I am trying to do things "by the book" and set up something that resembles
batterboards. I have read to make the corners square to use the 3 4 5
method.
So I put a stake in the corner and measure out three feet and put a stake
one way and then measure four feet the other and put a stake and if I did it
right the hypotenuse, or the distance between the two stakes at the three
and four feet marks is five feet and this insures what? That I have
straight lines going both directions? And I also read another way was
simply to measure diagonals of the alleged square to see if they are equal.
Do you guys do any of this? Thanks.
In article <ywS4b.327094$o%2.151232@sccrnsc02>, jm
<[email protected]> wrote:
> I am trying to do things "by the book" and set up something that resembles
> batterboards. I have read to make the corners square to use the 3 4 5
> method.
>
> So I put a stake in the corner and measure out three feet and put a stake
> one way and then measure four feet the other and put a stake and if I did it
> right the hypotenuse, or the distance between the two stakes at the three
> and four feet marks is five feet and this insures what?
That the three foot line is perpendicular to the four foot line, i.e.
the corner is square.
In article <[email protected]>, Robert Bonomi
<[email protected]> wrote:
> In article <[email protected]>,
> LRod <[email protected]> wrote:
> >On Tue, 2 Sep 2003 19:59:34 +0000 (UTC), [email protected]
> >(Robert Bonomi) wrote:
> >
> >
> >[major snippage that should have been done before]
> >
> >>Disproof by "real-world" counter-example:
> >
> >Unfortunately, incorrect.
>
> Unfortunately, _you_ demonstrate a lack of understanding of spherical
> geometry.
Unfortunately, _you_ don't realize that the original problem has
nothing to do with spherical geometry.
In article <[email protected]>, [email protected] (LRod) wrote:
[snip]
>
>In the context of measuring diagonals, a square is a special form of a
>geometric figure (rectangle) that will have angles equaling 90° if the
>diagonals (distance from opposing corners) are equal. This does assume
>that opposing sides are equal (if they aren't, then the figure isn't a
>rectangle; it's a trapezoid; a subset of a geometric figure called a
>quadrilateral).
>
Close but no cigar.
If the diagonals of a four-sided figure are equal, the figure is a rectangle,
but it is not necessarily a square. If the diagonals are equal, and *adjacent*
sides are equal, the figure is both a rectangle and a square. In either case,
the angles at the corners are right angles (the literal meaning of the word
"rectangle").
If *all* opposing sides are unequal, it's a quadrilateral (literally, "four
sides") and *not* a trapezoid. A trapezoid is a four-sided figure consisting
of two parallel but unequal sides, and two equal but non-parallel sides, i.e.
___
/___\
Finally, opposing sides being equal defines a parallelogram, not a rectangle.
A parallelogram is a rectangle if and only if its angles are right angles. And
before anybody jumps on me to say that a parallelogram is defined by
*parallel*, "not" equal, opposing sides, let me point out that it's trvial to
show that the two are equivalent: each implies the other.
--
Regards,
Doug Miller (alphageek-at-milmac-dot-com)
On Tue, 02 Sep 2003 01:38:38 GMT, "jm" <[email protected]>
wrote:
>I am trying to do things "by the book" and set up something that resembles
>batterboards. I have read to make the corners square to use the 3 4 5
>method.
>
>So I put a stake in the corner and measure out three feet and put a stake
>one way and then measure four feet the other and put a stake and if I did it
>right the hypotenuse, or the distance between the two stakes at the three
>and four feet marks is five feet and this insures what?
That the angle between the 3 foot line and the 4 foot line is exactly
90°, or "square."
>That I have straight lines going both directions?
The tension on the string does that.
>And I also read another way was simply to measure diagonals of the alleged
>square to see if they are equal.
You seem to be confusing the word "square." In the context of your
original question, "square" means a right angle; an angle measuring
90°. If the meeting of two boards, two pieces of steel, two lines of
string, etc., is 90°, then that sub-structure is square.
In the context of measuring diagonals, a square is a special form of a
geometric figure (rectangle) that will have angles equaling 90° if the
diagonals (distance from opposing corners) are equal. This does assume
that opposing sides are equal (if they aren't, then the figure isn't a
rectangle; it's a trapezoid; a subset of a geometric figure called a
quadrilateral).
Finally, although you didn't further muddy the waters by mentioning
it, a square is a device that is precisely manufactered to gauge
angles at 90°.
I hope this helps.
LRod
Master Woodbutcher and seasoned termite
Shamelessly whoring my website since 1999
http://www.woodbutcher.net
In article <[email protected]>, [email protected] (Robert Bonomi) wrote:
>In article <[email protected]>,
>Doug Miller <[email protected]> wrote:
>>In article <[email protected]>,
>>[email protected] (LRod) wrote:
>>[snip]
>>>
>>>In the context of measuring diagonals, a square is a special form of a
>>>geometric figure (rectangle) that will have angles equaling 90° if the
>>>diagonals (distance from opposing corners) are equal. This does assume
>>>that opposing sides are equal (if they aren't, then the figure isn't a
>>>rectangle; it's a trapezoid; a subset of a geometric figure called a
>>>quadrilateral).
>>>
>>Close but no cigar.
>
>You don't even get a Lewinsky.
>
>>
>>If the diagonals of a four-sided figure are equal, the figure is a rectangle,
>
>FALSE TO FACT.
>
>Disproof by counter-example:
> 1) draw an equilateral triangle.
> 2) 'cut off' the top, by scribing a line parallel to the base.
> 3) you now have a 'symmetric' trapezoid, which has equal-length diagonals.
> 4) none of the angles are right angles.
> therefore, it is _not_ a rectangle.
> Q.E.D.
>
:-( You're quite right, of course -- that's what I get for posting before
coffee. I *meant* to say, if diagonals are equal *and* opposite sides are
parallel...
>
>If the diagonals are equal, it means _only_ that the object is 'bilaterally
>symmetric". *IF* the object is a 'parallelogram', then the symmetry forces
>the corner angles to 90 degrees, and guarantees a rectangle. If the object
>is -not- a parallelogram, all bets are off. :)
>
>>but it is not necessarily a square. If the diagonals are equal, and *adjacent*
>>sides are equal, the figure is both a rectangle and a square. In either case,
>>the angles at the corners are right angles (the literal meaning of the word
>>"rectangle").
>>
>>If *all* opposing sides are unequal, it's a quadrilateral (literally, "four
>>sides") and *not* a trapezoid. A trapezoid is a four-sided figure consisting
>>of two parallel but unequal sides, and two equal but non-parallel sides, i.e.
>> ___
>>/___\
>
>FALSE TO FACT. The non-parallel sides do *not* have to be equal:
>
> +--+
> / |
> +----+
>
>is also a trapezoid.
I stand corrected.
>
>*ANY* parallelogram is also a trapezoid. (parallelograms are a 'strict subset'
>of the class of objects that are trapezoids.)
>
>The *ONLY* requirement for a trapezoid is that _one_ pair of edges be parallel.
>
>>Finally, opposing sides being equal defines a parallelogram, not a rectangle.
>
>FALSE TO FACT. Opposite sides being _parallel_ defines a parallelogram.
"FALSE TO FACT" yourself -- the proof that these are equivalent is absolutely
trivial. Given a quadrilateral with opposing sides equal, construct a
diagonal. The two triangles formed are congruent; hence, the interior angles
at the diagonal are congruent, and the sides are parallel. Alternatively,
given a quadrilateral with opposing sides parallel, again construct a
diagonal; again, the triangles formed are congruent, and therefore their
corresponding sides are equal.
>
>>A parallelogram is a rectangle if and only if its angles are right angles. And
>>before anybody jumps on me to say that a parallelogram is defined by
>>*parallel*, "not" equal, opposing sides, let me point out that it's trvial to
>>show that the two are equivalent: each implies the other.
>
>*ONLY* in the limited scope of "Euclidean" geometry. <grin>
>
>The distinction _is_ important, when one gets to advanced (non-Euclidean)
>geometry, because the two concepts are *not* interchangable, and the -defining-
>characteristic _is_ that the sides are 'parallel'. (consider that in
>'spherical geometry', for example, one can have 'parallel lines that
> intersect')
And the distinction is of absolutely *no* importance to any *practical*
considerations whatsoever, unless you happen to inhabit a non-Euclidean
universe.
--
Regards,
Doug Miller (alphageek-at-milmac-dot-com)
"Doug Miller" <[email protected]> wrote in message
news:[email protected]...
> >*ONLY* in the limited scope of "Euclidean" geometry. <grin>
> >
> >The distinction _is_ important, when one gets to advanced (non-Euclidean)
> >geometry, because the two concepts are *not* interchangable, and
the -defining-
> >characteristic _is_ that the sides are 'parallel'. (consider that in
> >'spherical geometry', for example, one can have 'parallel lines that
> > intersect')
>
> And the distinction is of absolutely *no* importance to any *practical*
> considerations whatsoever, unless you happen to inhabit a non-Euclidean
> universe.
>
>
> --
> Regards,
> Doug Miller (alphageek-at-milmac-dot-com)
Ah, but we do, whenever we use a level to determine if something is "Flat".
Granted, it would have to be something rather big that you are making to
make much of a difference...
-Jack
In article <[email protected]>,
Doug Miller <[email protected]> wrote:
>In article <[email protected]>,
>[email protected] (Robert Bonomi) wrote:
>>In article <[email protected]>,
>>Doug Miller <[email protected]> wrote:
>>>In article <[email protected]>,
>>>[email protected] (LRod) wrote:
>>>[snip]
>>>>
>>>>In the context of measuring diagonals, a square is a special form of a
>>>>geometric figure (rectangle) that will have angles equaling 90° if the
>>>>diagonals (distance from opposing corners) are equal. This does assume
>>>>that opposing sides are equal (if they aren't, then the figure isn't a
>>>>rectangle; it's a trapezoid; a subset of a geometric figure called a
>>>>quadrilateral).
>>>>
>>>Close but no cigar.
>>
>>You don't even get a Lewinsky.
>>
>>>
>>>If the diagonals of a four-sided figure are equal, the figure is a rectangle,
>>
>>FALSE TO FACT.
>>
>>Disproof by counter-example:
>> 1) draw an equilateral triangle.
>> 2) 'cut off' the top, by scribing a line parallel to the base.
>> 3) you now have a 'symmetric' trapezoid, which has equal-length diagonals.
>> 4) none of the angles are right angles.
>> therefore, it is _not_ a rectangle.
>> Q.E.D.
>>
>:-( You're quite right, of course -- that's what I get for posting before
>coffee. I *meant* to say, if diagonals are equal *and* opposite sides are
>parallel...
>
>>
>>If the diagonals are equal, it means _only_ that the object is 'bilaterally
>>symmetric". *IF* the object is a 'parallelogram', then the symmetry forces
>>the corner angles to 90 degrees, and guarantees a rectangle. If the object
>>is -not- a parallelogram, all bets are off. :)
>>
>>>but it is not necessarily a square. If the diagonals are equal, and *adjacent*
>>>sides are equal, the figure is both a rectangle and a square. In either case,
>>>the angles at the corners are right angles (the literal meaning of the word
>>>"rectangle").
>>>
>>>If *all* opposing sides are unequal, it's a quadrilateral (literally, "four
>>>sides") and *not* a trapezoid. A trapezoid is a four-sided figure consisting
>>>of two parallel but unequal sides, and two equal but non-parallel sides, i.e.
>>> ___
>>>/___\
>>
>>FALSE TO FACT. The non-parallel sides do *not* have to be equal:
>>
>> +--+
>> / |
>> +----+
>>
>>is also a trapezoid.
>
>I stand corrected.
>
>>
>>*ANY* parallelogram is also a trapezoid. (parallelograms are a 'strict subset'
>>of the class of objects that are trapezoids.)
>>
>>The *ONLY* requirement for a trapezoid is that _one_ pair of edges be parallel.
>>
>>>Finally, opposing sides being equal defines a parallelogram, not a rectangle.
>>
>>FALSE TO FACT. Opposite sides being _parallel_ defines a parallelogram.
>
>"FALSE TO FACT" yourself -- the proof that these are equivalent is absolutely
>trivial. Given a quadrilateral with opposing sides equal, construct a
>diagonal. The two triangles formed are congruent; hence, the interior angles
>at the diagonal are congruent, and the sides are parallel. Alternatively,
>given a quadrilateral with opposing sides parallel, again construct a
>diagonal; again, the triangles formed are congruent, and therefore their
>corresponding sides are equal.
Disproof by "real-world" counter-example:
Take a globe.
lines of lattitude are perpendicular to the lines of longitude,
therefore, by definition, they _are_ parallel to each other.
lines of longitude are perpendicular to the lines of latitude.
therefore, by definition, they _are_ parallel to each other.
Consider a quadrilateral with points at "0 W, 0 N", "60 W, 0 N", "60 W ,60 N",
and "0 W, 60 N" all four corners are 90 degrees, opposite sides _are_ parallel,
yet the 'north' side of the square is only 1/2 the distance of the equatorial
side.
And, when you construct your 'diagonal', the triangles formed are *not*
congruent. Despite the fact that all three angles, and two adjacent sides
_are_ the same measure.
In _plane_ geometry, opposite sides of a parallelogram being of like length
*is* an inescapable 'side effect' of the definition.
>>>A parallelogram is a rectangle if and only if its angles are right angles. And
>>>before anybody jumps on me to say that a parallelogram is defined by
>>>*parallel*, "not" equal, opposing sides, let me point out that it's trvial to
>>>show that the two are equivalent: each implies the other.
>>
>>*ONLY* in the limited scope of "Euclidean" geometry. <grin>
>>
>>The distinction _is_ important, when one gets to advanced (non-Euclidean)
>>geometry, because the two concepts are *not* interchangable, and the -defining-
>>characteristic _is_ that the sides are 'parallel'. (consider that in
>>'spherical geometry', for example, one can have 'parallel lines that
>> intersect')
>
>And the distinction is of absolutely *no* importance to any *practical*
>considerations whatsoever, unless you happen to inhabit a non-Euclidean
>universe.
You sir, "don't know what you don't know".
Anyone who engages in long-distance navigation on, or above, the surface
of the Earth has to deal with such matters on a day-to-day basis. Because
it _is_ non-Euclidean. "Spherical geometry", as a matter of fact. Where
many things 'everybody knows' are simply incorrect. e.g., the sum of the
angles in a triangle is always _more_ than 180 degrees. And the shortest
distance between two points is *not* a straight line (rather, it is a
'great circle' route).
In article <[email protected]>,
LRod <[email protected]> wrote:
>On Tue, 2 Sep 2003 19:59:34 +0000 (UTC), [email protected]
>(Robert Bonomi) wrote:
>
>
>[major snippage that should have been done before]
>
>>Disproof by "real-world" counter-example:
>
>Unfortunately, incorrect.
Unfortunately, _you_ demonstrate a lack of understanding of spherical
geometry.
>
>> Take a globe.
>> lines of lattitude are perpendicular to the lines of longitude,
>> therefore, by definition, they _are_ parallel to each other.
>
>Only at the equator.
>
>> lines of longitude are perpendicular to the lines of latitude.
>> therefore, by definition, they _are_ parallel to each other.
>
>Only at the equator.
>
>In fact, lines of latitude ARE parallel to each other, and the
>distance between 80N and 90N is precisely the same as the distance
>between 0N and 10S at any meridian (line of longitude) on the globe.
>The length of a line of latitude varies with its location.
If the of longitude are perpendicular to the equator, then they are
BY DEFINITION perpendicular, as well, to any lines that are parallel
to the equator. As you admit the other lines of latitude -are- parallel
to the one representing the equator, the lines of longitude are _also_
perpendicular to those other parallels.
>Lines of longitude, however, all pass through both poles (the
>definition of a meridian), and thus CANNOT be parallel to each other,
>by definition (parallel lines never meet).
That 'parallel lines never meet' is *NOT* part of the definition. And is
true *only* in "Euclidean plane geometry".
> The extreme examples are
>the distances between any two meridians (let's say 10W and 20W) at the
>equator (approximately 600 nautical miles, in this example) and at a
>pole (0 miles, any example).
>
>>Consider a quadrilateral with points at "0 W, 0 N", "60 W, 0 N", "60 W ,60 N",
>>and "0 W, 60 N" all four corners are 90 degrees, opposite sides _are_
>parallel,
>>yet the 'north' side of the square is only 1/2 the distance of the equatorial
>>side.
>
>Your example, proves my point. The lines from 0,0 to 0,60 and 60,60 to
>60,0 cannot, as meridians, be parallel.
They are both perpendicular to a common line, thus complying with the actual
mathematical DEFINITION.
In 'spherical geometry', the "ground rules" _are_ different. A lot of what
'everybody knows', from _plane_ geometry, simply "doesn't apply".