… claims for itself the greatest problem of all, which is: To Solve Every Problem” (Viète 1591, 9). In England, Thomas Harriot’s Artis Analyticæ Praxis and William Oughtred’s Clavis Mathematicae were important developments in the application of algebra to geometry.9
A critical difference between the Cartesian approach to geometry and its classical counterpart involves the issue of the homogeneity of magnitudes under multiplication. In the Cartesian scheme, multiplication of two lines yields another line, and likewise the multiplication of any number of lines produces yet another line. Thus, Cartesian geometry sees lines as homogeneous when subject to multiplication. In contrast, classical geometry takes the product of two lines to be a surface, so that multiplication yields a magnitude heterogeneous to that of the multiplicands. Similarly, classical geometry takes the product of three lines to be a solid, with the consequence that multiplication beyond the third (“cubic”) power has no geometric interpretation. Such powers were termed “biquadrate” (fourth power) or “quadrato-cube” (fifth power), etc., but they were not taken to have specific geometric meaning.
Hobbes’s disdain for analytic geometry was boundless. In part, his objections amount to little more than “aesthetic” complaints to the effect that algebraic symbols deface geometric demonstrations, as when he complains in SL Lesson 3 that “Symboles are poor unhandsome (though necessary) scaffolds of Demonstration; and ought no more to appear in publique, then the most deformed necessary business which you do in your Chambers” (EW VII.48). A more substantive complaint is that the employment of algebra in geometry is either methodologically suspect or (at best) irrelevant. If the application of algebra to geometry is warranted by underlying principles, these principles must ultimately be vindicated by appeal to geometric constructions, thereby making algebra superfluous. Yet if algebraic techniques are not grounded geometrically, there is no guarantee that an algebraic transformation corresponds to a legitimate geometric operation. Hobbes poses the dilemma this way in Dialogue 1 of Examinatio:
What else do the great masters of the current symbolics, Oughtred and Descartes, teach, but that for a sought quantity we should take some letter from the alphabet, and then by right reasoning we should proceed to the consequence? But if this be an art, it would need to have been shown what this right reasoning is. Because they do not do this, the algebraists are known to begin sometimes with one supposition, sometimes with another, and to follow sometimes one path, and sometimes another …. Moreover, what proposition discovered by algebra does not depend upon Euclid (Book VI, Prop. 16) and (Book I, Prop. 47), and other famous propositions, which one must first know before he can use the rules of algebra? Certainly, algebra needs geometry, but geometry does not need algebra.
(OL IV.9–10)
This sort of objection highlights a difficulty in the analytic approach, which is that it is not always clear how a problem is to be “unraveled” (in Descartes’s phrase), and it is occasionally problematic to discern whether the manipulation of symbols by algebraic rules reflects an acceptable geometric construction.
Hobbes further complains that the “intrusion” of algebra into geometry fails to recognize the fundamental difference between algebraic multiplication and the “drawing of lines into lines” that generates geometric objects. Among the “Manifest Principles” at the outset of his treatise Lux Mathematica, he numbers:
A square figure is made by the drawing of a straight line into a line equal to it and placed at right angles to it. This drawing of a line therefore describes a surface, that is a new species of quantity, and this surface drawn into another straight line equal to the first two and orthogonal to them, will describe a solid, that is the third and last species of quantity.
For geometry recognizes no quantity beyond the solid. For the sur-solids, quadrato-quadrates, the quadrato-cubes, and cubi-cubes are mere numbers, but not figures. Thus there is a great difference between what is made by the drawing of two lines into one another and what is made by the multiplication of two numbers into one another, truly as much as there is between a line and a surface or between a surface and a solid. These differ in kind, so much that one of them could by no multiplication exceed the other; and thus (as Euclid says) they can have no ratio to one another, nor could they be compared according to quantity. There is likewise a great difference between the root of a square number and the side of a square figure. For the root is a number, and an aliquot part of its square; but the side is not a part of a square figure.
(Lux, Introduction 8; OL V.96)
This amounts to a wholesale rejection of Descartes’s thesis on the homogeneity of geometric lines under multiplication, and it set the stage for Hobbes’s bizarre claim that algebraic calculation could never refute a supposed geometric deduction.
Hobbes’s resolute rejection of analytic geometry contrasts with his ambivalent attitude toward another important innovation in seventeenth-century mathematics – the “method of indivisibles” pioneered by Bonaventura Cavalieri. The method is based on the notion that we can reason about the areas of plane figures by comparing “all the lines” contained within them, which Cavalieri called “the indivisibles” of the figures.10 The method first appeared in Cavalieri’s 1635 Geometria indivisibilibus continuourm nova quadam ratione promota, and its fundamental concept is illustrated in Figure 4.1. Take the circle ACEG and the ellipse JLNP, of equal height and both based on the line RS (which Cavalieri terms the regula). If the regula is understood to move parallel to itself upward until it reaches the position TU, its intersection with the two figures will produce “all the lines” in each. Cavalieri then proceeds to argue that “all the lines” in a figure can be treated as magnitudes and are subject to ordering relations. So, in Figure 3.1, the fact that “all the lines” of the circle exceed “all the lines” of the ellipse means that the area of the circle must be greater than that of the ellipse.
Figure 3.1 Cavalieri’s indivisibles.
Cavalieri used the method to investigate the areas of figures and volumes of solids, and specifically to establish ratios between pairs of such magnitudes. In the two-dimensional case, he first shows that a specific ratio holds between the indivisibles of two figures, and then concludes the areas stand in the same ratios as the indivisibles. As he put it
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 of 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.
(1635, 115)
This procedure leads to the theorem now known as “Cavalieri’s Principle”: if two figures have equal altitudes and sections made by lines parallel to the bases at equal distances are always in a given ratio, then the areas of the figures stand in the same ratio.
At first sight it may seem that Cavalieri takes continuous magnitudes (figures) to be composed of infinite collections of indivisibles (lines). He steadfastly denied this, however, and insisted that his method did not run afoul of traditional strictures against the use of infinitely small magnitudes. Other mathematicians who followed Cavalieri were not so cautious. Most notable for our purposes is Wallis’s pronouncement that “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” (1693–1699, 1:297). Wallis was hardly alone in this interpretation of Cavalieri: Evangelista Torricelli and Gilles Personne de Roberval
