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Journal of the History of Ideas 57.2 (1996) 217-231
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Squaring the Circle: Hobbes on Philosophy and Geometry

Alexander Bird


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 [End Page 217] 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. [End Page 218]

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 knowing first their effects." 4

This definition brings to the fore two essential elements in philosophical knowledge; (1) its content, viz., knowledge of cause and effect, the causal relation, and (2) its method, viz. true ratiocination. Hobbes initially describes "true ratiocination" somewhat opaquely as "addition and subtraction of ideas." This is made clearer in the chapter on "logique," where the various forms of the syllogism appear as the paradigms of true ratiocination. The essential feature of the latter thus illustrated is that from true premisses only true conclusions follow. In modern jargon, true ratiocination is thus purely deductive.

This formalism is a vestige of Aristotelianism in Hobbes's thought. It contrasts with Bacon's criticism of the syllogism: "For in all ordinary logic almost all the work is spent about the syllogism.... I on the contrary, reject demonstration by syllogism as acting too confusedly and letting nature slip out of its hands." 5

Hobbes's emphasis on the syllogism is a direct consequence of his doctrine of names and propositions. Names come in two varieties, proper and common. Proper names, which include definite descriptions and demonstratives, serve to pick out specific individuals, e.g., "he that writ the Illiad, Homer, this man, that man." Common names do not pick out specific individuals, but any individual belonging to a class. They are "names of things taken severally, but not collectively or all taken together." These are universals, but "universal" does not name anything to be found in nature; it only names a word or name showing that the latter is a name common to many things. The working of common names depends on our ability (the imagination) to identify particulars of which this common name is a name.

At this point Hobbes says, "One universal name is imposed on many things, for their similitude in some quality, or other accident." 6 We should note Land's comment: "But this is a thinly disguised return to the philosophy of realism: the features by virtue of which particulars receive their common names are the universals whose real existence was initially denied." 7

A proposition is, for Hobbes, the conjunction of two names by a copula. That the proposition is true lies in this: the extension of the second named includes the extension of the first named, e.g., "man is a living creature," the class of living creatures includes the class of men. [End Page 219]

Since the truth of propositions is simply a matter of "class inclusion" reasoning can be performed without reference to the entities signified by the terms. At this point reasoning becomes mathematical, on precisely those considerations which stimulated the logic of Boole and Venn or, to use Hobbes's term, just "reckoning." The classic form of such reckoning is clearly the syllogism. (A is a subset of B and B is a subset of C implies A is a subset of C, and A is a subset of B and C is an element of A implies C is an element of B, etc.)

Cause is "the sum or aggregate of all such accidents, both in the agents and in the patient, as concur to the producing of the effect propounded, all of which existing together, it cannot be understood but that the effect existeth with them; or that it can possibly exist if any of them be absent." 8 That is, the cause is that set of accidents which are together sufficient and individually necessary to my conceiving of the effect's existence. Although Hobbes does not propose anything like a Humean projectionism with respect to causes, it is clear that his nominalism inclines him to put emphasis on our conceptions rather than on ineffable (causal) connexions between universals.

Hobbes's term for the causal relation is "generation." The operation of cause to produce an effect is the generation of that effect. We cannot conceive, says Hobbes, but that causes operate through motion; and so the subject matter of philosophy is body, for it is body of which there is generation, which has appearances (effects) and which has motion.

Bodies, it happens, come in two varieties, natural and political (the body politic is the commonwealth) and hence the division of philosophy into natural and political. (Hobbes adds a third intermediate subject, Man.) In such cases the concern of philosophy is precisely the operation of cause and effect between the various bodies; and though its operation in political bodies may be through subtle and complex paths, exact knowledge of this operation requires knowledge of the motions of those bodies and those mediating between them.

Thus far Hobbes's doctrines are perspicuous. His views on method, which is "the shortest way of finding out effects by their known causes, or of causes by their known effects," 9 lacks perspicuity to the degree of not only being confusing but confused as well. His conception is this: method comes in two processes, analysis and synthesis, which operate on facts or ideas. The former is resolutive, of singular entities into more general ones; the latter is a matter of composition, of general principles to form ideas or conceptions of specific, singular, and complex things. Examples give the analysis of man as figurative, rational, and animal; of square as equilateral, quadrilateral, and right angled; gold as solid, visible, and heavy, etc. In such cases analysis employs the definitions of the terms involved; eventually we end up with the most general ideas, ideas of body and motion. [End Page 220]

The natures of these most general ideas are governed by general principles, usually specifying a cause or manner of generation. Using these principles and applying the syllogism, we can work our way back along the analysed path, to the singular object under consideration. Thus this method will reveal the relevant properties of this object and is synthetical. This process of analysis then synthesis is the general method of science. The general principles are either truths which are manifest ("known to nature") or definitions according to cause. As mentioned these will in general have to mention cause or manner of generation, for the aim of science is to give the cause of (all) things, and details of cause will occur in the conclusion of our process of syllogistic reasoning only if found in the initial premisses.

The process of analysis, unlike synthesis, does not, however, appear always to be one of syllogistic ratiocination. It plays a role also in the invention of general principles from sense and experience. For example, we may seek the cause of light, and in so doing, we observe the things about us, seeing that whensoever light appears there is a source of light etc. In general: "we must examine singly every accident and ... see whether the ... effect may be conceived to exist, without the existence of any of those accidents." 10 This aspect of analysis requires the operation of the faculties of imagination and sense and is not purely "analytic" in a Russellian sense. To this extent Hobbes's view of science is neither purely conventionalist, as is claimed by Madden, 11 nor thoroughly rationalist as is, for instance, that of Leibniz.

In fact Hobbes equivocates as to how wide the notion of "definition" is to be taken. Above I distinguished between definition and general principles of cause or generation. But Hobbes also tells us that "such [universal] principles are nothing but definitions ..." having just told us that universal principles are known to us "by nature." 12

In parallel with the notion of analysis, we cannot make a clear distinction between definitions as purely verbal postulates and universal principles known to us by nature (and thus to some extent a posteriori). Definitions, we are told, are nothing but the explications of our simple conceptions. They can be faulty, by failing to raise a clear idea in the mind, of the thing defined. Refusal to acquiesce in a definition is equated not with refusal to enter the arena (language) of debate but with a refusal to be taught--implying that a definition should have an educative content.

Returning to the definition of philosophy, we note that true ratiocination gives us knowledge of effects from causes and of causes from effects. Since definitions are in terms of causes ("generation"), there is an inbuilt asymmetry between these two arms of ratiocination. If we know the cause, we know precisely the effect: e.g., the circumduction of a point produces a circle. It is [End Page 221] important to note that to know the effect is to know that name of the effect, that is, what is supplied by the generative definition.

But on the other hand, given any effect, say a drawn figure, there is no such precise knowledge of cause. If our figure is something that looks like a circle, there is no telling from its perceived form what caused it and thus whether in fact it is a circle. Thus we need at least to see that effect as it is generated in order to find its cause. Even this is not sufficient, since we are not immediately aware which aspects (accidents) of the generation are relevant to the production of the required effect. This, as remarked above, requires experience and imagination. This asymmetry, of deductive reasoning from cause to effect but non-deductively from effect to cause, accounts for the noted asymmetry between synthesis and analysis. It is also the asymmetry which accounts for the distinction between demonstrable and non-demonstrable arts.

The typical form of general principles and definitions is hypothetical: "if this is a man, it is animal, rational, figurative"; "if there is a cause c there follows effect e." This is simply a consequence of the doctrine of propositions, whereby a proposition consists of two names, one of which is posited to include the other. By employing the syllogism we generate a further hypothetical proposition connecting two names previously unconnected. This is defined as demonstration ("a syllogism or a series of syllogisms devined and continued, from the definitions of names to the last conclusion") and is clearly synthetic.

I have shown how scientific knowledge for Hobbes is primarily hypothetical and demonstrative. The hypotheses are from cause to effect, the result of which is, if we know the causes, we know by deductive demonstration, the effect. When do we know the cause, when a cause is operating, and what it is? Only, Hobbes argues, in cases of human agency, for only in such cases do we not need to seek the cause--it is "in us." In cases of natural agency we do need to seek the cause, and as argued above, seeking causes is analytic not synthetic, i.e., not by demonstration. This doctrine, that the only true knowledge is knowledge possessed by the maker of his creation, is aptly summed up by Hobbes in the dedicatory epistle in the Six Lessons:

Of arts some are demonstrable, others indemonstrable; and demonstrable are those construction of the subject whereof is in the power of the artist himself, who, in his demonstration, does no more but deduce the consequences of his own operation. The reason whereof is this, that the science of every subject is derived from a precognition of the causes, generation, and construction of the same; and consequently where the causes are known, there is place for demonstration, but not where the causes are to seek for. Geometry therefore is demonstrable, for the lines and figures from which we reason are drawn and described by ourselves; and civil philosophy is demonstrable, [End Page 222] because we make the commonwealth ourselves. But because of natural bodies we know not the construction, but seek it from the effects, there lies no demonstration of what the causes be we seek for, but only of what they may be. 13

Clearly philosophy is prescriptive not descriptive. It tells how we may bring something about but not how something we merely observe (pace Boyle) occurs. This dovetails nicely with Hobbes's view as to the value and end of philosophical thought. This end is the commodity of human life, rather than the discovery of obscure "truths" or the production of intellectual pleasure. "The scope of all speculation," he says, "is the performing of some action, or thing to be done." 14 Natural philosophy finds its end in "measuring matter and motion; of moving ponderous bodies; of architecture; of navigation; of making instruments for all uses; of calculating the celestial motions, the aspects of the stars, and the parts of time; of geography etc." 15 Correspondingly the aim of civil philosophy is the preservation of peaceful order and the prevention of civil strife. In De Cive Hobbes remarks that if moral philosophy had made similar progress to that achieved by geometry, then human happiness would have been increased, for example, by the prevention of wars. (Hobbes's claim in De Corpore is that he himself has originated civil philosophy in De Cive.) Overall then, we may say that philosophy is technology put to the improvement of the human lot.

III. Our interest in Hobbes's geometry must primarily be with the natures of bodies and their motions. Prior to our understanding of body must come a grasp of the concept of space. For Hobbes defines body as "that, which having no dependence on our thought is coincident or coextended with some part of space." 16 Bodies are real in being independent of thought, but space is not; it is imaginary, "the phantasm of a thing existing without the mind simply." 17 This is what he calls "imaginary space." This is distinguished from "real space," which is simply the magnitude of some object as measured.

There are three possible approaches to understanding imaginary space.

(1) Imaginary space is Cartesian psychological space; that is, it is the inner visual field, the stage for Cartesian ideas to appear upon. Support for this interpretation lies in Hobbes's Cartesian definition of body as extension and [End Page 223] in Hobbes's own interest in the relation between extension in psychological space, i.e., the size of visual images, and the extension of the objects themselves. 18 The problem with this Cartesian view is that it does not explain how, while space may be imaginary, body may be real. For just as space is a Cartesian idea, so is any body; and in so far as a veracious God guarantees the real existence of bodies, he also guarantees the real existence of space.

(2) Brandt suggests that space is imaginary in so far as when it is conceived by vision it is only motion in the observer not an accident in the thing perceived; it is not real. 19 But then any quality or thing for that matter, bodies included, as conceived by vision will be simply motion or some such in the observer. Again the difficulty lies in saying what makes space imaginary while allowing bodies to be real.

(3) Space may not be real, for this is to allow the possibility that parts of it may not be filled by bodies. It must thus be imaginary; the reality of space is no more than coextension with (the magnitude of ) some body.

This last argument leaves unexplained what "imaginary" is supposed to mean. It does, however, suggest an account like this: what makes body real is its being mind independent; it is not simply an idea or conception. What makes space imaginary is the converse; it is simply a conception and as such is dependent on some thought. It is the conception of extension, specifically the extension of some body. It is an abstraction and so, considered in itself, is imaginary; considered as the extension of some body, it is real. For the extension of a body is not mind dependent. Real and imaginary space are not two sorts of space but the same space, the extension of a body considered in two different ways: as the magnitude of a body (real) or as the abstracted extension (not necessarily with any body) and so existent only in thought (hence imaginary).

Hobbes's "definition" of body, given above, is a consequence of this explanation. Since space, real or imaginary, is the extension of some body, bodies are coincident or coextended with some part of space. Furthermore, bodies are real, that is, independent of any thought.

Having defined body, Hobbes is in a position to define the concepts of geometry. A point, he says, is a body considered without its magnitude. The path of a body through space considered without its breadth is a geometrical line. A superficies (surface) is the space made by the motion of a body considered as a line (i.e., considered without its depth). 20 These definitions are characteristically Hobbesian. They contrast most obviously with definitions 1.3 and 5 of Euclid: "A line is a breadthless length," and "a surface is [End Page 224] that which has length and breadth only." 21 Hobbes's rejection of these traditional definitions reflects his nominalism. As traditionally defined "point," "line," etc., could not be names, singular or general, since no thing could have no parts or have length but not breadth. Just as Berkeley was to argue against the Lockean abstract idea of a triangle, so in Hobbes's view, these Euclidean definitions of abstractions, being incomplete (and defined to be incomplete), could not be instantiated. As such, the Euclidean terms were not names but meaningless symbols. By contrast Hobbes's points were not incompletely defined. Like anything, they have magnitude. It is simply that bodies as points are considered without their magnitude.

Quite clearly these points do have instantiations: in any body we care to consider without its magnitude, typically those bodies used in actual geometrical figures, or other bodies considered geometrically (e.g., the planets). Euclidean quantities, leaving aside their meaninglessness, have no instantiations; a fortiori such geometry has no application. However, a point, being a body considered without its magnitude, is not real but imaginary, just as space is imaginary, being a body's extension considered without the body; for both are things considered in a particular way and hence dependent on our thought.

Another aspect of Hobbes's philosophy, in addition to his nominalism, which is illustrated by these definitions, is his motionalism or generationalism: that the things of which we have scientific knowledge are things generated--this knowledge being knowledge of their manner of generation or cause, and this generation being produced by motion. Hence the line is defined in terms of how to make a line, viz., by moving a body.

About Euclid's fifth definition--"a circle is a plain figure comprehended by one line which is called the circumference, to which circumference all the straight lines drawn from one of the points within the figure are equal to one another" 22 --Hobbes remarks:

This is true. But if a man had never seen the generation of a circle by motion of a compass or other equivalent means, it would have been hard to persuade him that there was any such figure possible. It had been therefore not amiss first to let him see that such a figure might be described. Therefore so much of geometry is no part of philosophy which seeketh the proper passions of all things in the generation of the things themselves. 23

In response to Hobbes's rejection of Euclid's definitions, Wallis objected that it is up to geometers to choose the definitions they please. Just as it is proper for a grammarian or logician to define the words peculiar to grammar [End Page 225] or logic, so it is proper for a mathematician to define his own terms, which definitions are called mathematical. 24 This right Hobbes would not concede. There are no such mathematical definitions, for definitions are not part of any mathematical art but are prior to it, belonging to the learning of language and the apprehension of common notions. These are not the notions of those who call themselves mathematicians, but notions common to all men: "it is not the work of a geometrician, as a geometrician, to define what is equality, or proportion, or any other word he useth, though it be the work of the same man, as a man." 25

If a definition does not give the generation of the definiendum, it fails to be a principle of science. Any reasoning based upon it will be unscientific, for as discussed above, all scientific knowledge and definition needs concern the manner of generation. If a definition fails to conform to our natural and simple conceptions, our common notions, the defined terms will be empty and meaningless or at best misleading.

Hobbes had not only mathematicians in view here but also theologians, in particular the university divines, such as John Wallis, Seth Ward, and their colleagues. In Hobbes's view, they paradigmatically practised a pseudo-science and employed meaningless or misleading locutions. It was such empty definitions, he held, which legitimated doctrines of separated essences and of immaterial souls upon which the authority of the ecclesiastics and school divines was founded. In essence, then, the authority of the divines was baseless, resting upon their assumed right to coin their own terms. For by what right does a man call himself a mathematician? If it is by practising mathematics and that mathematics is founded upon definitions he himself makes, then his right to be called a mathematician, his authority is entirely assumed. And so it was with the theologians. To deny them this right, which Wallis repeatedly claimed, was to deny them their authority.

The dispute over the right use of language and symbols reached another climax in Hobbes's rejection of algebra, and in particular the use of algebra in geometry. Following the publication of the Examinatio et Emendatio (1660), his criticism of Wallis's Mathesis Universalis, Hobbes produced, anonymously in Paris, his solution to the problem of the duplication of the cube (known as the Delphic problem). Wallis rose to the bait and attacked the anonymous solution. In the Seven Philosophical Problems (1662), Hobbes's conversants, A and B, have this to say:

A: Have you seen a printed paper sent from Paris containing the duplication of the cube, written in French? [End Page 226]

B: Yes. It was I that writ it, and sent it thither to be printed, on purpose to see what objections would be made to it by our professors of algebra here. 26

Hobbes mentions that he has seen some of the "confutations," by algebra, of his demonstration. Only one he says, was rightly calculated, and this one was also false, since the units or dimensions used are not constantly the same, the numbers being sometimes so many lines, sometimes so many planes, solids, etc. The dialogue continues with a discussion of the "Detection of the Absurd Use of Arithmetic as it is now applied to Geometry." With a passing reference to Wallis's Civil War career as a parliamentary cryptographer, A remarks:

What a deal of labour has been lost by them that being professors of geometry have read nothing else to their auditors but such stuff as you have here seen. And some of them have written great books of it in strange characters, such as in troublesome times a man would suspect to be a cypher.

A concludes:

I see you have wrested out of the hands of our antagonists this weapon of algebra, so as they can never make use of it again. Which I consider as a thing of much more consequence to the science of geometry, than either of the duplication of the cube, or the finding of two mean proportionals, or the quadrature of a circle, or all these problems put together. 27

Wallis's achievement in De Sectionibus Conicis was precisely an advance in the algebraicization of geometry--the translation of geometric figures into algebraic equations and geometric problems into those involving finding solutions to simultaneous algebraic equations. Until this point the parabola had been regarded literally as a conic section, the edge of a cone where it had been cut by a plane parallel to one of its sides. Wallis showed that this figure might be described by the equation p 2 =l.d on the cartesian plane. Indeed this is sufficient for a definition of the parabola. The parabola is no longer essentially a conic section. Wallis explains that although the first part of his De Sectionibus Conicis follows traditional lines, this is only because it is commonly supposed that parabolas and other conic sections have their origin in the cone. He says he could have done the same work without any reference to the cone at all and to do so would be to work with the [End Page 227] same figures, not to invent new ones (although that might have been suspected had the first part been ommitted). 28

To this Hobbes remarks:

Your treatise of the Angle of Contact, I have before confuted in a very few leaves. And for that of your Conic Sections, it is so covered over with a scab of symbols, that I had not the patience to examine whether it be well or ill demonstrated. 29

Hobbes's first complaint is that "symbols, though they shorten the writing, yet they do not make the reader understand it sooner than if it were written in words." He remarks that there is then a double labor in understanding, first reducing the symbols to words, then to attend to the ideas thereby signified. But as we have seen, Hobbes was to realize that the significance of symbols, or "certain figures, as if a hen had been scratching there" was more than simply that they were a shorthand which he found tedious and inefficient. 30 Rather, like the terminology of the school divines, these symbols were a cloak for more devious goings-on. Signifying at one moment a length and at another a volume, the use of these symbols appeared to license trains of reasoning, such as "spurious" confutations of Hobbes's duplication, trains of reasoning, according to Hobbes, which were in fact false.

One major facet of this false use of symbols is contained in Wallis's claim that the conic sections might alternatively be defined by characteristic algebraic equations. The implication of this is that geometry may be pursued, its concepts defined, its problems solved as with equations such as p 2 =l.d, entirely without reference to any method of generating the geometrical figures themselves. No possible cause is given, nor is any desirable. Indeed what could be a cause of the equation "p 2 =1.d?" Clearly the employment of algebra in geometry ran counter to Hobbes's central philosophical and scientific principle, the generative principle of knowledge. That algebra typified the devious and empty use of terms by the mathematician-divines and denied the generative nature of geometry and science generally explains why Hobbes regarded the extirpation of algebra from geometry as far more significant than any of his attempted demonstrations.

Hobbes admits then, that the quadrature, duplication, and other claims are in themselves relatively insignificant. He does appear, however, to be convinced of their correctness and persisted in keeping them at the center of the dispute for almost twenty years. It was not simply that Hobbes was an [End Page 228] argumentative old man with a bee in his bonnet nor that Wallis was motivated by spiteful pride. Rather, the quadrature and duplication claims were signs of their differences over the deeper issues mentioned.

Wallis too was fully aware of the wider significance of the dispute and its relevance to theological as well as mathematical concerns. In 1659 he wrote to Huygens:

Our Leviathan is furiously attacking and destroying our Universities ... and especially ministers and the Clergy and all religion, as though the Christian world had not sound knowledge ... and as though men could not understand religion if they did not understand Philosophy, nor Philosophy unless they knew Mathematics. Hence it seemed necessary that some mathematician should show him ... how little he understands the Mathematics from which he takes his courage; nor should we be deterred from doing this by his arrogance which we know will vomit up poisonous filth against us. 31

Wallis himself undertook the part of this mathematician, a task all the more urgent for Hobbes's immense reputation, particularly abroad, as a man of science. Nor was he mistaken in his opponent. Hobbes recognized Wallis's attack as directed at the root of his system and at his own objections to school divinity. The dispute over algebra was a central case in point, for Wallis's algebraic and arithmetical approach to geometry represented everything Hobbes was against. In particular, Wallis's approach gave different results from Hobbes's own constructive and generative method; Wallis's algebra said that Hobbes's constructions yielded false quadratures and duplications. To have conceded that his constructions were in error would have been to concede the superiority of the algebraic method; and to have conceded this would have been to concede that the mathematician-divines had the right to make their own definitions, that they did not use terms emptily, and so that understanding of generation was not necessary to science--too much, clearly, for Hobbes to concede.

IV. Hobbes's "proofs" are many. They vary from the downright crass to the scarcely detectably flawed. Of the approximations they give, some are almost reasonable, others are ancient chestnuts, e.g. pi= square root of 10, given a new flavor; none are an advance over the best contemporary values or approximations.

Corresponding to the more difficult proofs are exceedingly complicated constructions. Though the proofs themselves changed over time and under criticism, the associated constructions remained constant. The quadrature of [End Page 229] De Corpore was retained, with an altered demonstration, in the much revised English edition. The same construction reappears in the Six Lessons, the Principia et Problemata and in the Lux Mathematica. The Archimedean spiral is straightened with the same construction in De Corpore, its translation and in the Six Lessons. A different quadrature construction appears in the Rosetum Geometricum and again in the papers addressed to the Royal Society. The construction of the "solved" Delphic problem occurs in the original Paris publication, in the Dialogus Physicus, in a paper presented to Pell, the Problemata Physica, and the Seven Philosophical Problems, again with various proofs.

Clearly Hobbes distinguished between, on the one hand, the construction which gave a way of drawing a straight line equal to a given curved one, or equal to the side of a cube doubly large than a given one, or whatever the problem required, and, on the other hand, the demonstration or proof that the constructed line was in fact such a line with those magnitudes on the other. One might easily give a correct method of generating the desired line while erring in the proof.

This distinction, a common one for the time, allowed Hobbes tacitly to accept and to correct fallacies in his argument while sticking to his quadrature and duplication claims. These were founded on the unchanged constructions, which, of course, embodied the all important generation of the desired quantities. The constructions, independently of the proofs, were embodiments of Hobbes's generative principle.

Once the controversy was under way it was polemically important for Hobbes to stand by his constructions, for to have abandoned them in the face of Wallis's objections would have appeared to be an acknowledgment of the superiority of Wallis's algebraic and arithmetical treatment of geometry and of the deficiency of his own generative definitions and methods. This does not explain why Hobbes did in fact think these constructions the correct ones, especially when initially devised and before they entered the arena of dispute. For his generative and other principles did not especially license these particular constructions and none other.

Hobbes did in fact possess a method of quadrature, one perfectly general in its application, not approximate, and accurate in so far as any physical act of measurement is ever accurate. Given that (for reasons discussed above) geometry was for Hobbes concerned not with abstract or platonic entities but with real and physical ones, that its essence lies in its application, then a quadrature method which fulfills the criteria of the best possible measurable result is all and no more than a quadrature method should be. Discussing the values found for ¼ by Archimedes, Van Cullen, Snell, and other, Hobbes remarks:

Nevertheless, if we consider the benefit, which is the scope at which all speculation should aim, the improvement they have made has been little or none. For any ordinary man may much sooner and more [End Page 230] accurately find a straight line equal to the perimeter of a circle, and consequently square the circle, by winding a small thread about a given cylinder, than any geometrician shall do the same by dividing the radius into 10,000,000 equal parts. 32

The utility of finding a quadrature, he says, lies in its providing a method of sectioning any given angle. Indeed, Hobbes's quadrature in De Corpore and elsewhere purports to provide a method of doing just this, by projecting a curved line onto a straight one. The latter may be sectioned accordingly and the sections projected back onto the curve so that these sections now section the angle at the center of the curve. This procedure is precisely analogous to that which employs a thread wrapped around the cylinder, is unwound until straight, sectioned, then wrapped again around the cylinder whose circumference is thereby sectioned proportionately.

Given that Hobbes mentions the method of thread and cylinder, one might tentatively surmise not only that the analogy is conscious in his ruler and compass construction, but that he used the thread and cylinder method to check which constructions actually gave reasonable results.

Because Hobbes contributed little or nothing to the history of geometry, the disputes arising from his endeavors in this field have largely been ignored, dismissed as a splenetic eccentricity. I hope to have shown that on the contrary the dispute with Wallis, more even than the other disputes of Hobbes's career, reveals the fundamental differences on issues which underpin their methods and philosophies.

In particular the dispute highlights the significance to Hobbes of the maker's knowledge argument, the generative principle of knowledge, and crucial issues of the right use of signs and symbols and of the right to make definitions. On the resolution of these debates rested not simply the precise results of Hobbes's geometrical theorems but indeed the validity of the method and assumptions of his life's work on man, society, and government.

University of Edinburgh

Notes

I would like to thank Simon Schaffer, David Sherry, and Ian Bostridge for their kind and valuable advice during the preparation of this paper.

1. Thomas Hobbes, The Life, Written by Himself (1680), 13.

2. Thomas Hobbes, Elementorum Philosophiae Sectio Prima de Corpore (1655). Henceforth De Corpore (English trans.). References are to part, chapter, and section, and, for Molesworth's edition of Hobbes's English Works (11 vols.; London, 1839-45), henceforth EW, by volume and page number.

3. In John Wallis, Elenchus Geometriae Hobbianae (1655).

4. Thomas Hobbes, De Corpore (1655), part I, ch. 1 §3; EW, I, 3.

5. Francis Bacon, The New Organon (1620); quoted in Stephen Land, The Philosophy of Language in Britain: Major Theories from Hobbes to Reid (New York, 1986).

6. Thomas Hobbes, Leviathan (1651), part I, ch. 4; EW, III, 21.

7. Stephen Land, op. cit., 20.

8. Thomas Hobbes, De Corpore (1655), part I, ch. 6, §10; EW, I, 77.

9. Ibid., part I, ch. 6, §l; EW, I, 66.

10. Ibid., part I, ch. 6, §10; EW, I, 77.

11. Edward Madden, Theories of Scientific Method (Seattle, 1960).

12. Thomas Hobbes, De Corpore (1655), part I, ch. 6, §§12, 13; EW, I, 80-83.

13. Thomas Hobbes, Six Lessons (1656), Epistle Dedicatory; EW, VII, 183-84. The Maker's Knowledge Argument and Hobbes's use of it along with Vico and others in a "mentalistic" constructivist stance is discussed in Antonio Pérez-Ramos, Francis Bacon's Idea of Science and the Maker's Knowledge Tradition (Oxford, 1988), 186-92.

14. Thomas Hobbes, De Corpore (1655), part I, ch. 1, §6; EW, I, 7.

15. Ibid., part I, ch. 1, §7; EW, I, 7.

16. Ibid., part II, ch. 8, §1; EW, I, 102.

17. Ibid., part II, ch. 7, §2; EW, I, 94.

18. This discussion, which anticipates George Berkeley's An Essay Towards a New Theory of Vision (1709), is to be found in particular in Hobbes's insertion in Marin Mersenne's Ballistica (1644); see Thomas Hobbes, Opera Philosophica (the Latin works in 5 vols., ed. Molesworth [London, 1839-45]), V, 309-18.

19. Frithiof Brandt, Thomas Hobbes's Mechanical Conception of Nature (Copenhagen, 1928), 251-52.

20. Thomas Hobbes, De Corpore (1655), part II, ch. 6, §12; EW, I, 111-12.

21. Euclid, The Elements (Cambridge, 1926), Bk. I, 153-70.

22. Ibid., 154.

23. Thomas Hobbes, Six Lessons (1656), EW, VII, 205.

24. John Wallis, Due Correction for Mr Hobbes (1656), 25. Wallis's title page quotes from Hobbes's Leviathan: "Who is so stupid as both to mistake in Geometry and also to persist in it, when another detects his error to him?" (part I, ch. 5).

25. Thomas Hobbes, Six Lessons (1656), lesson II; EW, VII, 222.

26. Thomas Hobbes, Seven Philosophical Problems (1662), ch. 8; EW, VII, 59.

27. Ibid., ch. 8; EW, VII, 68.

28. John Wallis, De Sectionibus Conicis (prop. XXI) in Operum Mathematicorum, Pars Altera (Oxford, 1656), 48 and in Opera Mathematica, Volumen Primum (Oxford, 1695), 320.

29. Thomas Hobbes, Six Lessons (1656), EW, VII, 316.

30. Ibid., EW, VII, 329.

31. Letter from Wallis to Huygens. Huygens, Oeuvres complètes (La Haye, 1888), ii, 296 (1 January 1659). Quoted in J. F. Scott, The Mathematical Work of John Wallis (London, 1938), which contains a useful summary of the dispute (166-72).

32. Thomas Hobbes, De Corpore (1655), part III, ch. 20, §1; EW, I, 288.

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