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.
The second “I say” of Proposition 15, by emphasizing the relationship of GH to ED, prepares the reader to consider a pair of conjugate diameters in which the second diameter is only defined by its relationship to its conjugate. It subtly connects Propositions 15 and 16 and facilitates the transition between the two. While the two diameters in Proposition 15 are conjugate diameters, Apollonius does not label them as such, and waits until the reader has adjusted to the idea of the possibility of two diameters in the beginning of Proposition 15. He uses the second “I say” to carry the reader into Proposition 16, when the reader will be ready to consider conjugate diameters according to their definition. In the second “I say” he states, “I say then that also, if produced to the other side of the section the straight line GH will be bisected by the straight line DE” (Apollonius I.15). The second “I say” confirms that GH is an ordinate of ED, but Apollonius does not label it as an ordinate in order to prevent drawing the reader back to the language of Proposition 13. Rather than focusing on the relationship of the diameter ED to the square on a line labeled as an ordinate, he lets the original definition of GH as the line parallel to the original diameter AB stand, which emphasizes the angular relationship of GH to AB, the original diameter. This shifts the reader’s focus from the established role of diameters in relation to the parameter, square on the parameter, and abscissa to the relationship between the two diameters and the lines parallel to them. Drawing from the reader’s understanding of an ordinate’s relationship to its diameter, the emphasis on the parallel relationship of GH to AB also opens the reader’s mind up to the possibility of an ordinate having a specific relationship to another diameter as well as its own diameter. GH’s parallel relationship to AB expands the reader’s understanding of ordinates and prepares the reader to clearly understand the definition of conjugate diameters as it is actualized in Proposition 16.
Proposition 16 brings to completion the foundations laid in Proposition 15 by simultaneously challenging the reader to consider diameters in a new context and offering newfound certainty in its application of the definition for conjugate diameters. Apollonius finally uses the label of conjugate diameters in Proposition 16, when the reader is ready to consider a second diameter in a new way in an unfamiliar context: in a hyperbola and drawn outside of the sections. Unlike the second diameter from Proposition 15, the new diameter XCD does not include any of the qualifications about relationships to squares on parameters, and does not have a parameter. Instead of leading the reader to recognize XCD as a diameter because of those relationships, Apollonius uses Definition 5 to prove it as a diameter because it has bisected GH, which is parallel to AB. This challenges the reader to grow in an understanding of diameters because the reader must accept a second diameter solely on its relationship to the first diameter AB and the lines parallel to AB, with no other established relationships to the rest of the figure.
In the midst of this growth in the reader’s understanding, Apollonius also provides new certainty when he finally uses the sixth definition to label the two diameters of Proposition 16 as conjugates. Definition 6 reads, “The two straight lines, each of which, being a diameter, bisects the straight lines parallel to the other, I call the conjugate diameters (συζυγεῖς διαμέτροι) of a curved line and of two curved lines” (Apollonius I. Def 6). Having proved that each of the straight lines XCD and AB are diameters, Apollonius labels the relationship between XCD and AB to offer a clear example of conjugate diameters. That exact relationship, expressed in Definition 5 as well, is the only way he defines XCD as a diameter, which highlights what makes diameters conjugates without dividing the reader’s attention with other relationships. This shift in the way of considering diameters in combination with its isolated context allows the reader to discover a new appreciation for the importance of a diameter’s relationship of bisection to its ordinates and the relationship of its ordinates to other diameters in the reader’s definition and recognition of a diameter. Proposition 16 provides the reader with clarity on conjugate diameters while stretching the reader’s understanding of diameters outside of the context of parameters, abscissas, and squares on parameters. Flowing from this balance of growth and certainty, the cycle begins over again as the reader moves forward in the text.
Apollonius’s expansion of the reader’s understanding of the diameter through Propositions 15 and 16 opens up new applications and possibilities for growth in the way the reader thinks about diameters. Already new questions arise from Apollonius’s use of the fifth definition in Proposition 16 and the placement of Propositions 15 and 16 right before the second set of definitions: Are transverse sides always transverse diameters? Can there be a transverse diameter or upright diameter without its conjugate? Can that be true for transverse diameters, but not for upright diameters because they would not have a reference point, like the diameter in Proposition 16? Are upright sides related to upright diameters? How does the εῖδος mentioned in the eleventh definition relate to the εῖδος of Proposition 12, where hyperbolas are first defined? The reader’s recent growth in understanding conjugate diameters prepares and propels the reader to embark upon an exploration of these questions. Each learned step in certainty leads to more growth, and more questions. This engaged relationship with mathematics through geometry reflects the human capacity for learning, develops the mind, and equips the reader to participate in wonder and inquiry. The process that Apollonius models and gifts to his readers provides the learning space for these questions in tandem with an exercise of the mind-- not spinning in the contained circles of sets of skills and rules, but spiraling like a nautilus shell.
Apollonius. Conics. Translation by R. Catesby Taliaferro and Michael N. Fried. Santa Fe: Green Lion Press, 2013. Print.