Take a look at A.W. Fuller's article Universal Rectilinear Dials in the 1957 Mathematical Gazette. He says:
"I have repeatedly tried to evolve an explanation of some way in which dials of this kind may have been invented. Only recently have I been satisfied with my results." The rest of the article is dedicated to developing his idea. Note that it's only speculation - he can't point to any actual historical proof. That's the problem with this whole endeavor; there is no known early proof for this form of dial - either in universal or specific form. (It seems that the universal form probably came first.) It was published in 1474 by Regiomontanus without proof. He does not claim it as his own invention and in fact refers to an earlier unidentified writer. There has been speculation that he got it from Islamic scholars - but nothing has been found in Islamic research that would qualify as a precursor. The dial is somewhat similar to the navicula that may have originated in England - but that dial is only an approximation to correct time. In discussing this history, Delambre says: "All the authors who have spoken of the universal analemma, such as Munster, Oronce Fine, several others and even Clavius, who demonstrates all at great length, contented themselves with giving the description of it without descending, as Ozanam says, to the level of demonstration." "At this one need not be surprised, seeing that it rests on very hidden principles of a very profound theory, such that it seems that it was reserved to [Claude Dechalles] to be able to penetrate the obscurity." So Dechalles gave what was evidently the first proof in 1674 - 200 years after Regiomontanus' publication. But as Delambre further notes: Dechalles’ proof … is long, painful and indirect, … without shedding the least light on the way by which one could be led to [the dial’s] origin. So - pick whichever proof makes sense for you. Fred Sawyer
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