This article discusses the applicability of the various calibration formulae to archaeological finds. In other words, whether or not the formulae were actually followed in practice.
Washers and field-frames Edit
A number of field-frames from the Imperial Roman period have been found. If calibration formulae were adhered to, we could reasonably expect that the field-frames would have roughly the relative dimensions. This does not seem to be the case:
|Ballista||Ring hole Ø||Frame height||Ratio||Washer Ø||Bundle height||Ratio|
The data is taken from various articles by Baatz (1974, 1977, 1978, 1981) and from the cheiroballistra manuscript. The data for the Sala field-frames (Boube-Piccot 1994; see Iriarte 2003) is missing at the moment. When measurements are given in textual format they're used as is. Those measurements which are not given have been estimated from the accompanying diagrams using a CAD program. Provided the original diagrams don't have any distortions or errors, the accuracy of the measurements should be at least 90%. Note that numbers for the cheiroballistra are simply for comparing the ratios, because 1 dactyl does not equal to 20 mm, but somewhat less (possibly 19.3 mm), depending on how we choose interpret a dactyl in that particular manuscript (see Iriarte 2000).
As can be seen, the Gornea field-frames seem very squat and the ones from Cheiroballistra very tall, whereas the rest are in roughly the same category. Of course, the ratio could be adjusted by modifying washer wall thickness and washer height, but the discrepancy between, say, Gornea and Orsova field-frames can't be explained away very easily using this method. The data for the Sala field-frames (Boube-Piccot 1994; images also in Iriarte 2003) is missing.
There are two archaeological finds where washers and field-frames have been preserved, and where the ratio of spring height and diameter can be calculated. In these cases (Lyon and Hatra) the difference is less pronounced than in the field-frames, but still significant.
All in all, the discrepancy in these numbers is too great in my opinion to suggest that any single formula was used.