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The paradox of the title is that while most historians of medieval architecture agree that a combination of geometric and arithmetic methods was generally used to lay out a medieval church, there has been little consensus on the specifics of the process in relation to the design of any particular edifice.
I begin by identifying four premises which underlie the debate. A case study uses newly generated point cloud data from a laser scan of the choir of Beauvais Cathedral.
Here is my paradox. All students of Gothic architecture would surely agree that our great churches were laid out using some combination of geometric and arithmetic methods — methods that must leave their traces in the finished edifices. Yet attempts to define the process more closely in any given building, to establish patterns of practice common to many buildings, or to speculate upon the significance of numbers and shapes often result not in consensus or productive scholarly exchange, but rather in rancorous accusations of unacceptable methodology, sloppy measuring, wishful thinking, or skullduggery.
In order to crack the underlying code, Fernie insists, the investigator must actually measure the building and work with the numbers, or with accurate digitally scanned shapes. It is not enough to superimpose thickly limned geometric figures upon small-scale plans or sections. Why, the skeptical student might wonder, would the builder adopt such extraordinarily complicated design strategies as these?
What advantage would have been gained? How could users or visitors have begun to comprehend the system? Zenner proposes a series of greater and lesser circles; Addiss favors a modular system based upon a known foot unit.