The Marriage of Growth and Certainty in Conjugate Diameters
Mathematics are often adored for their straightforward, factual nature. While problems can be challenging, mathematical calculations provide a comfortable logic in which terms are defined and one can choose from a toolbox of operations to work towards a clear goal. The well-educated high school math student learns the skills and information to carry out calculations within established orders, such as algebraic rules and coordinate planes. Such a student would be able to graph conic sections on a Cartesian plane and know how to recognize and work with the algebraic equations that describe them. But mathematics have more to offer to the development of the mind. The challenge of grappling with mathematical relationships without the comfort of a synthesized and ordered curriculum of rules calls for a different kind of certainty and an openness to growth. In Conics Apollonius explores the ways to cut a cone from the unique perspective of geometry without the established rules of the Cartesian plane. This approach offers insight into the underlying geometric relationships of particular aspects of the conic sections, and Apollonius guides the reader through shifts in understanding their qualities that challenge the reader to consider new perspectives. In particular, his use of emphasis and definitions in Propositions 15 and 16 of Book 1 facilitates the expansion of the reader’s understanding of the diameters of conic sections, shifting from the established relationships of diameters in earlier propositions to include a new way of understanding diameters through the conjugate diameters of a hyperbola.

In Proposition 15, Apollonius builds on the reader’s understanding of the ellipse’s diameters from Proposition 13 to construct a second diameter for the first time, and then uses a second “I say” statement to prepare the reader for the intellectual jump to a new way of considering diameters in Proposition 16. The second diameter ED in Proposition 15 is constructed by drawing a line ordinatewise from the midpoint of an ellipse’s diameter AB, and has a parameter DF. Apollonius puts forth the result of this construction in his first “I say” statement: “I say that the straight line GH [parallel to AB] is equal in square to the area DL which is applied to the straight line DF

[parameter]

, having as breadth the straight line DH [abscissa] and deficient by a figure LF similar to the rectangle contained by ED [new diameter] and DF [parameter]” (Apollonius I.15). This particular relationship between the diameter ED and parameter DF matches the qualifications for an ellipse’s diameter in Proposition 13, where Apollonius first establishes the relationships that define an ellipse. The reader can recognize and affirm that ED is a diameter because it passes the test of having the same relationship respectively to its parameter, square on the parameter, and abscissa as all previous ellipse diameters have had to theirs. While the reader does have to adjust to the new idea of a second diameter, the familiarity of the terms and context bridge the transition by allowing the reader to focus in on understanding how the number of diameters has changed, without the distraction of a change in how diameters are understood or described. However, before Apollonius finishes Proposition 15, he thoughtfully prepares the reader to make the next jump: to consider a diameter without these previously established relationships.