The literature of Art and Architecture makes much of the importance of correct proportions. It turns out that the most referred-to proportions have been popular since the time of the ancient Greeks. The Greek philosophers, in particular Pythagorus, spent a lot of effort on the subject of geometry. They arrived at a set of pleasing proportions that can be created using simple geometric figures and therefore by ordinary drawing instruments.
Could a geometric method have been used by a latter-day draftsman to construct his dial drawing ready for use by the screen-printing department?
Pythagorus came up with three kinds of proportions: Arithmetic, Geometric and Harmonic. You can read more about those here. These proportions or "means" can be calculated but, more importantly, they can be drawn using simple drawing instruments such as rulers or compasses. If we were to consider the numbers 1 and 2 and find out what the three possible means are, that would give us three easily-drawn ratios. They work out as follows:
The arithmetic mean of 1 and 2 is 1.5, a proportion of 2:3. Easy enough to draw: Make a square and divide it into quarters. Draw a line through the diagonal of one of the quarters and continue until it meets the extension of one side of the square (see construction at right).
The geometric mean of 1 and 2 is 1.414: Draw a square with a side equal to the required width. Then use dividers to measure the diagonal and use them to mark the required height.
The harmonic mean of 1 and 2 is 1.333. To be exact, it's 1-1/3 and can also be represented by the proportion 3:4: Split a square in two and then split one of the resulting rectangles in two along it's length and "remove" the outer part.
Be that as it may, I then measured the aspect ratios of the outside edges of the watch dial minutes chapters for which I have pictures. I measured the chapters because these are probably both the starting point and the main feature of any dial design. I leveled them up in PhotoShop and then measured the minutes chapters width and height in pixels. Dividing the height by the width gave me an aspect ratio, i.e. the proportional mean. Whilst bearing in mind experimental errors with this method, here are the results:
These two have the arithmetic proportions; the Jurgensen measured at exactly 2:3
This one was close-ish to geometric at 1.39:
This one was exactly harmonic:
These non-pythagoreans came in at an easily drawn 1.25 which, by the way, is a popular aspect ratio for photographic printing, e.g 8 by 10:
The one below, at 1.7, looks skinny and appears not to fit any known arty-crafty criterion, not even the oft-quoted golden ratio. However, 1.7 is extremely close to 1.666, i.e. a proportion of 3:5 which can be drawn similarly to the previous method:
Note also that the numbers 3,5 are sequential members of the Fibonacci series which, at it's limit, does indeed equal the Golden Ratio.
I conclude that many designers used easily-drawn geometric proportions for dial art-work but did not necessarily adhere to just the three classic Pythagorean means. In Japanese architecture, rooms were often sized by the fit of standard tatami mats which themselves were 3ft by 6ft i.e. 1:2. Different arrangements and numbers of mats used resulted in various proportions, including our 2:3, 3:4 and 4:5 but apparently not 3:5. However, the Italian architect Andrea Palladio (1508-1580) did include the proportion 3:5 in his "The Four Books on Architecture" as one of the seven "most beautiful and proportionable manners of rooms". Not to be outdone, the 20th century French architect Le Corbusier came up with a "Modulor" tool that allowed the drafting of dimensions that were all proportioned according to the Golden Ratio.