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The Puzzle:

Source material taken from here

As discussed on Numberphile, here is John Herschel's doodle, sent to his friend Charles Babbage.

John H.

John Herschel was an English polymath, mathematician, astronomer, chemist, inventor, and experimental photographer, who also did valuable botanical work. Herschel originated the use of the Julian day system in astronomy. He named seven moons of Saturn and four moons of Uranus. He made many contributions to the science of photography, and investigated colour blindness and the chemical power of ultraviolet rays; his Preliminary Discourse (1831), which advocated an inductive approach to scientific experiment and theory building, was an important contribution to the philosophy of science.

Info on This Sketch:

Herschel said: "Interpret this hieroglyphic, it contains a great discovery."

The figure at the centre is described as: "Dionysius the God of Functions, alias the genius of abstract numerical magnitude."

Just silliness between two friends, or something more?

Pictures:

Whole Page More Devil Middling
(Dionysus, the god of functions, alias, the genius of abstract numerical magnitude)
Face Top Bit (new notation) (Muriate of exchequer bills) Stack Tail Staff Tree

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  • $\begingroup$ The image is very low res, can you replace it with a readable one? A summary of the video would be appreciated as well. In general, everything you need to fully understand the puzzle should be included in the post itself, not just linked, in case of link rot. $\endgroup$ – Mike Earnest Jul 1 '16 at 0:49
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    $\begingroup$ This is currently on -2. That seems rather harsh. $\endgroup$ – Gareth McCaughan Jul 1 '16 at 12:22
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    $\begingroup$ @Engineer Toast Thank you very much for adding the extra pictures, I mean to do that but hadn't had time until now. $\endgroup$ – Ovi Jul 1 '16 at 19:38
  • $\begingroup$ @engineerToast Is there a significant difference between emage 4 and 5? The contrast seems to be a bit different, but I don't see anything better because of it. $\endgroup$ – BmyGuest Jul 22 '16 at 8:58
  • $\begingroup$ @BmyGuest Good catch. I must've uploaded the same picture twice. The one I should have uploaded didn't turn out to provide much more detail as it isn't any larger in size, but at least now they match the source material. $\endgroup$ – Engineer Toast Jul 22 '16 at 11:34
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I'm not sure if the answer is relevant, but could it be continued fractions ?

At least, that's what the "serie" we see in this picture makes me think.

EDIT :
That's being said, the formula in the top right of this picture looks like the definition of $i$ : $\sqrt{-1}$

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  • $\begingroup$ And why do you think so? $\endgroup$ – ABcDexter Aug 3 '16 at 17:43
  • $\begingroup$ Edited the post, my answer wasn't really clear indeed ! $\endgroup$ – Frédéric Aug 3 '16 at 21:13
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The answer could be under that piece of tape. If one looks closely, they could see that there is ink underneath!

Not only this, but the page appears to have had the writing stripped off of it, you can see in the foruth picture, bright yellow words. What is strange about these, is that in the seventh picture, one can see that d/dx aligns almost perfectly with the washed out writing.

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  • $\begingroup$ If you select the 'yellow' text, mirror image it and rotate it slightly, it is clear that it is just the text on the previous page transferred onto the next page (presumably either 'cos the ink wasn't completely dry at the time of through some chemical reaction over the years). For example: the 'symbol' just above the Sqrt(-Sqrt(-)) symbol in the tree is actually the wort 'to' in "to turn to" and the three blobs and curved structure just below the exp(sqrt(+/-)) to the bottom right of the tree is the 'day' of "Tuesday". Unfortunately I can't show you 'cos I can't put an image in a comment... $\endgroup$ – Penguino Aug 4 '16 at 5:12
  • $\begingroup$ But you can put a link in your comment, @Penguino. $\endgroup$ – elias Aug 5 '16 at 6:12
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Material That May Prove Helpful

On a Remarkable Application of Cotes's Theorem
John F. W. Herschel
Phil. Trans. R. Soc. Lond. 1813 103, 8-26, published 1 January 1813
http://rstl.royalsocietypublishing.org/content/103/8.full.pdf
May provide insight in to how Herschel wrote mathematical notations

A History of Mathematical Notations: Vol. II
Florian Cajori
Cosimo Classics. Originally published in 1929.
Annoyingly Long link to Google Books
May provide insight in to how Herschel wrote mathematical notations

Reddit Thread Discussing the Same Material
/r/Remedan, et al
Reddit.com https://www.reddit.com/r/math/comments/3csjhe/a_mysterious_doodle_by_john_herschel_does_anybody/
May provide threads of investigation we have not yet thought of yet

Descendants of John Herschel
WikiTree
http://www.wikitree.com/genealogy/Herschel-Descendants-17
I have just now attempted to contact one of them. I may try more in the future. It is unlikely they any of them will have any insight but it's worth a shot.

Depressing Outlook

It is unlikely that we will be able to conclusively figure out any meaning in this message. It's possibly a joke between friends instead of a riddle. It's possible that Herschel was simply showing up a new notation he wanted everyone to adopt. One perspective I found from @OnThisDayinMath (Twitter) is as follows:

The God of Functions is suspected to refer to Dionysius Lardner who taught Mathematics at Trinity College at that time. Many of the symbols seem to have no relevance to mathematics of the period. They may have been ideas tossed around between the two men in their attempts to modernize British Mathematics.

They add a piece of information about Dionysius Lardner but speculate that the meaning behind this letter may not be of great value.

I will provide an update if I hear back from any descendants.

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I think Herschel's "Great Discovery" is:

The Devil is full of crap. Each mathematical concept is sorely misplaced, a bit like the Devil.

Except for the $x^y, y^x$ reference on the Devil's Horns:

Face,

which might mean that Herschel has solved $x^y=y^x$, but there is no supporting evidence in the picture.

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