Re: Making money from Java

In article <b2kof.1248$lv3.1229@clgrps12>,
Oliver Wong <owong@xxxxxxxxxxxxxx> wrote:
>
><docdwarf@xxxxxxxxx> wrote in message news:dnqo50$iq3$1@xxxxxxxxxxxxxxxxxxxx
>> In article <Jk_nf.256568$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.
>
> Euclid contributed an axiomatic system which we now call "Euclidean
>geometry". Not sure if Euclid himself mentions what the angles in a triangle
>add up to in his text, but it IS using his system that we have derived the
>above fact.

Once again, Mr Wong... as I was taught it Euclid gave (in the sense that
'Homer' gave the Illiad and the Odyssey) a set of works beginning with
Definitions, Postulates and Common Notions; using these he generated
Propositions and the whole of these constitute the Geometry. I am not
sure how this is to be reconciled to what you are calling 'an axiomatic
system' but to state 'since Euclid's system is used to derive facts about
sums of angles in a triangle the source of these facts is Euclidean
geometry'; this appears similar to a logical extension along the lines of
'Joe builds walls; since within these walls a plumbing system is
constructed the source of the plumbing system is Joe's walls'.

>
>>
>>>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.
>
> Yes, it sounds like what you are calling "Postulates", I would (in a
>more rigorous context) called "axioms". I believe axioms can be further
>subdivided into "assumptions" and "tautologies". Tautologies are those
>statements which are true by their own definition, and I figured none of the
>Euclid's five "postulates" falls under that category (though I don't
>actually remember all five of them), so I called them assumptions.

Mr Wong, what I am calling Postulates are also called Postulates in
various translations of The Elements; the Heath edition begins this
section with 'Let the following be postulated'. The Definitions, in
general, concern themselves with nouns ('a point', 'a line', 'an angle',
'trilateral figures', etc.) while the Postulates concern themselves with
actions ('to draw', 'to describe') and conditions (4. 'That all right
angles are equal one another', 5. 'That, if a straight line falling on two
straight lines makes... the two straight lines, if produced
indefinitely...') It is a habit of my training to try and make use of as
much original material as possible, including the terms used... perhaps I
should find a laundromat for nuns and try to deal with this unclean habit.

[snip]

>>
>> (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.)
>
> In Euclidean geometry, the parallel line postulate says (or is
>equivalent to):
>
> Given a line L and a point P which is not on L, there exist exactly one
>line which crosses P but which does not cross L. This line is said to be
>"parallel" to L.

Equivalences are one thing, Mr Wong, texts are another. The Heath
translation renders Postulate 5 as 'That, if a straight line falling on
two straight lines makes the interior angles on the same side less than
two right angles, the two straight lines, if produced indefinitely, meet
on that side on which are the angles less than the two right angles.'; it
may be beneficial to view this in light of a preceding Definition (23),
which he translates as 'Parallel straight lines are straight lines which,
being in the same plane and being produced indefinitely in both
directions, do not meet one another in either direction.'

>
> In spherical geometry, there does NOT exist any such line, and in
>hyperbolic (or Lobachevskian) geometry, there exists infinitely many such
>lines.

Euclid's geometry, being constructed first in a plane and later in a
series of intersecting planes, just might, possibly, not be applicable to
what happens on the surface of spheres, no... and Lobachevskian geometry
is another delightful matter, entire.

>
>>> 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 .
>
> Yes, I may have been "skipping a few steps". =)

Caution has been advised about doing such things, Mr Wong; Aristotle is
reported to have told Alexander that 'there is no royal road to geometry'.

DD

.

Relevant Pages

  • Re: Examining Mathematical Approaches
    ... straight lines makes the interior angles on the same side less than two ... > statements about the validity of non-Euclidean Geometry were erroneous. ... > could be contained in a Euclidean geometry of N+1 dimensions. ...
    (sci.physics.relativity)
  • Re: Making money from Java
    ... we have a whole system of geometry ... >>that the sum of all the angles in a triangle is 180 degrees. ... Euclid contributed an axiomatic system which we now call "Euclidean ... > Postulates and Common Notions. ...
    (comp.lang.cobol)
  • Re: is sci.math slow? correction to my post to the AP thread
    ... Consider a simple model of a hyperbolic geometry: ... Straight lines are the same as the Euclidean ... angles on the same side less thatn two right angles, ...
    (sci.math)
  • Re: Making money from Java
    ... >called "Euclidean geometry" that, for example, tells us ... >that the sum of all the angles in a triangle is 180 degrees. ... Postulates and Common Notions. ...
    (comp.lang.cobol)
  • Re: Why didnt ancient Greek Mathematicians use a string instead of a compass for their constructions
    ... It is for this reason geometry IS NOT mathematical. ... Of course straight lines (tightening the rope, ... A line as a set defines mathematics. ...
    (sci.math)