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Discursive Image
The Two-Dimensional Orthographic Projection of a Semicircle onto an Epicycle of Cones
Mark Ericson
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In “Observation Nine” of chapter three in the fourth tractate of Guarino Guarini’s treatise Architettura civile (1735), he describes the process of constructing the drawings of a vault that intersects a cylinder at an oblique angle. He provides a template for a semiellipse and instructs the reader to make a drawing instrument out of “strong card” using the template as a guide. Guarini utilized an argument from Euclid to create an instrument that draws curves specific to the angle of incidence. The drawing included in this publication is part of series that translates the written and drawn instructions from the fourth tractate of Architettura civile into a set of repeatable procedures. This particular drawing uses the techniques of “Observation Nine” to orthographically project a semicircle onto cones moving about the minor and major orbits of an epicycle. The cones vary in height, orientation, and radius over time, producing variable conic sections. Whereas Guarini utilized a single template, this drawing utilizes many. Importantly, like Guarini’s work, this drawing is confined to the use of points and lines on a two-dimensional plane. There are no three-dimensional objects, surfaces, or curves of any kind. The drawing is an orthographic projection.

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