Would you read this?

by jemma margaret

Dear Readers,

I know you are all secretly following along with the hope of learning more history of mathematics. Over the past few weeks I have been researching and writing an article sized object that may one day be part of the Dissertation. At about the halfway (wishful thinking) point, I find myself floundering for structure. So here is an abstract draft, with that in mind. Specific suggestions are of course appreciated–but do bear in mind there are certain constraints on how much of a popularization (vulgarisation, as they say here) this could be. And never mind the grammar or the use of strange tenses…

Even if you aren’t interested in this sort of nonsense, you can surely appreciate the beautiful diagrams of Poncelet’s Traité, non? I very much appreciate the design of this book, where you can unfold the diagram in the back so it can be viewed simultaneously with whatever page you are reading. Unfortunately, the people who make copies for Google books did not bother to properly unfold the paper.

And speaking of my discipline in the real world, I had a really painful conversation last night about someone who thought Descartes had postulated values for x-cubed plus y-cubed is z-cubed for three positive integer values and the enormous numbers satisfying this equation had only been discovered recently using supercomputers. Well, in fact, it was Fermat who had postulated the impossibility of such integer values for any power greater than 2 (so cubed, or to the fourth, or to the fifth, etc), which had been proved correct by supercomputers in 1994.  It’s impossible, I repeated, and slowly the significance of the “it” began to change.

Sigh.

Abstract: In the early nineteenth century like in ancient Greece, geometry was the study of relationships between abstract figures in the plane and in space. Below this superficial continuity, geometers in France and Germany invented and promoted a variety of different means, systematized as “methods”, with which to approach old and new problems and theorems.  To better understand the set of techniques associated with proposed methods in geometry, this paper considers a set of problems that served as a battleground for methodological shows of power. A good method would lead to simplicity, elegance, fruitfulness, or generality. We will explore problems rather than theorems. Up through the nineteenth century, problems concerned constructing objects meeting specified constraints. In the context of geometry, solutions to problems were instructions to graphically render such objects, usually with a compass and straightedge, or some other set of specified tools. To verify that this solution will always work, the construction should be proved by referring back to earlier problems, theorems, definitions, and the like. The proof can employ a wide range of methods, but the solution must be theoretically realizable in geometric space. Consequently, the same construction might be proved in widely different manners, and constructions can even remain unproved–although this is generally frowned upon. Although the most famous problems of the nineteenth century (the problems of Apollonius and Malfatti, which were each solved over 100 times) will be discussed, the primary investigation focuses on a broader range of research with a narrower set of participants. The problems of inscribing or circumscribing a second order curve (conic sections: circle, ellipse, hyperbola, parabola) to a given polygon through given points or tangent to given lines can be as seemingly simple as putting a triangle in a circle. The problem generalizes along three primary paths: the number of sides of the polygon, the number of dimensions of the figures or the types of surfaces considered, and the variety of curves. We will encounter all three variations and see how approaches that might excel along one path were grossly inadequate in another. We open with Brianchon, who happily promoted a variety of methods through his career and is best known for proving a theorem about hexagons which became a very powerful tool in later applications. Then, we will meet Gergonne, an outspoken supporter of analytic geometry as well as the editor of the Annales de mathématiques pures et appliquées in which many geometrical problems were posed and solved. Within the pages of this journal, we will also be introduced to Poncelet, who redesigned pure geometry to achieve the generality of analysis. On the side advocating pure geometry as it once was, stand Durrande and Steiner, at least when it’s convenient. Finally, Plücker eliminated elimination from analytic geometry, enabling a form of coordinate representation free from tedious computations. This problem is rooted in antiquity and variations appear in contemporary textbooks, but we will primarily focus on the twenty-one year span of Gergonne’s Annales (1810–1831), the forum in which much of this debate took place.

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