in Six Lessons to the Professors of the Mathematiques (hereafter SL)
That which is indivisible is no Quantity; and if a point be not Quantity, seeing it is neither substance nor Quality, it is nothing. And if Euclide had meant it so in his definition, (as you pretend he did) he might have defined it more briefly, (but ridiculously) thus, a Point is nothing.
(EW VII.201)
Hobbes held that the Euclidean definitions of point, line, and surface can be remedied by resolving the ambiguity. He asserted in Dialogue 1 of Examination that understanding the point as a divisible body whose magnitude disregarded implies that “the quantity of a point is not nothing, but rather not computed. Nor is the point itself nothing, or indivisible, but undivided” (OL IV:33).
The flaws in other Euclidean definitions cannot be so easily mended. In particular, Euclid’s definition of ‘ratio’ is entirely uninformative, reading; “A ratio is a sort of relation in respect of size between two magnitudes of the same kind” (Elements, Book V, Def. 3). Hobbes dismisses this in SL Lesson 2 as “insignificant” and amounting to no more than saying a ratio “is a whatshicalt habitude of two Quantities” (EW VII.229). In its place he offers a definition that takes a ratio to be the comparison of the relative magnitude of two bodies. He claims in De corpore XI.3 that a ratio is the “Excess or Defect” of one body when compared with another (EW I.133). The ratio between two bodies therefore expresses the amount by which they differ in some quantity. As it happens, there are various kinds of quantities associated with bodies: volume, mass, velocity, temperature, density etc. Moreover, the difference in these magnitudes can be expressed in alternative ways. Consider two bodies, one with a weight of 5 kg, the other weighing 3 kg This difference could be expressed as the arithmetical ratio of 2 kg, corresponding to 5 – 3, or it could be expressed as the geometric ratio 5:3, arising from the division of five by three. In general, it is the geometric ratio that is of interest when volumes, arc lengths or areas are compared. Although many kinds of quantities can stand in geometric ratios, Hobbes holds that any ratio can be “exposed” or “exhibited” by displaying two straight lines whose lengths stand in the desired ratio.
Where Hobbes dismissed the Euclidean definition of ‘ratio’ as uninformative, he took the definition of ‘same ratio’ to be both overly complex and capable of definition from more basic principles. In Euclid’s presentation, the definition reads
Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of the latter equimultiples respectively taken in corresponding order.
(Elements, Book V, Def. 5)
This definition attracted a great deal of commentary over the centuries,6 and Hobbes was not alone in thinking that it was too intricate and obscure to be numbered among the first principles of geometry.
In Hobbes’s view, sameness of ratio can be properly defined only in terms of the motion of bodies. Thus, rather than following Euclid and explicating this concept in terms of order preservation under arbitrary equimultiples, he seeks to find a more general principle from which the Euclidean definition can be derived. As he puts the matter in PRG chapter 12: “I define “the same ratios” to be “those which uniform motion exposes in two straight lines in equal times”; or more universally, “those things are in the same ratio which are determined by some cause producing equal effects in equal times” (OL IV.420). Hobbes’s “more universal” definition cited here comes from De corpore (2.13.6), and he bragged that Euclid’s definition of ‘same ratio’ could be demonstrated from it. Speaking of himself in the third person in the second dialogue of Examinatio, he insists that the Euclidean definition “is true, I say, but not a principle, because it is demonstrable and demonstrated by Hobbes in article 12 of chapter 13 of the book De corpore, but demonstrated from a definition of the same ratio by generation that is different from this of Euclid” (OL IV.76–7).7
Hobbes also critiques the Euclidean definition of the circle, which defines it as “a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another” (Elements, Book I, Def. 15). In Hobbes’s estimation, this has the virtue of stating something true about the circle, but it fails to be a proper definition because it does not identify its causal generation, as he claims in SL Lesson 1:
But if a man had never seen the generation of a Circle by the motion of a Compass or other æquivalent means, it would have been hard to perswade him, that there was any such Figure possible. It had been therefore not amiss first to have 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.
(EW VII.205)
In the Hobbesian scheme of things, the correct definition of the circle – found in De corpore XIV.4 – is in terms of the rotational motion of a straight line about one of its termini (EW I.180–1). Such a definition not only identifies the cause of the circle, but also enables the geometer to investigate its properties.
Hobbes’s methodology holds that demonstrative knowledge must be based on definitions that identify the causes of things. This is summarized in the Dedicatory Epistle to the SL, where he insisted that “where there is place for Demonstration, if the first Principles, that is to say, the Definitions contain not the Generation of the Subject; there can be nothing demonstrated as it ought to be” (SL Epistle). As he explains a greater length in PRG 12:
”But”, you will ask, “what need is there for demonstrations of purely geometric theorems to appeal to motion?” I respond: first, all demonstrations are flawed, unless they are scientific, and unless they proceed from causes they are not scientific. Second, 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 are not contained in the definitions of motions by which figures are described.
(OL IV.421)
In Hobbes’s view, once proper causal definitions are in place, the development of a science is a matter of relative routine. The guiding thought here is that knowledge of causes should provide easy access to knowledge of effects. In particular, since geometric objects are brought into being by motions through which the geometer literally constructs the object of investigation, so that “in his demonstration [the geometer] does no more but deduce the Consequences of his own operation” (EW VII.183–4). This approach to demonstration led Hobbes to believe that, once proper geometric definitions were in place, the solution of any geometric problem should follow with ease. Thus, the failure of previous generations of geometers to solve such problems as the quadrature of the circle did not arise from intractability of the problems, but from flawed first principles. Once the Euclidean definitions have been replaced by Hobbes’s causal definitions, the royal road to the solution of geometric problems was open. Or so Hobbes believed. As we will see, this confidence in the efficacy of his geometric first principles ultimately led Hobbes to serious error.
3.1.3 The Status of Algebraic and Infinitesimal Methods
Hobbes’s approach to geometry clearly set him against the traditional view of the subject, but he also opposed some of the innovations that were the most important advances in seventeenth-century mathematics. In particular, he was quite hostile to the algebraic methods characteristic of Descartes’s analytic geometry, and he found fault with some presentations of the “method of indivisibles” that is an important precursor to the calculus. A brief summary of these issues can bring our investigation into the