Re: Making money from Java

In article <Jk_nf.256568$ir4.37132@edtnps90>,
Oliver Wong <owong@xxxxxxxxxxxxxx> wrote:

[snip]

>For example, we have a whole system of geometry
>called "Euclidean geometry" that, for example, tells us (among other things)
>that the sum of all the angles in a triangle is 180 degrees.

With all due respect, Mr Wong, the Euclid I studied (Heath edition) made
no mention of degrees... it may be that somewhere in the Geometry it is
concluded that the sum of the angles of a triangle is equal to the sum of
two right angles but that is, I believe, a proposition which is
demonstrated.

>Euclidean
>geometry is based on 5 assumptions (which I won't list here, you can look it
>up if you don't know and are interested), and the "angle in triangle adds up
>to 180" statement is only true IF you accept those 5 assumptions.

Ummmm... the Euclid I studied (Heath edition) began with Definitions,
Postulates and Common Notions. Definitions were things like 'a point is
that which has no part' and 'a line is breadthless length'... there were a
whole bunch of these, more than five.

Five was the number of Postulates, which included things like 'to draw a
straight point between two lines' and 'to produce (continue) a straight
line continuously in a straight line'... and the Fifth Postulate was the
Parallel Postulate.

[snip]

>Euclide had those 5
>assumptions for his system which he hoped were sufficiently obvious, and for
>a long time they were. But recently people have been looking at his
>assumption about parallel lines and wondered if it was nescessarily true.
>They removed that assumption and derived a new set of rules to form set of a
>system collectively referred to as "Non-Euclidean geometry". Not sure, but I
>believe the most popular non-euclidean geometry is Lobachevskian geometry,
>and it is now believed that on the macroscopic scale (e.g. the size of the
>universe), our space-time conforms more closely with Lobachevskian geometry
>than Euclidean geometry. In other words, it turns out that the angles of a
>triangle don't really add up to 180 degrees in "real life".

Hmmmmm... in another posting I referred to Popper's theories from the
1930s as 'barely a century old'... Lobachevski put together/plagiarised
his work in the early-mid 19th century... nice to see that called
'recent'.

(And... my memory is, admittedly, porous but I recall that if one attempts
to conctruct a geometry on the surface of a hypersphere then the Parallel
Postulate holds... but that's for another time, perhaps.)

[snip]

> Instead, Euclides start with assumptions like "Assume any two points can
>be joined by a straight line". These assumptions were so "obvious" to the
>audience that they accepted them without question. And FROM these
>assumptions, Euclide derived the rest of geometry.

Not as I recall it... first Definitions, then Postulates, then Common
Notions, *then* Propositions. But... enough about my memory. Consider
http://aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html .

DD

.

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