results. To have admitted defeat on this front would have entailed, at least in his mind, the outright demolition of his entire system of philosophy.

The irony is that readers of both Hobbes and Wallis seem not to have shared this view of Hobbes's philosophical project. Hobbes's reputation as a philosopher and political theorist survived, but it did so because his readers were prepared to disregard his claims to have delivered a systematic, unified treatment of first philosophy, geometry, natural philosophy and politics. Citizens of the seventeenth-century Republic of Letters were prepared to detach Hobbes's geometry from his political theory, and in doing so they made his geometric defeat far less significant than it would otherwise have been. In other words, Hobbes's intellectual reputation survived his battle with Wallis, but it did so because his readers took his philosophy in a manner rather different from what its author had intended.^{64}

## Footnotes

↵1 Thomas Hobbes,

*Elementorum Philosophiae Sectio Prima: De corpore*(Andreæ Crook, London, 1655) is universally known as*De corpore*, as is the English translation*Elements of Philosophy, the First Section, Concerning Body*(Printed by R. & W. Leybourn for Andrew Crooke, London, 1656). I quote from the latter text.↵2 The quadrature or squaring of the circle is the problem of constructing a square equal in area to a given circle; the problem of arbitrary angular section is to divide a given angle into any number of equal angles; the duplication of the cube is the problem of constructing a cube double in volume to a given cube. These are all now known to be unsolvable with the resources of Euclidean geometry. For details, see Douglas Jesseph,

*Squaring the circle: the war between Hobbes and Wallis*(University of Chicago Press, Chicago, 1999), pp. 16–32.↵3 ‘Christiaan Huygens to John Wallis, 5/15 March 1656, letter 72’, in

*The correspondence of John Wallis*(*1616–1703*) (ed. Philip Beeley and Christoph Scriba), 4 vols (Oxford University Press, Oxford, 2003), vol. 1, p. 176.↵4 John Wallis,

*Due Correction for Mr. Hobbes; or Schoole Discipline, for not saying his Lessons right*(Printed by Leonard Lichfield for Tho: Robinson, Oxford, 1656), p. 3.↵5 Thomas Hobbes,

*Six lessons to the professors of the mathematiques one of geometry the other of astronomy, in the chaires set up by the noble and learned Sir Henry Savile in the University of Oxford*(Printed by J.M. for Andrew Crook, London, 1656), pp. 63–64.↵6 Richard Westfall,

*Science and religion in seventeenth-century England*(Yale University Press, New Haven, 1958), p. 18.↵7 See Jesseph,

*op. cit.*, (note 2), p. 353 for an account of Wallis's long list of polemical and controversial publications.↵8 Thomas White,

*Exercitatio geometrica. De geometria indivisibilium, & proportione spiralis ad circulum*(London, 1658).↵9 For an account of the controversy engendered by White's failed quadrature, see Howard W. Jones, ‘A seventeenth-century geometrical debate’,

*Ann. Sci.***31**, 307–333 (1974).↵10 ‘Wallis to Huygens, 22 December 1658/1 January 1658’, letter 181 in Beeley & Scriba,

*op. cit.*(note 3), vol. 1, p. 539.↵11 John Wallis, ‘Preface’, in

*Opera Mathematica*(E Theatro Sheldoniano, Oxford, 1695), vol. 1, sig. A4v.↵12 Thomas Hobbes,

*Leviathan*(ed. Noel Malcolm), 3 vols (Oxford University Press, 2012), vol. 2, ch. 11, p. 158.↵13

*Ibid.*, vol. 2, ch. 30, p. 538.↵14 For details on Puritan criticisms of English education, see Alan Debus,

*Science and education in seventeenth-century England: the Webster–Ward debate*(MacDonald, London, 1970). All translations are my own unless otherwise attributed.↵15 Austin Woolrych,

*From commonwealth to protectorate*(Oxford University Press, Oxford, 1982), at pp. 191–194.↵16 For additional discussions of Hobbes and the universities as they relate to the dispute with Wallis, see Ted H. Miller,

*Mortal gods: science, politics and the humanist ambitions of Thomas Hobbes*(Pennsylvania State University Press, University Park, PA, 2011), ch. 7; Jon Parkin,*Taming the Leviathan: the reception of the political and religious ideas of Thomas Hobbes in England, 1640–1700*(Cambridge University Press, Cambridge, 2007), ch. 3.↵17 After a detailed rebuttal to Hobbes's charges against the universities in

*Leviathan*, Ward offered that ‘I have heard that M.*Hobbes*hath given out, that he hath found the solution of some Problemes, amounting to no lesse then the Quadrature of the Circle, when we shall be made happy with the sight of those his labours, I shall fall in with those that speake loudest in his praise’; Seth Ward,*Vindiciæ academiarum containing some briefe animadversions upon Mr Websters book stiled, The examination of academies : together with an appendix concerning what M. Hobbs and M. Dell have published on this argument*(Printed by Leonard Lichfield … for Thomas Robinson, Oxford, 1654), p. 57.↵18

*Ibid.*, p. 58.↵19

*Ibid.*, p. 52. Wallis expressed a similar sentiment in the ‘Epilogue’ to the*Elenchus*, attacking Hobbes for supposedly claiming that ‘all of the ancient or recent universities were never of any use, nor will they ever be, unless they are taught by you’. John Wallis,*Elenchus Geometriae Hobbianae : Sive, Geometricorum, quae in ipsius Elementis Philosophia, à Thoma Hobbes Malmesburiensi proferuntur, Refutatio*(Johannis Crook, Oxford, 1655), p. 135.↵20 In 1654 he took part in a public disputation on the question of church governance, maintaining the Presbyterian view that proper Christian ministers can exercise their authority beyond the confines of a specific congregation. His contribution was published in 1657 under the title

*Mens Sobria Seriò Commendata*(Leonard Litchfield for Thomas Robinson, Oxford, 1657). It is interesting to note that Wallis addressed the dedicatory epistle to his*Elenchus*to Owen. This fact suggests that he saw significant utility in making common cause with the Independents against the dreaded Hobbes. For more on Wallis and religion, see Philip Beeley and Siegmund Probst, ‘John Wallis (1616–1703): mathematician and divine’, in*Mathematics and the divine: a historical study*(ed. Teun Koetsier and Luc Bergmans), pp. 441–457 (Elsevier, Amsterdam, 2005).↵21 Whether Hobbes's materialist philosophy can be consistently combined with theism is considered in detail in Jesseph, ‘Hobbes's Atheism’,

*Midwest Stud. Philos.***26**, 140–166 (2002).↵22 Hobbes,

*op. cit.*(note 12), vol. 2, ch. 12, p. 60.↵23 Hobbes,

*op cit.*(note 12), vol. 3, ch. 46, p. 1076.↵24 Wallis,

*op. cit.*(note 19), p. 90.↵25 Hobbes,

*Στίγμαι, Αγεομετρίας, Αγροικίας, Αντιπολιτείας, Αμαθείας*;*or Markes of the Absurd Geometry, Rural Language, Scottish Church-Politicks, and Barbarismes of John Wallis Professor of Geometry and Doctor of Divinity*(Andrew Crooke, London, 1657), p. 13.↵26 Although Wallis's approach to natural philosophy (and particularly mechanics) was a clear departure from the Aristotelian tradition, his view of the foundations of mathematics owed a significant intellectual debt to the tradition. His use of infinitesimal techniques seems to fall between ancient and modern conceptions. For further information on this point see Katherine Hill, ‘Neither ancient nor modern: Wallis and Barrow on the composition of continua, part I: mathematical styles and the composition of continua’,

*Notes Rec. R. Soc. Lond.***50**, 165–178 (1996) and Katherine Hill, ‘Neither ancient nor modern: Wallis and Barrow on the composition of continua, part II. The seventeenth-century context: the struggle between ancient and modern’,*Notes Rec. R. Soc. Lond.***51**, 13–22 (1997).↵27 Joseph Glanvill, ‘Preface', in

*Scepsis scientifica, or, Confest ignorance, the way to science in an essay of The vanity of dogmatizing, and confident opinion : with a reply to the exceptions of the learned Thomas Albius*(Printed by E. C. for Henry Eversden, London, 1665), sig. B1.↵28 Ibid.

↵29 Wallis,

*op. cit.*(note 19), p. 4.↵30 Wallis,

*op. cit.*(note 19), epistle, sig. A3.↵31 This is not to say that the Restoration brought peace and harmony to the universities. Many (such as Ward and Wallis) who owed their positions to the Protectorate had to chart a careful course through a perilous professional and political landscape. Wallis, in particular, had to worry that his service as cryptanalyst for the Parliamentary forces would come back to haunt him. Nevertheless, Wallis had no need to fear that Hobbes's particular critique of the universities posed any threat after 1660, as there was no move for a radical reworking of the universities along Hobbesian lines. This accounts for the fact that the discussion of the role of universities vanished entirely from the polemical exchanges between Hobbes and Wallis after 1657.

↵32 The persistence of the controversy beyond the Restoration seems to have been due in part simply to inertia, with neither party willing to abandon the fight even after the status of the universities and the Church of England had been resolved. The conflict also spilled over into issues of grammar and philology, as the disputants traded charges of stylistic inelegance, which are most notably catalogued in Hobbes's

*Στλίγμαι*,*op. cit.*(note 25). Miller (*op. cit*. (note 16)) argues that Hobbes's mathematical interests were fully consistent with his status as a ‘humanist’, and the level of detailed philological and grammatical invective in*Στλίγμαι*abundantly confirms that point. The continued wrangling over issues of demonstration and mathematical method became central to the dispute, however, as these ultimately concerned the validity of Hobbes's putative quadratures and the viability of his entire mathematical programme.↵33 Euclid,

*The thirteen books of Euclid's ‘Elements’ translated from the text of Heiberg with introduction and commentary*(ed. and trans. Thomas L. Heath), 3 vols (Cambridge University Press, Cambridge, 1925), vol. 1, bk. 1, definitions 1, 2, p. 153.↵34 Christopher Clavius,

*Christophori Clavii Bambergensis E Societate Jesu Opera Mathematica V Tomis distributa*, 5 vols (Reinhard Eltz, Mainz, 1612), vol. 1, p. 13.↵35 Wallis,

*op. cit.*(note 11), p. 18.↵36 Thomas Hobbes,

*Elements of Philosophy, the First Section, Concerning Body. Written in Latine by Thomas Hobbes of Malmesbury and now translated into English*(R. and W. Leybourn for Andrew Crooke, London, 1656), part II, ch. 8, sect. 12, p. 81.↵37 Wallis,

*op. cit*. (note 19), p. 6.↵38

*Ibid.*, p. 7.↵39 As Hobbes argued in the second of his

*Six Lessons*: ‘To them therefore that deny a Point to have Quantity, that is, a Line to have Latitude, the forenamed Principles are not possible, and consequently no proposition in Geometry is demonstrated, or demonstrable. You therefore that deny a Point to have Quantity, and a Line to have Breadth, have nothing at all of the Science of Geometry’;*Six Lessons to the Professors of the Mathematiques, one of Geometry, the other of Astronomy*(Printed by J.M. for Andrew Crook, London, 1656), p. 12.↵40 This ‘Quaestio de Certitudine Mathematicarum’ is discussed in detail in Paolo Mancosu,

*Philosophy of mathematics and mathematical practice in the seventeenth century*(Oxford University Press, Oxford, 1996), ch. 1.↵41 Wallis,

*op. cit.*(note 11), p. 24.↵42 Isaac Barrow,

*The mathematical works of Isaac Barrow, D. D.*(ed. William Whewell), vol. I (Cambridge University Press, Cambridge, 1860), p. 93. This way of conceptualizing the causality of mathematical demonstration is examined in Helena Pycior, ‘Mathematics and philosophy: Wallis, Hobbes, Barrow and Berkeley’,*J. Hist. Ideas***48**, 265–286 (1987).↵43 David Sepkowski (2005) has drawn attention to Hobbes's ‘constructivism’ and his ‘nominalism’, or the rejection of the notion that the objects of mathematical investigation are non-physical, ideal objects; Sepkowski, ‘Nominalism and constructivism in seventeenth-century mathematical philosophy’,

*Hist. Math.***32**, 33–59 (2005).↵44 Thomas Hobbes,

*De principiis et ratiocinatione geometrarum*(Andream Crooke, London, 1666), at p. 28.↵45 Wallis,

*op. cit.*(note 19), p. 10.↵46 Hobbes,

*op. cit.*(note 44), pp. 50–51.↵47 In Chapter 23 of

*De principiis et ratiocinatione geometrarum*, Hobbes declared: ‘if these things of mine are correctly demonstrated, there are some further things that you should consider. First, the greater part of the propositions that depend upon proposition 47 of book I of Euclid (and there are many) are not yet demonstrated. Second, the tables of sines, tangents, and secants are entirely false’;*op. cit.*(note 44), p. 61.↵48 Analogously, the transit of a plane

*regula*through a solid would produce ‘all the planes’ of the solid, or ‘the indivisibles’ of the solid. I will consider only the two-dimensional case, however.↵49 Thus, a fundamental result in the

*Geometria*is the third theorem of the second book: ‘All the lines … of any plane figures, and all the planes of any solids, are magnitudes having a ratio to one another’, Bonaventura Cavalieri,*Geometria indivisibilibus continuorum nova quadam ratione promota*(Ex Typographia de Ducijs, Bologna, 1635), at p. 108. For details on the method, see Kirsti Andersen, ‘Cavalieri's method of indivisibles’,*Arch. Hist. Exact Sci.***28**, 292–367 (1985).↵50 Cavalieri,

*ibid.*, p. 115.↵51 Wallis,

*op. cit.*(note 11), vol. 1, p. 297*.*↵52 Wallis,

*op. cit.*(note 11), vol. 1, p. 383.↵53 Thus, in his

*Treatise of Algebra*, Wallis contended: ‘The Method of Exhaustions (by Inscribing and Circumscribing Figures, till their difference becomes less than any assignable,) is a little disguised, in (what hath been called) Geometria Indivisibilium … which is not, as to the substance of it, really different from the Method of Exhaustions, (used both by Ancients and Moderns,) but grounded on it, and demonstrable by it: But is only a shorter way of expressing the same notion in other terms’;*A Treatise of Algebra, Both Historical and Practical*(Printed by John Playford, for Richard Davis, London, 1685), p. 285.↵54 Hobbes,

*op. cit*. (note 39), p. 46.↵55 Hobbes,

*op. cit*. (note 39), p. 36.↵56 This is summed up in Hobbes's

*Principia et problemata aliquot geometrica*(J.C. pro Gulielmo Crook, London, 1674), ch. 13. The chapter is entitled ‘On the Infinite’ and rejects the possibility of correct reasoning about infinite totalities.↵57 Hobbes,

*Examinatio et emendatio mathematicæ hodiernæ*(Andreæ Crooke, London, 1660), pp. 47–48.↵58 For more on this aspect of

*De corpore*and Hobbes's debt to Cavalieri, see Jesseph, ‘Hobbes on the ratios of motions and magnitudes: the central task of*De corpore*, part III’,*Hobbes Stud.***30**, 58–82 (2017).↵59 Hobbes,

*op. cit.*(note 36), part III, ch. 17, art. 2, p. 182.↵60 In his words, ‘the proportion of the complement BEFC to the deficient figure ABEFC, is all the proportions of DB to OE, and of DB to QF, and of all the lines parallel to DB terminated in the line BEFC, to all the parallels to AB terminated in the same points of the line BEFC. And … therefore the deficient figure ABEFC which is the aggregate of all the lines HF, GE, AB, &c. is triple to the complement of BEFCD made of all the lines QF, OE, DB, &c.’,

*op. cit.*(note 36), part III, ch. 17, art. 2, pp. 182–183.↵61 Wallis catalogued a dozen of Hobbes's quadratures, each of which yields a different result for the value of π in his

*Hobbius Heauton-timorumenos; or a Consideration of Mr. Hobbes his Dialogues in an Epistolary Discourse Addressed to the Honourable Robert Boyle, Esq.*(A. & L. Lichfield, Oxford, 1662), pp. 104–111. This list does not include others that Hobbes would later produce, giving values for π that range from 3.1547 to √10 to 3.2.↵62 In his posthumously published

*Vita*, an unrepentant Hobbes declared that he had solved a variety of great geometrical problems, including ‘to exhibit a straight line equal to the arc of a circle and a square equal in area to a circle, and this by various methods in several books’; Hobbes,*Thomæ Hobbes Angli Malmesburiensis Philosophi Vita*(Eleutherium Anglicum, London, 1681), at p. 15.↵63 Hobbes,

*op. cit.*(note 39), epistle.↵64 This paper continues my work on the Hobbes–Wallis controversy that appeared in

*Squaring the circle, op. cit.*(note 2). In the ensuing decades, I have come to see that study as deficient in some respects, and I attempt to correct them here. First,*Squaring the circle*depicts the Hobbes–Wallis controversy as (in its initial stages) concerned almost exclusively with the status of the universities. I now believe that approach is too narrow: the university debate was significant, but Wallis was at least as motivated by a concern to defeat the religious and moral dangers he saw in Hobbes's philosophy. Second,*Squaring the circle*treats much of Hobbes's mathematical practice as guided by a ‘method of motion’ that he claimed to have invented, and consequently takes the method of indivisibles as decidedly secondary. I now hold that the method of indivisibles is more significant for Hobbes's mathematics. Not only did he employ it far more frequently (from the 1650s through the 1670s), but he claimed that the only possible foundation for the method was in his thoroughly materialistic conception of mathematics. Third,*Squaring the circle*has relatively little to say about the causality of mathematical demonstration, except as it concerns the distinction between analytic and synthetic methods. I now believe that Hobbes's insistence on the necessity of causes is a central piece of his philosophy of mathematics. Further, the difference between Hobbes and Wallis on how to understand the causality of mathematical demonstrations puts their opposing programmes in sharper relief than I had previously understood.

- © 2018 The Author(s)

Published by the Royal Society.