Kant on conic sections

Abstract

This paper tries to make sense of Kant's scattered remarks about conic sections to see what light they shed on his philosophy of mathematics. It proceeds by confronting his remarks with the source that seems to have informed his thinking about conic sections: the Conica of Apollonius. The paper raises questions about Kant's attitude towards mathematics and the way he understood the cognitive resources available to us to do mathematics.

Keywords::

• Kant
• Apollonius
• conic sections

Acknowledgements

I would like to express my thanks to Vincenzo De Risi, Stephen Menn and Roshdi Rashed for kindly taking the time I know they did not have to give me extensive feedback on earlier versions of this paper. I have done my best to take this feedback into account. The remaining infelicities are my fault.

Notes

^1. No doubt his reason for doing so is that the circle is the figure that results from passing a cutting plane through a cone parallel to its base. It should be noted, though, that Kant is departing in this respect from Apollonius – his point of reference for conic sections. Apollonius never speaks of circles in the Conica, but always of circumferences of circles, namely as traced on a conic surface by the cutting plane.

^2. All translations in this paper, from Kant and other sources, are my own. Quotations from Kant's works are by volume number of the Academy edition, page number and line number. Passages from the first Critique indicate page number from the first (A) or second edition (B), as the case may be.

^3. The long-winded title may be translated as ‘concerning a discovery that is supposed to show that every new critique of pure reason is made dispensable by an earlier one’. The work was brought out by the publishing concern of Friedrich Nicolovius.

^4. By ‘ordinate’ I mean all those lines parallel to one another, meeting the section at two points, bisected by the diameter or axis of the section. By ‘abscissa’ I mean that part of the diameter or axis of a section lying between the vertex of the section and the point of intersection between the diameter and one of its ordinates. By the ‘latus rectum’ of the parabola, I mean the length upon which is formed the rectangle whose width is the abscissa and the magnitude of whose area is equal to the square on the ordinate.

^5. It is important not to assume that these two planes are orthogonal to each other. For precisely in the case when they are not orthogonal, it will turn out the angle of inclination of the ordinates to the diameter will not be a right angle.

^6. This diameter of the circular base is itself the line of intersection between the circular base and the first cutting plane. This line of intersection has to be a diameter of a circle, because the first cutting plane passes through the axis of the cone, which joins the summit of the cone to the centre of the circular base.

^7. Proposition 46 says that, in any parabola, any line parallel to the first diameter is a diameter. Proposition 47 says that, in any ellipse or hyperbola, any line passing through the centre is a diameter. Proposition 48 says that, in any double-branched hyperbola, any line that joins a point on one branch to a point on the other branch and that passes through the centre is a diameter.

^8. Perhaps, we may regard this thought as on display in the problems of determination in Conica, Book Two. Propositions 44 and 45 in the Arabic translation (they are treated as a single problem in Eutocius' edition) address the problem of finding a diameter of any conic section. The analysis in Proposition 44 turns on the fact that ordinates are defined relative to a certain diameter, as I have indicated. The synthesis then involves constructing parallel lines inside a given conic section, finding their mid-points, joining the mid-points and extending the line joining the mid-points to the section. It follows that this line is a diameter from Book Two Proposition 28 and the relation between ordinates and diameters invoked in the analysis of Proposition 44.

^9. There is a serious and difficult question here about what could have been Eberhard's source. Eberhard explicitly refers to Borelli, the editor of the books of Apollonius that survive only in Arabic (translated into Latin by Abraham des Echelles in 1661). Indeed, Eberhard purports to quote from Borelli's Admonitio prefaced to the Apollonius edition (1789, 159). But Eberhard later retracts the attribution and says that the quoted passage is from Claudius Richardus. At least, this is what one learns from the note of the Academy edition to 8.191.27–29 of Kant's works. I have not yet been able to consult the Richardus. It remains a question, in my mind at least, what could be the source of Eberhard's information on Apollonius. There is an obvious question related to this one, namely what is the nature of Kant's access to Apollonius. Vincenzo de Risi suggested to me that Kant knew Apollonius through the Borelli edition – at least partly. The plausibility of the idea rests on the fact that Kant himself mentions the edition in the passage from the Reply to Eberhard at issue, and indeed quotes part of the passage quoted by Eberhard. But given that Eberhard himself retracts the attribution and given that Kant himself did not catch the original mistake of attribution, one may doubt whether Kant had been reading the Borelli edition or its preface.

^10. The relevant treatises are lost, but we learn these things from remarks in Pappus' Collectio and Eutocius' commentary on Apollonius. Pappus' report is reprinted from Hultsch's edition as Appendix II and translated into German in Zeuthen (1966). For Eutocius‘ commentary, see Apollonius (Heiberg 1893, 168–170).

^11. Prior to Apollonius, the property used to characterize both the ellipse and the hyperbola seems to have been one of the two proportions established by Apollonius in Proposition 21 of Conica, Book One. Let AΓ be the latus rectum of an ellipse or hyperbola, and let AB be the diameter. Let ZH be an ordinate of AB. Let AH be the abscissa. Then, we have AΓ:AB :: square ZH, ZH: rect.AH, HB. At any rate, Archimedes uses this proportion to characterize conic sections other than the parabola in Conoids and Spheroids; he seems to be a witness to the older theory.

^12. Indeed, whereas the equation states both necessary and sufficient conditions to be satisfied by points lying on the curve, Apollonius’ geometrical characterization states only the necessary conditions – so too his geometrical characterization of the ellipse and the hyperbola. It would not be hard at all for him to have stated the sufficient conditions as well. But he does not do this. Moreover, he uses the converse of the proposition that gives the geometrical characterization of the parabola (Book One, Proposition) in subsequent propositions of Book One, like the problem of constructing a parabola in a given plane under the conditions spelled out in Proposition 53. Roshdi Rashed points this out in his commentary on the Arabic translation of the Conica. In particular, he draws our attention to the sensitivity to this point of later mathematicians writing in Arabic, responding to Apollonius and contributing (or at least aware of) advances in algebra. Thus, Nāir al-Dīn al-ūsī (1201–1274) points out in his gloss on the Conica the use of the converse of Book One, Proposition 11 in Book One, Proposition 27 and Proposition 53. Notes Complémentaires 36 and 68 of Rashed's edition of the Arabic translation. See too Rashed (2005).

^13. Zeuthen's Die Lehre von den Kegelschnitten im Altertum is a classic. But the idea that Apollonius was presenting fundamentally algebraic insights, in geometric guise, is no longer considered respectable. See, for example, Saito (2004, 139–169) and Fried and Unguru (2001, 17–56).

^14. Gordon Brittan may well reject my reading of this passage. He takes Kant to be saying that the algebraic mathematicians do not need the relevant constructions in Apollonius to secure the objective reality of their treatment of conic sections. Brittan goes on to argue in light of a passage in the first Critique at A716/B744 that, for Kant, algebra is shored up by a different kind of construction, namely symbolic construction and that this is why the geometrical construction is not needed. I take Kant to be saying that it (the geometrical) is indeed needed (for the theory of conic sections), but not necessarily in the actual presentation of the theory. Brittan is no doubt right to say, in effect, that Kant needs to say more about algebra. To be sure, Kant says that, even though the algebraic mathematicians do not bother showing how to generate parabolas from cones, their results are objectively real. But he also says pretty clearly that they do not need to do so, because ‘they are always completely conscious … of the pure, merely schematic construction along with [the relevant definition expressed as an algebraic formula]’ (8.192.11–12). By ‘pure, schematic construction’ Kant means the construction we carry out by ‘mere imagination according to a concept a priori’ (8.192.26–27) as opposed to a construction we carry out by applying some instrument to some kind of material like paper or copper, if we are preparing copper plates. Apollonius' construction of the cone and the parabola would count, for Kant, as pure, schematic constructions. On balance, the passage here is best read as saying: the algebraic mathematicians do not have to explicitly present Apollonius' construction in their treatises, precisely because they are always conscious of it. See Brittan (1992, 315–339).

^15. As indicated in the previous footnote, there is an obvious and immediate question about the source of the objective reality of the algebraic treatment of curves other than conic sections. This will be especially urgent in the case of the curves characterized as ‘mechanical’ by Descartes. I do not have anything special to say about this question, beyond appealing, as everybody does at this point, to the passage in the first Critique at B745 where Kant claims that algebra is based on a ‘symbolic’ construction in intuition where ‘the constitution of the object is completely abstracted’. I am struck by his use of the word ‘symbolic’. Where there is a symbol, there must be something symbolized. If algebra has objective reality by virtue of its underlying symbolic construction, perhaps this is by virtue of whatever has been symbolized and the way it has been symbolized. In the case of the equation for the parabola, discussed in our passage from the ‘Reply to Eberhard’, the thing symbolized is the relevant geometrical construction. Perhaps, in the case of the algebraic treatment of ‘mechanical’ curves, in Descartes' sense, the thing symbolized is the set of rules needed to mechanically generate the curve. I realize that Kant polemicizes in our passage from the ‘Reply to Eberhard’ against anybody who could think that knowing how to ‘mechanically’, i.e. ‘graphically’, produce a curve really knows anything about the curve in question. But I do not think that this really affects my suggestion, because the sense of ‘mechanical’ is not the same. In any case, this paper raises another difficult question for Kant: the question that I raise at the end of Section 3.

^16. The only hitch of this kind I might conceivably encounter is if I tried to construct something impossible. That can and does happen. Typically, it happens in Euclid in the course of a proof by reduction. But we discover (or rig up) the auxiliary construction in such a way that the attempt to produce it in the intended way produces a contradiction instead. That is not the fault of space, understood as a reservoir of every that I might be given or need for the purposes of solving constructive problems in geometry.

^17. I try to spell out this idea in detail and as precisely as possible in the chapter on Section 17 of the B-Deduction in my forthcoming book Kant's Transcendental Deduction: A General Cosmology of Experience.

^18. Perhaps, this title might be translated into English, thus ‘Concerning a Lofty Tone Lately Struck in Philosophy’.

^19. The full title of the book was Plato's Briefe, nebst einer historischen Einleitung und Anmerkungen, Königsberg bey Friedrich Nikolovius1795.

^20. It is amusing to note that Schlosser himself seems to have pre-emptively read Kant as an intellectual drudge, i.e. as someone determined to put everybody to work in the factory of reason. This can be seen in the first footnote to Schlosser's translation of what we usually refer to as Plato's ‘seventh letter (Schlosser prints it as the sixth). The note is a comment on the famous passage of this letter where Plato tries to explain to the friends of Dion why it is impossible for him (Plato) or for anybody to write up true philosophy: there is something ultimately ineffable in the objects of philosophical enquiry that cannot be communicated to others in this way (342ff). The passage gives Schlosser an opportunity to reflect on the current state of German philosophy. He happily seizes the opportunity and proceeds to excoriate Kant, without mentioning him by name (the illusions to a ‘critique of reason’ and to the ‘purifying of reason’ are among the many tip-offs that Kant is the intended target.) Thus, Schlosser is led, by a chain of reflections that are not always easy to follow, to say that ‘a critique that would deny reason [the right to proceed by analogies] would not so much purify as unman it. It even seems to me that a philosophy that would, through such a purification, so sequester itself from reason, would itself run the risk of soon transforming into a mere factory production of forms [Formgebungsmanufactur] ….’ (1795, 183). Kant replies to this tit with a tat: ‘This dismissive way of writing off the formal aspect of our knowledge (which aspect is indeed the chief concern of philosophy) as pedantry, by calling it a ‘factory production of forms’, confirms the suspicion of a secret motive: to hang out the philosopher's shingle and thereby banish all philosophy, while giving oneself out as victor by disporting over it [sc. philosophy] in a lofty way’ (8.404.3–8).

^21. It is worth pointing out the Plato himself stresses the significance, value and unavoidability of intellectual labour in the Seventh Letter. See especially 340b–c. This passage must have resonated with Kant. Perhaps, that is why Kant tries to claim Plato for himself in a footnote. Plato himself is naturally guilty of believing in some kind of intellectual intuition. But Kant claims that he almost discerned the essential points of the critical philosophy (8.391.30ff – the footnote that continues to the next page).

^22. I am here ringing the changes of Roshdi Rashed's observations on Proposition 17 in his commentary to the Arabic translation of the Conica (Apollonius 2010, 314–316).

^23. One thing really should be said here, though briefly. It is obvious that, in the case of mathematical concepts like the one on display in Apollonius III.17, the story about concept formation Kant is understood to be telling in the Jäsche logic is totally inadequate. The idea in Section 6 is that three ‘logical acts’ of the understanding produce ‘concepts according to their form’ (9.94.20). The example given in the footnote to this section is that of the concept Tree. We apparently get this concept by comparing, say, the foliage of a fir tree and a linden tree in a meadow, then by reflecting on what they have in common and then by abstracting from this reflection the concept Tree. Even if that is adequate as an account of how we form empirical concepts like that of Tree, it clearly cannot account for the way Apollonius and his predecessors formed the general concept Conic Section. It may well be that there are different moments in the Conica that might be characterized as comparisons, reflections or abstractions. But nothing much will have been gained by using these words unless we also try to understand precisely what Apollonius was doing mathematically in the Conica. Whatever he was doing, it was different from what I do when I form my concept Tree.