20070911, 11:24  #12 
"Lucan"
Dec 2006
England
2×3×13×83 Posts 
The similar result which is most worth remembering
is that the surface area of a sphere radius R between two parallel planes a distance h apart is 2*PI*R*h. Last fiddled with by davieddy on 20070911 at 11:27 
20070911, 17:17  #13 
Cranksta Rap Ayatollah
Jul 2003
641_{10} Posts 
we welcome wellformed, intelligently stated puzzles.

20070911, 17:23  #14  
Cranksta Rap Ayatollah
Jul 2003
281_{16} Posts 
Quote:
This doesn't even qualify as a lemma. It's patently obvious, and your showboating and windbaggery border on offensive. 

20070911, 18:59  #15 
"William"
May 2003
New Haven
2·7·13^{2} Posts 
We are talking about two concentric circles, and a chord of the larger circle that is tangent to the smaller circle. Is it really "patently obvious" that the area of the annulus is the same as the area of a circle with the chord as diameter?
It's a simple consequence of the Pythagorean Theorem, but I didn't see it until I added the triangle to the sketch. Is there some even simpler method that I missed, or does this qualify as patently obvious in your lexicon? 
20070911, 20:42  #16  
"Lucan"
Dec 2006
England
2·3·13·83 Posts 
Obvious or not, I stand by my original post in which I
pointed out that this lemma didn't really help solve the problem: Quote:


20070911, 21:35  #17  
"Lucan"
Dec 2006
England
194A_{16} Posts 
Quote:
that Newton/Leibnitz invented calculus to make finding volumes difficult David PS I don't even count quoting (4/3)piR^3 as "avoiding calculus" (Unless you have a "purely logical" way of deriving it.) Mind you I suppose Archimedes knew this formula. Last fiddled with by davieddy on 20070911 at 22:06 

20070912, 05:25  #18  
Cranksta Rap Ayatollah
Jul 2003
641 Posts 
Quote:
Granted, the windbaggery and showboating still holds, since that's a result that I've known about since I was like, 9 or 10. 

20070912, 09:34  #19  
"Lucan"
Dec 2006
England
2×3×13×83 Posts 
Quote:


20070912, 09:44  #20  
"Lucan"
Dec 2006
England
2·3·13·83 Posts 
Quote:
the remainder from a hole of height H is more memorable, and use this to deduce the volume of the caps. 

20070912, 12:15  #21  
Bronze Medalist
Jan 2004
Mumbai,India
804_{16} Posts 
Worth remembering!
Quote:
Well! Well! Davie old boy! You state you can’t digest my long posts (attention deficiency?) so I’ll oblige and make it short but not necessarily sweet! Your ‘brilliant’ remark confuses and mystifies one more than it enlightens. You have done a grave injustice to the greatest mathematician Archimedes, The Father of Calculus! Yes! long before Newton and codiscoverer Leibniz and Gauss was a Greek called Archimedes. I had to study mathematical history that far back as some of our Indian mathematicians predate even Pythagoras. I don’t want to go into priority of knowledge, and who taught who, as this is a very controversial subject. Recent Excavations reveal that Budhayan used the Pythag. Theorem as far back as 1000 B.C. Evidently he knew it too! You have isolated a result stemming from two of the most profound theorems enunciated by Archimedes which can be represented by one diagram. Archimedes himself considered it his greatest achievement, and wished that a replica be put on his tomb stone! Unfortunately he was unceremoniously hacked to death by a Roman soldier and his grave lies in obscurity. Without even a passing mention of a sphere inscribed in a cylinder and the relationship of area and volume between the two your ‘brilliant’ observation is open to question and proof. I can imagine the newbie’s scrambling for their calculus text books when there’s no need too. Archimedes demonstrated this very clearly and without ambiguity 300 years before Christ. Finding the area of a spherical cap is as easy as pie if you know Archie’s theorems. And I will stress that one does not need Calculus at all! Rereading your subsequent posts there is every evidence that you are not at all aware of this theorem. I sincerely hope that I am wrong. I can face ridicule, but I doubt if your ego will take it. What a great pity! Now this is when the great Silverman should Step in and call you all the names ‘wit’ ‘twit’ and my own to him ‘twat’ Bernard Shaw made a very wise observation when he quipped “Those who can, DO, Those who can’t, TEACH.” How very true! In my experience of flying British school children specials on their holidays to and from London, most of them can only go as far back in mathematical History as Newton. The simple reason I suppose is that the ‘over qualified’ teachers don’t study Mathematical history themselves! Need I say more ? Mally . 

20070912, 13:02  #22  
Nov 2003
1110100100100_{2} Posts 
Quote:


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