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. Thus, Eric Fernie opens his Beginner's Guide to the Study of Architectural Proportions and Systems of Length with a rueful reflection on the ”…almost pathological condition once described as pyramidiocy…,” which drives the idiot to explicate the forms and dimensions of the Great Pyramids through reference to geometrically encoded messages. After this admirably skeptical introduction, Fernie proposes a useful set of critical working principles: that behind the conception of the great building may lie relatively simple geometric manipulations, often involving a basic proportional relationship like 1:√2. 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. The work most often cited as a negative example is George Lesser's Gothic Cathedrals and Sacred Geometry with its small plans heavily overlaid with geometric figures.

Interestingly, in the other notable recent wide-ranging essay on medieval architectural design, The Wise Master Builder, Nigel Hiscock takes positions diametrically opposed to Fernie’s, insisting instead that one should work by imposing a limited range of geometric shapes on existing plans since the direct involvement of the investigator in measuring the building will, or so he claims, introduce an unacceptable level of subjectivity. Hiscock argues that older studies tended to place too much emphasis on the application of the manipulative mechanism of the square root of two: he suggests that we should concentrate instead upon the three “Platonic figures”: square, equilateral triangle and pentagon. Nobody would doubt the importance of these figures in the builder's design tool box, but Hiscock's bewildering geometric tangles inscribed upon small plans may leave the reader baffled. Why, the skeptical student might wonder, would the builder adopt such extraordinarily complicated design strategies as these?