Talk:Gabriel's horn

message: Well, it seems as though in the context of the horn, the paradox mentioned defeats itself. A finite volume of paint could have an infinite surface area, if a piece of metal can also have those two quantities... Either I am missing something, or the original creators of the paradox were missing something (I shall not say what). Does anyone have any external information about the "paradox"? The time argument seems to be of a completely different paradox. btw, I am changing nothing as this is my own personal opinion... --Llamatron 06:54, 18 Aug 2004 (UTC)
thread_title: Not paradox

message: It is not needed for the proof as we have 
:A = 2\pi \int_1^a \frac{\sqrt{1 + \frac{1}{x^4}}}{x}\mathrm{d}x > 2\pi \int_1^a \frac{\sqrt{1}}{x}\ \mathrm{d}x = 2\pi \ln a 
but some people might be interested to note that 
:A = 2\pi \int_1^a \frac{\sqrt{1 + \frac{1}{x^4}}}{x}\mathrm{d}x = 2\pi \ln a + \pi \left[\ln \left( 1 + \sqrt{1 + \frac{1}{x^4}} \right)  - \sqrt{1 + \frac{1}{x^4}} \right]_1^a 
--Henrygb 01:43, 8 May 2006 (UTC)
replies: text: Of course you will have trouble evaluating that expression in brackets at 1. --Spoon! 22:30, 12 September 2007 (UTC)
replies: text: Why? It looks to me like [ln(1+sqrt(2)) - sqrt(2)] or [-0.5328...] 00:35, 24 January 2008 (UTC)
thread_title: Exact integral

message: A limit was evaluated as being equal to infinity which is nonsensical; I rephrased it for correctness.
Jake 06:52, 27 November 2006 (UTC)
thread_title: Mistake corrected

message: The article could be improved by adding a bit of history to it. Perhaps a whole section with it. When and where did Evangelista Torricelli first describe Grabriel's Horn? And the like. --Bilgrau 08:28, 28 March 2007 (UTC)
thread_title:  History 

message: At the end of Mathematical Definition it says "...it was considered paradoxical as, by rotating an infinite _area_ about the x-axis, an object of finite volume is obtained.". I changed it to read 'curve' instead of 'area', because building a solid of revolution is rotating a curve, not an area, the area is a mere consequence of the rotating curve. If anyone disagrees I'd like to hear their opinion, please.
Christophe Lasserre 16:23, 16 November 2007 (UTC)
thread_title:  Minor change 

message: They would be better if they ranged x>=1 rather than x>0. I don't have the tools to fix these. -- SGBailey (talk) 22:28, 11 July 2010 (UTC)
thread_title: Diagrams

message: I'm fairly certain the bit about the cissoid of Diocles is wrong. If the cup has infinite height, it must have infinite area, just like Gabriel's Horn. The book referenced is on Google Books, although the images have been removed which doesn't help. This Stack Exchange discussion also throws doubt on the claim of finite area. 109.155.174.230 (talk) 21:01, 15 October 2011 (UTC)
replies: text: As far as I can tell the point with the Huygens/de Sluse analysis of the cissoid of Diocles is that the area between the cissoid and the line x=1 is finite.  If this area for non-negative y is rotated about the line x=0, then the resulting volume is also finite and looks like a glass tumbler or jar infinitely extended in one direction. Meanwhile the volume inside the rotated cissoid is clearly infinite, and it is the comparison of these two volumes which is interesting. The area here is not a surface area as in Gabriel's Horn, but in effect a cross-sectional area.--Rumping (talk) 01:41, 26 January 2012 (UTC)
replies: text: and now there is an image of this container Rumping (talk) 09:08, 9 November 2021 (UTC)
thread_title:  Cissoid of Diocles 

message: The material comes from elsewhere. Paradoctor (talk) 23:43, 14 November 2013 (UTC)
thread_title:  New Further reading section 

message: The title and most of the external links spell "Horn" capitalized. The article lowercases it as "horn." I think we should settle on one spelling. What should it be? EpsilonCarinae (talk) 18:50, 13 June 2020 (UTC)
thread_title:  Capitalization of "horn" 

message: As far as I can determine, only authors of recreational mathematics books, and the odd textbook, use the name "Gabriel's horn".
Even that is a late 20th and 21st century affectation; there being nothing before the 1980s that I can find.
It isn't known as that in most places; in the 17th to 19th century everyone just called it the "solidum hyperbolicum acutum" ("solido hyperbolico acuto" being the ablative case, of course) or translations of the same into Italian/French/English/German/whatever.
We've got a published source that has researched the names into the 18th century, and I can confirm from my own findings that mathematics dictionaries into the 19th century continued likewise.
Barlow's 1814 New Mathematical and Philosophical Dictionary has "Hyperbolicum Acutum", for example.
"acute hyperbolic solid" is even given in Julian Havil's controversial-to-this-talk-page book, on page 83.
(I hope that this conotroversy is addressed now, by the way, by the mathematics professor that not only made the same error as Havil, but documented making that error.)
The original Latin "solidum hyperbolicum acutum" can be found even in this century in non-recreational mathematics books; Steven G. Krantz and Brian E. Blank's 2006 book on single variable calculus uses that name (on page 620), for example.
It is somewhat saddening that the two names first introduced for this are not the thing's proper name, promoting a false notion of what its primary name is, and a supposed connection to Christianity that isn't even right.
(The people who back in the 17th century tried to connect this with religion almost certainly knew their own religion well enough to know that Gabriel isn't stated to have a horn.  They never called it that, or a trumpet.)
Jonathan de Boyne Pollard (talk) 12:08, 27 September 2021 (UTC)
thread_title:  Title 

message: The section Apparent paradox is not well written and would greatly benefit from rewriting. 2601:200:C000:1A0:B513:B27B:407D:C1BB (talk) 18:27, 5 April 2022 (UTC)
thread_title:  Section "Apparent paradox" needs rewriting