Here goes . . .
An London founded in London in about 1836 by a seven individuals with an interest in the London Clay including, in particular, the fossils found therein. The founding members were
- Dr James Scott Bowerbank (1797-1877)
- Frederick E. Edwards (1799-1875), author of The Eocene Mollusca (Palaeontography Society)
- Searles Valentine Wood
- John Morris
- Alfred White (zoologist)
- N. T. Wetherell, surgeon of Highgate (1800-1875)
- James de Cane Sowerby
Korean mythology describes a unit of measurement of time that is defined as follows:Imagine a mountain, made of solid granite, exactly one mile high. Once every thousand years an angel flies down from heaven and brushes the summit of the hill with her wings. The unit of time represents the number of years it would take for the angel and her summit-brushing wing to erode the mountain down to sea level.Source:
- a nearly literal transcription of the footnote on page 69 of The Map That Changed The World / William Smith and the Birth of Modern Geology, by Simon Winchester (© 2001 Simon Winchester; published by Harper Collins; ISBN 0-06-019361-1)
I first encountered this proof when I was a student in junior high school (about 14 years old). It is still my favourite proof.
Here we go:Assume that a rational number exists such that r2 = 2. Represent this rational number as a/b where a and b are integers which have no factors in common other than 1 (i.e. a and b are mutually prime).In summary, any rational number a/b which is the square root of 2 can never be expressed in its simplest form (i.e. a and b will have common factors). This is absurd and hence, the square root of 2 isn't rational.
- If a/b (with a and b mutually prime) is the square root of 2 then 2 = (a/b)2 = (a2)/(b2).
- Therefore 2b2 = a2.
- Therefore a2 is divisible by 2.
- If a2 is divisible by 2 then a is also divisible by 2 (remember that a is an integer).
- Therefore 4 is a factor of a2 which implies that 4 is also a factor of 2b2 (because 2b2 = a2).
- Therefore 2 is a factor of b2 and hence b is divisible by 2.
- Therefore a and b aren't mutually prime (since they have a common factor of 2).
- But we assumed that a and b had no common factors! Contradiction.
- Therefore, the square root of 2 can't be represented as a rational number a/b.
Here's an even simpler proof:Again assume that there is a rational number a/b where (a/b)2 = 2. The Fundamental Theorem of Arithmetic (and common sense) states that any integer can be uniquely represented by the product of a set of prime numbers.
- Assume that there is a rational number a/b where (a/b)2 = 2 (note that a/b need not be a rational number in its simplest form).
- Therefore 2b2 = a2.
- Factor a into its unique set of prime factors and b into its unique set of prime factors.
- Re-write step 2 by replacing a and b with their respective unique set of prime factors giving 2*(prime factors of b)2 = (prime factors of a)2.
- The set of prime factors of a2 will have exactly twice as many elements as the set of prime factors of a. Similarily, the set of prime factors of b2 will have exactly twice as many elements as the prime factors of b.
- Therefore, the set of prime factors of a2 will have an even number of elements as will the set of prime factors of b2.
- Therefore the prime factors of 2b2 has an odd number of elements. But 2b2 = a2 and a2 has an even number of prime factors. Contradiction.
- Therefore, there is no rational number a/b where (a/b)2 = 2.
Now this one is a truly twisted idea. The Malbolge programming language was created with the idea that programming should be hard (after all, if it was easy then everybody would want to do it). Malbolge is named after Dante's Ninth Circle of H*ll. In fact, the creator was so successful that nobody seems to have managed to create a valid Malbolge program (yet?). The creator of Malbolge has also created a second language called Dis which tries to walk the fine line between inhumanely difficult to use and actually impossible to use.
More information is available here ( this page along with the software is 'mirrored' here in case the original page goes away (the page and the software was released into the public domain by the original author with an encouragement that people create their own homegrown versions of Malbolge and Dis in order to encourage the sort of portability problems normally associated with major languages)).
A hyperlink will have to do in this case - click here for a good discussion of this point. The conclusion is that glass isn't a liquid. Rather, glass is what is known as an amorphous solid. He also debunks the myth that the glass in very old stained glass windows shows signs of having flowed downwards over the centuries.Daniel Boulet