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Bessler's use of the Golden Section

Johann Bessler clearly knew much about the properties of the pentagon, pentagram and other geometrical shapes, as is evidenced by his use of the pentagram in most of his drawings and he admitted as much in his biographical details in ĎApologia Poeticaí. So it seemed to me that there might be more information to be extracted from his drawings than I had already found. One of the principle properties of the pentagram widely discussed and used in Besslerís time and before, is the presence of a concept known as the golden mean, golden ratio or golden section.

Since long before the Renaissance, many artists and architects designed their works to accomodate the golden ratio in various forms - and the most popular has been the so-called golden rectangle, in which the ratio of the longer side to the shorter side observes the golden ratio. It was thought that this proportion was more pleasing to the eye, and mathematicians have studied the golden ratio because of its unique and interesting properties.

In its simplest form, it is the division of a line in two sections, where the ratio between the smallest section and the largest section is identical to the ratio between the largest section and the entire length of the line. The ratio is about 1/1.618.

A line, A to B, is divided at point X, so that the ratio of the two parts, the smaller XB to the larger AX is the same as the ratio of the larger part AX to the whole AB.
line golden ratio1
So BX to AX is as AX is to AB and the ratio is equal to 1.618033988749895... or Phi (note the capital P). This ratio is most famously found in abundance in the pentagram. Each segment of each line is found to accord to the same golden ratio.  In the drawing below the blue horizontal line demonstrates the golden ration when compared to the shorter green line. The green line is also in the same ration when compared with the red extension to the same line. There are many examples of the same ratio in the drawing.

pentagram_lengths2

This same ratio can be applied to a rectangle and this is what Bessler did in his drawings.

Draw a square (blue lines) and on the mid point on one side (A), place the sharp point of a pair of compasses. Extend the pencil point downwards to one of the opposite corners of the square (B), and then draw a curve (red) from B to C. The extended red line at the top of the square AC is in the golden ratio with the blue part of the same line. 
golden rectangle1
Looking at Besslerís first and second drawings of his wheel we can see something that appears to have no explanation unless we assume it is to draw our attention to the presence of a square within the rectangle of the whole picture. I have placed the drawing which also contains the hidden pentagram below, and below that an enlargement from each drawing, of the feature Iím referring to.

GB wheel

merseberg wheel  original small1

If you look at the enlargement below you will see that there is a small extension to the base of the left pillar supporting the pendulum. I have identified it with a red arrow. This does not seem very noticeable but it is when you relate it to the whole picture. The line in question forms the right side of a square of the left side of the whole picture, leaving to the right a rectangle. 

2 wheel GR enlargement
In the next picture I have extended the line (red) upwards to the top of the picture. I have also drawn in blue, the horizontal line at the top edge of the drawing, and the lower horizontal line. Due to the age of the drawing the lines do bend slightly out of true, however assuming that the first vertical red line is true, I have taken the measurement of its distance from the left edge to the vertical red line and found that it matches exactly the length of the red vertical line and thus forms a perfect square. I then added the second blue vertical to mark the mid point of the square. Notice that it is aligned perfectly with the left edge of the number 4 pillar, on the right side of the edge-on wheel. The blue diagonal shown in the drawing is the same length as the distance from C to B and confirms that this drawing is a golden rectangle and that the golden ratio is present because AC to CB is as CB is to AB.

Grundlicher Bericht GR 1wheel2

The same applies to the second version of the wheel in Das Triumphirende. However there are differences. For instance in the first drawing the two number 1s (highlighted in green circles) are lower than they are in the second version. In the second drawing (below) they are shown placed slightly higher and now align (pink line) with the supporting strut at the top of the stamper unit. It may be that the first picture did not draw a clear enough connection with that line in which case it may assume importance later on in our research.

Merseberg GR2
Also there is an interesting pair of parallel lines. I drew the diagonal (green) from upper right to lower left in the right side rectangle and found that it ran precisely parallel with the green line drawn from the point of the padlock up through the centre of the wheel, through the exact point where the rope emerges from behind the wheel and up to the point where the pulley supporting beam is connected to the roof.

I got the same result with the first drawing, except that the upper point matched the lower side of the point where the pulley supporting beam is attached to the roof.

To avoid a succession of bewildering lines, I shall in future drawings on this page, omit some of those already described.

In the picture below, there appear to be two datum points, they have very clearly marked short horizontal lines on the top of the two pendulum-supprting pillars numbered 12. They are positioned on either side of the central pillar which supports the wheel. I have drawn a horizontal line (red) across them. 

The following lines all align with a feature in the drawing.

From the way the enlarged piece is drawn I have assumed that it also refers to the right side of the picture - the rectangle. Notice that the rectangle is exactly bisected by the vertical green line which coincides with pillar (number 12) on the right side of the central pillar.

Next I drew in the same pink line I referred to above which runs precisely between the two 1s.

Notice that blue horizontal line also bisects the two squares occupying the upper part of the for-shortened rectangle.

The yellow line bisects the right half of the rectangle and coincides with the right edge of the window wall.

The dark green lines bisects the left half of the same rectangle, and is aligned on the centre of the main supporting pillar.

There are two grey lines which run in perfect alignment with the two short arms supporting the pendulum, I have reddened them where they align. One grey line runs from the upper right of the complete drawing down to the left bottom. The other grey line runs from the top left of the rectangle to the bottom right of the same rectangle.
Merseberg wheel 2nd GR
This confusion of lines leads one to one conclusion and that is that the whole picture is composed on a set of gridlines. See below. The original rectangle on the right is filled with 28 squares and the original square on the left is filled with 28 rectangles.

Merseberg wheel grid
It seems clear that Bessler was completely aware that artists of his time and earlier often designed and set their works of art on a grid composed within the rules of the golden section. It is widely recognised today, that surviving works of art can be shown to follow this rule and proves that Bessler was extremely well educated in this sphere as many others.

What does it all mean? I donít know. It clearly shows that Bessler was fully cognisant of the use in art of the golden mean and all its ramifications, but why was it included here? Is it another link to the pentagram and is that a reciprocal feature merely confirming each otherís existence ... only?

Copyright © 2011 John Collins

 

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