Here's another little coded example. Please be aware that it has been abbreviated from my own writing and I have omitted some detail but the facts are there for anyone to check.
I noticed early on that there seemed to be an excess of numbering in the wheel drawings in Grundlicher Bericht and Das Triumphans. It looks as though some of the items are ‘over-numbered’. By that I mean that Bessler seems to have labelled the parts with a particular number more than seems necessary. For example the main pillar supporting the wheel is numbered 4, three times. The slimmer pillars are numbered 12, and two of them to the left are numbered twice each, and the other two are only numbered once each.
Some numbers appear more often than others and not just because they are attached to more similar pieces. After number 18 the rest of the numbers are lone examples. I speculated that this was done to achieve a certain total, and having identified each part once with its number, Bessler then sought to add to the total by labelling the same parts more than once. Obviously the higher numbers would make the jumps toward his desired total too big too quickly so he labelled everything once and having acquired a total, he added more of the smaller numbers until he had achieved his desired end. There are other peculiarities in the labelling and why this should have been done was unclear to me at the time.
I noticed early on that there seemed to be an excess of numbering in the wheel drawings in Grundlicher Bericht and Das Triumphans. It looks as though some of the items are ‘over-numbered’. By that I mean that Bessler seems to have labelled the parts with a particular number more than seems necessary. For example the main pillar supporting the wheel is numbered 4, three times. The slimmer pillars are numbered 12, and two of them to the left are numbered twice each, and the other two are only numbered once each.
Some numbers appear more often than others and not just because they are attached to more similar pieces. After number 18 the rest of the numbers are lone examples. I speculated that this was done to achieve a certain total, and having identified each part once with its number, Bessler then sought to add to the total by labelling the same parts more than once. Obviously the higher numbers would make the jumps toward his desired total too big too quickly so he labelled everything once and having acquired a total, he added more of the smaller numbers until he had achieved his desired end. There are other peculiarities in the labelling and why this should have been done was unclear to me at the time.
There are discrepancies between the two drawings which I shall discuss in a later post but for now be aware that in the first drawing the numbers, composed from
59 numbers, add up to 649 which is, interestingly, equal to 59 x 11 (both prime
numbers). In the second drawing the
numbers add up to 633, which is 16 short of the 649. In the second drawing the numbers 5 and one
of the 11s has been omitted, which is why the second drawing does not match the
649 of the first drawing (NOTE 5 x 11 =55). In both
drawings the picture cuts off the left hand end of the drawing and in the
process cuts off one of the number 11 weights.
If, in the first drawing, this is added to the 649 of the first drawing
it produces the number 660, and because we then have 60 numbers, 660 divided by
60 equals 11, but more interestingly, 660 divided by 12 equals 55. 55 is a number we shall see many times during these posts. This choice of the number twelve to introduce yet another example of the number 55, may seem too speculative, however, fascinating proof that it is the right assumption will appear in my next post
All the drawings in Das Triumphans contain similar number manipulations, the 'Andere Figura' and its companion, 'Secunda Figura', use the numbers from one to ten. There is obviously a case of overlabelling in the right picture, with four number eights.
The numbers in the
left picture add up to 28; those in the right, 62, to total 90. There are 15 numbers used and 90 divided by
15 is six. This does not seem to be a significant number, however
knowing that Bessler’s favourite number was 5, I realised that it divided 90
exactly 18 times – the ubiquitous pentagonal numbers again. Secondly the numbers used, 1 to 10, add up to
a total of 55 – the other Bessler number.
There are 39 numbers, running from 1 to 20, totalling
355. This does not seem significant
until you discover that one of the letter ‘e’s representing the ropes which run
around the spokes on the axle, has been omitted in the left side picture. The one ‘e’ missing could, if replaced,
increase the numbers to total 20 in each picture, and the total from 355, to 360. 360 divided by 20 eqals 18, our favourite
pentagonal number again - of course 360 divided by the missing 5 equals 72, another pentagon number.
So the first drawings have 24 numbers, apart from an apparent hiccup over the number 24 getting transposed to number 42 which was deliberate, as I shall show in a later post. The Andere figures use ten numbers, and the waterwheel uses 20.
There is so much more than these simple examples, but clearly there is a reason other than blinding us with mathematical mystification. It has to be something useful to us for reconstruction his wheel.
JC
The wheel drawing containing the archimedes pump (see above) also uses overlabelling to achieve a specific number. One of the differences between this drawing and the other
ones is the fact that in this one the parts are labelled with letters rather
than numbers. However there is one
labelled part which is strangely ambiguous and that is the main supporting
column which supports the wheel. It
looks like a ‘W’ however it can also be mistaken for the number ten, but this
cannot be right because the other parts are labelled with letters. The answer lies in the attached list of
labelled parts; here the list is entirely in letters except for the last item
which is undoubtedly labelled 10. You can see the
ambiguity in the expanded detail below, which has two examples of the number
‘10’, or the letter ‘W’.
Why then is the last item called item 10? The solution seems obvious; the intention is
that the reader should replace all the letters with numbers. The letters run
from ‘A’ to T’, plus the letter/number 10. Since
10 is the last item on the list one might suppose that it would represent the
letter ‘U’ as ‘T’ was the last letter, but in fact it represents the letter
‘J’. We know this for the simple reason
that ’J’ is omitted from the list of parts and does not appear in the drawing.
Bessler's use of the letter 'W' was often used as a way of implying the presence of the number ten, consisting as it does of two letter V's or Roman numerals to produce two more 5s. He wrote it in the style shown below, which was taken from one of hos many handwritten examples. The letter 'J' it replaces is the 10th letter of the alphabet.
So the first drawings have 24 numbers, apart from an apparent hiccup over the number 24 getting transposed to number 42 which was deliberate, as I shall show in a later post. The Andere figures use ten numbers, and the waterwheel uses 20.
There is so much more than these simple examples, but clearly there is a reason other than blinding us with mathematical mystification. It has to be something useful to us for reconstruction his wheel.
JC
