Pretty alright throughout. The one joke thought is that trigonometry is useless - t would be better to have something more solid for its use rather than just signs and wacky academics.
Intro:Good, I would say however that it seems a tad long winded compared to the length of the rest of the article - would shorten it or integrate material in to subsequent sections; it is currently kind of top heavy. I think you could briefly cover the entire article in introductions "Say it, Say it more, Say it again is the golden rule. The intro could be more integrated and reference could be made to developments that are discussed further own in our article.
Moment of clarity: Good, I like this section a lot, and I think it’s much snappier and would be the kind of thing that would work well in the introduction. I also think that having the characters here works well. I like Stodemen and I think that his part here is a little short, and his character could be filled out more, I like the Archimedes reference here.
Gibson: This section does not work too well, the lit is not really all that good, also I thin that t just isn’t all that funny, you also have an unnamed "Laurite which comes across as random. Lists are generally a no go and it does not seem to work to well here, personally I think I would b a lot more funny to perhaps add uses that actually work, but fatally go wrong - for example thought you have "trigonometry is useless" I think there could be lot more humour potential in something that suggested great uses that were hampered by some outside force from either the mathematicians or the math = something like a curse that seems to perpetuate some great evil, There’s this really cool book called Candide where there’s this ridiculous philosophy being parody about "best for all possible worlds" - by following this madness all the protagonist get themselves into quite mess and many humorous adventures. I thin that having something more like this will open the article out more.
Rutherford: This is a good section, and probably the best I the article, I like how you bring philosophy in, it would be quite cool to see more fields brought in further up like this. The following section seems tacked on and leads to closely on from this; I suggest integrating Rutherford’s conclusions within you prose. Overall the Rutherford section could do with a more degenerate close; it would be quite hysterical to here of Rutherford dying in an opium den with Kleenex box shoes trying still to find the mystical use of Trigonometry. It’s just an idea to liven the humour levels up. Overall the article seems to be a little weak on humour levels, a bit of wacky randomness could work quite well
It's a bit of a one joke article "Trigimetry is useless" - this is not necessarily bad, but I would package this up more wit some more material, similarly the progression is created allot though lists: admittedly this is part of the title, but I think that they don’t really add too much humour, and to my knowledge their use is generally discouraged. You keep a great deal f focus on trigonometry - I don’t think it would hurt broaden this out a little with some other fields of mathematics.
Prose and formatting:
Other than the list’s very well written throughout, and I could see no spelling/grammar errors - there could be more links however. I mentioned the lists, as with the introduction its not always as snappy as it could be - this is likely to be sentence and paragraph length which could be shortened throughout to give some of the joke s little more impact
Nothing to shout about here, possibly to few, a bout three would fit in its current form. I don't think they're all that funny. The first one is reasonable though, the second one seems a little awkward and not really adding to much. With further content/characterization it will probably become more apparent what would fit in
This could turn out to be something quite special with some time spent on it - for me I would add some more creative elements which will broaden the concept out into something a lot more funny.
I am very tempted to whore here, but I won't. Should you need anything, do not hesitate to leave a note on my talkpage;)