The publication mentioned in the title of this post, Maschinen Tractate, was never published by Bessler but was found in his possessions after his death. I have producedca digital copy and a printed version available from the side panel of this post. It includes an English translation of Bessler’s handwritten notes, which were difficult to read, but still useful.
Bessler’s Maschinen Tractate (MT) consisted of 141 illustrations designed to lead one to the discovery of Bessler’s Perpetual Motion machine. Originally intended to provide material for his planned school for apprentices, it unfortunately lacks the final illustration depicting the solution. I called the final page, ‘The Toys Page’, because it includes a brief reference to ‘children’s games’. This final page has the numbers 138, 139, 140 and 141 added to the bottom left of the picture. The page immediately before the toys page was numbered MT137, which was the logical number for the preceding page.
There is a possible explanation for the inclusion of those numbers. Firstly 141 is only divisible by 3 and 47. Euclid’s 47th proposition shows how to construct a pentagram and I’ve shown that most his construction method can be seen in his two pictures of his Merseberg and Weissenstein wheel.
But interestingly adding together all four numbers written at the foot of the ‘Toys’ page produces a total of 558, and 55 we know is one of Bessler’s most favourite numbers, but the 8 is not so easily explained. However adding these three numbers brings the total to 18, the key number in the pentagram. So here we have a typical Bessler move designed to make us think and seek an explanation - suggesting the pentagram again.
As I have pointed out previously MT137 contains the musical ‘circle of fifths’ diagram, a guide for musicians. It takes the form of a dodecagram, a twelve pointed star. See adjacent picture of MT137.
Some of you may be aware of the work I've published on www.theorffyreuscode.com .Three of the pages refer to the dodecagram on MT 137. I wrote that Johann David Heinichen, 1683-1729, a German musician, introduced the concept known as the ‘circles of fifths’ in 1711 (he called it Quintenzirkel). I suggested that MT 137 being similar to his quintenzirkel was designed to point to the circle of fifths, thus being another pointer to the number five.
I was drawn to this illustration, MT137 because it looked like a random addition but I knew that nothing in Bessler’s books was devoid of purpose. We know that Bessler was fascinated by the history and the relationship between numbers and letters and their hidden meanings and of course all the popular codes of the era, and it seemed to me that he had included the number quite deliberately, even if it was somewhat roughly executed. I believe that he added the Toys page later in life and replaced some of the existing pages which gave the secret of the wheel. My guess is that he also added the MT 137 at the same time.
MT 137, is the only illustration in the MT which doesn’t appear to show any mechanisms. One reason for this was, as I suggested above, to provide a hint towards the circle of fifths, but as Bessler usually included two or even three pieces of information in each of his clues I felt there could be something additional, that was invisible to me.
I was intrigued by the possibility that the number 137 was recognised to have special properties in Bessler’s time. There are websites devoted to such things as the properties of the three main pyramids which allude to the number 137, plus the Kabbalah, numerology, Freemasons, etc. I’m inserting an interesting link about the number 137, the GOD particle.
“Leon Max Lederman is an American experimental physicist who received the Nobel Prize for Physics in 1988. In his book, “The God Particle”, he writes:
“One hundred thirty-seven is the inverse of something called the fine-structure constant.”
But I discovered that Bessler was hinting at the relationship between 137 and the golden angle or the golden mean, well known to the ancient Egyptians and the Greeks who called it phi, after the Greek sculptor Phideas. Phi, the golden ratio, is equal to 1.618, plus an unending succession of numbers. Plato discussed the subject at length in his Timaeus and of course there are the Leonardo Fibonacci series of numbers, and the laws of nature also dependant on the gold mean!
In geometry, the golden angle is the smaller of the two angles created by dividing the circumference of a circle according to the golden ratio, thus creating two arcs so that the ratio of the length of the smaller arc to the length of the larger is the same as the ration of the larger arc to the full circumference of the circle.
I must thank Trevor, a long time correspondent for pointing out the error in the above illustration. 360/1.618 is of course 222.5 and not 225.5. I originally posted the illustration back in 2020, but nobody noticed then, me included. 😃
This provides two radii with angles of two particular degrees. The golden angle is 137.508. I suspect that using the number 137 for his dodecagram seemed like a good idea to the inventor, but he couldn’t name it MT 137.5, that would be too obvious. Bessler used the golden ratio routinely in his drawings and it was more commonly integrated in works of art than it is today.
If you use two radii to divide a circle according to the golden ratio it yields sectors of approximately 137° (1.618, the golden ratio) and 222°, hence it being the numbered 137.
To be accurate 360/1.618 = 222.5 and 360-222.5=137.5 Curiously 1/137.5 = 00727272727 etc. and 5x72=360, the basic numbers of the pentagram again.
But it is also interesting that 137.5/55 = 2.5 exactly because in his musical circle of fifths Heinichen explained that the circle of fifths gets its name from the fact that you travel across the circle from one point to another 5/12th away, or 2.5 segments away, in order to find circle of fifths. This is a way of organising pitches as a sequence of perfect fifths.
In the above illustration the circle with its 137.5 degree angle also mimics a clock at five o’clock, which is a good pointer to the circle of fifths. But in truth with the hands showing five o’clock the angle would be 150 degrees (5x30) not 137.5. To show the angle as 137.5 with the hour hand at five, the minute hand is in fact closer to the number one on the clock, or 12.5 degrees nearer,
Check out my web site at www.theorffyreuscode.com
JC
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