On Mon, 02 Aug 2004 20:35:17 GMT, George E. Cawthon
It's the kind of thing that comes up in formal proofs.
"Prove that points A, B, and C are colinear"
The usual approach would be to construct a few line segments, then talk
about them using formal language. Here's the first and last lines of
such a proof:
Suppose, on the contrary, that points A, B, and C are NOT colinear, then
angle ABC with vertex B . . .
(lotsa other stuff)
. . . meaning that angle ABC is a straight angle. Therefore points A,
B, C are colinear.
But I agree with you that this is something that won't come up much in
"Hey, Fred! Pass me that straight angle. I need to cut this Sheetrock"
On Mon, 02 Aug 2004 20:35:17 GMT, "George E. Cawthon"
Now I know I've been told in my lifetime I'm as "thick as two short
planks", but bear with me and I'll try to convince you.
Which is ..."the measure of rotation of one line [segment] to another
...which can also include 180. When the segments line up, they can be
called a single line.
Thinking backwards... every line is a set of points, and all you need
to do is to choose one to form line segments, which are then at an
angle [of 180]. You can then freely rotate one with respect to the
other forming other angles. So decrease it [from 180] to get other
angles less than 180, or increase it. But 180 fits right in there!
The finite number system, and the geometry is continuous. There is no
Fence to table = 90. Fence to try-square = 90. when they meet [90 +
90], the table is the base line, the fence and try-square forming the
vertical. The 90 from the fence and the 90 from the try-square add to
the 180 for the table. I can say it only so many times. Take two
square sheets of pylwood. Assume dead-square. Place them next to
each other and there is no gap. The corner of one and the corner of
the other [90 + 90] combine to form the line at the bottom ...180.
On Sun, 01 Aug 2004 06:01:53 GMT, "George E. Cawthon"
A good point, but wrong, I'm afraid. The bevel is also a tool. You
are thinking of a "bevelled edge" being also called a "bevel", and a
"bevelled cut". The word "bias" also comes to mind.
I taught math for over thirty years [and it comes in *really* handy in
woodworking]. A straight line is not only an angle, but it can be
used as the basis for measuring angles:
Pi radians = 180 degrees.
That is used [constantly] to change back and forth between radian and
degree measure of angles.
My dictionary also says bevel is an adjustable tool for drawing
angles. But if you go to a tool store and as for a "bevel" the tool
many will as "bevel what?" The dictionary also has bevel square which
is the more common (maybe the correct) term for an adjustable tool
used for drawing angles and adjusting work.
Practical or abstract, geometry or trig? Which of those are you
A line that include the points A and C and also includes B is still
just a line even if a-b and b-c are segments. My only point through
this whole thing, is that fact. When you rotate a line around a
coordinate system, when you get to 180 degrees you have just one
line. And when you continue to rotate to 360 (or 0) you just have one
line because two lines cannot exist in the same space. Do you want to
debate that? Neither do I.
On Mon, 02 Aug 2004 00:14:07 GMT, "George E. Cawthon"
Now I don't know if you are referring to my post, or the prior one
when you ask for a debate. I've never argued against the facts you
state below, but against the insistence that a line is not a
measureable angle. The arguments by others about zero are absurd.
Without the number zero there is no solid finite arithmetic. Some
people do stop when they get to ten though.
About "practical and abstract": I see them as the same. You can draw
an ellipse, for example, from two concentric circles. But I'd hate to
try to describe the math here that lies behind it. A table of
compound angles is useful. The math to get there will do any other
angles in between as well. The math behind perspective drawing is
awesome. The relatively simple "layout" technique that evolvesfrom it
is even more awesome. It was once a military secret in old France
when first invented. Incidentally, trig is based on similarity in
geometry. They're all interconnected.
Hey, I'm here to help if I can. I get a lot out of this group, once
the OTs are ignored, and would love to put as much into it. With math
I can help.
Whoa, abstract and practical are the same? I hated geometry in high
school because it was never clear how to get from one concept to the
next. One of the reasons was that the basic definitions were not
stated or emphasized so the student tended to start off in water where
he couldn't put his feet down. Another problem was the confusion over
abstract and practical definitions of a line. The abstract
mathematical concept of a line is that it can't be seen, has no
beginning and no end, is continuous, and has no width.
As for mathematical definitions of angles, the definition is the
inclination of one line to another (thus two lines are required for an
angle). When they intersect at 180 degrees it is called a "straight
angle." Now that's a real oxymoron! If the angle is 360 degrees it
is a "perigon" while 0 degrees is called a "zero angle." These terms
may be useful in some mathematical disciplines but 360 degrees and 0
degrees are the same and "zero angle" is another oxymoron. I suppose
it is easier to say zero angle rather than day there is no angle. If
you apply the abstract mathematical concept of a line, i.e., without
beginning and end, to these angle definitions, then there really is a
single line when two intersecting lines form a "straight angle" or
"zero angle." (note we are assuming all lines lie in the same
plane.) All of which has essentially no practical value to woodwork
but I will continue to assume that my straight edge is a single line
and does not include a straight angle.
Bill Rogers wrote:
On Mon, 02 Aug 2004 08:14:40 GMT, "George E. Cawthon"
Sorry about that, but it *was* clear to others. I could lead you
precisely from one to the next [**]. That's the whole point of it,
the logical development. Part of the problem is that you could be
"given" the properties and try to commit them all to memory
[meaningless if there's no connection], or you need to develop them
and see where they come from [theory], and so understand them and
their interconnection. You might then find even better uses for them.
[**] Part of that development depends exactly on defining the line as
180 degrees. That is; a definition of a degree is 1/180 of a
It's called "completeness". The number system was invented [not
visible anywhere in the 'real world' people refer to.] The number
Zero was invented. It was necessary. It does not denote "nothing",
but is a symbol in counting just as good as is the symbol for other
quantities. What would you do instead without it? As simple as it
is, it's one of the great inventions of the mind. ["I saw a man upon a
A "straight angle" is not an oxymoron. It is still an angle ...a
"straight" one. It's still there; we can just now give it another
name as well suited.
By itself, a hammer is useless. It needs your skill to use it. By
itself math is useless, just a game of the mind. It also needs your
skill to use it. One of my favourites: IKEA designed a table, useful
in B&Bs for example. Take the top and rotate it 90 degrees, and flip
it on the dividing hinge, and you have a top twice the size. Where do
you put the pivot point so that it will sit square, centered on the
base. Second question: What shape [ratio of sides] must it be to
keep the same rectangular shape when opened up?
Hating math is like hating a hammer. Don't hate it, use it. It took
a whole lot of people over 3000 years to develop.
P.S. You can use your try-square to make sure everything in the table
lines up. :-)
Whoa, you are infering a lot of things I never said. I didn't say I
hated math, I said I hated geometry (hate is a little strong) and
really implying that I hated the way it was taught. The terminology
in most fields becomes a little ridiculous, and simple normal words
will do just as well. Those with lots of education in the field often
use the more complex terminology just to impress the less learned.
But some of the terminology is down right bad as it implies things
that were later found to be untrue. In the practical world, zero does
mean nothing. I've got zero dollars means I have no dollars. Nothing
is and always has been a clear concept in the real world. And having
grown up while zero was a concept in math, I have a hard time
understanding how it took so long to develop.
As to a hammer being useless without skill, tell that to a stone age
man. How much skill did it take to bash another man's head with a
Math isn't a game. Math is a language that tries to describe the real
world. As man's knowledge of the real world increase, math has had to
change and develop more complex words. Geometry has it uses but it is
pretty primitive and is useless in describing the more complex
phenomenon that we consider common knowledge nowadays.
What simple words for example? What do you call "normal"?
I do understand, you know. I'm from a blue-collar background, and
love working with my hands. But you shouldn't knock what others might
find useful even if you don't. Some fairly complex geometry and
other math is needed to produce the toys we take for granted, and what
we call a normal part of our everyday life. It really does require
some fairly hefty definition at times. A "chair" or a "seat", it's
just something to sit in and relax and smell the roses.
Botanists use the term leaf "margin" whereas you could more simply say
leaf "edge." A bit more technical is to talk about a meristem, which
means nothing to most people, but you could just say the growing
point. I'm not knocking useful terms, hell, I'm a botanist and use
technical terms as appropriate and technical terms are expected among
peers, but use of technical terms to a non-technical audience when a
common word will do is silly and destructive to communication. And we
are all a non-technical audience in some field. An example that I
especially like is the use of siphon by hydraulic engineers in the
description of water ways. The fact is that the structure is just a
pipe with a curve that starts high and end at a lower elevation (like
a J). True siphons in water ways are extremely rare, both because
filling the siphon to start the water flowing is difficult or
expensive and most conveyances would collapse at the high points due
to the negative pressures. Stand around and listen to a technical
discussion in a field that you know little about and soon you will be
saying, "Why didn't he just say.......?"
On Tue, 03 Aug 2004 06:21:56 GMT, "George E. Cawthon"
Yeah. I used to have the same problem with botany. :-)
Still and all, it's been an interesting discussion. Most of all it's
interesting to see those who know a little trying to say a lot [no
personal reference there!] That's why I keep my mouth shut when
around botanists talking about botany. I thought Krebs Cycle was
something he used to ride to work.
Sounds like you had a really incompetent geometry teacher.
The thing to understand about math is that it's a game, it has rules, as
long as you obey the rules anything goes.
With geometry, you have rules that are called "postulates" and "definitions"
and given those the game is to create "theorems" which are "proven" based
on the logical development from "postulates" and "definitions".
Going in a different direction, with compass and straight-edge you do
"constructions" of various shapes--these, before the availability of
accurately marked scales and protractors were of great importance in
drafting--since constructions are necessarily done using real
approximations of the ideal it is best to always remember this and provide
a check of one sort or another on each construction, as for example by
repeating it several times and taking the centerpoint of the resulting
group of marks.
Your geometry teacher should have made that clear to you at the outset. If
he did then you'd probably have had a lot less trouble with it.
Sounds like you got all your math from this same teacher. "Perigon" appears
to be one of those wonderful terms that was created during the Victorian
era, probably in an effort to standardize terminology (if the distance
around a circle is the "perimeter" then the angle subtended by the
"perimeter" must be the "perigon" . . .), and its utility is that if 360
degrees is a "perigon" then 180 degrees is a "hemiperigon" and 90 degrees
is a "semihemiperigon" and so on. Might still be used in the UK but it's
certainly not used in mainstream math, science, or engineering in the US.
That is a perfectly reasonable assumption since a straight edge is not used
for measuring angles unless it is attached to a protractor.
Reply to jclarke at ae tee tee global dot net
Most in highschool are incompetent, but it could have been the student
I found geometry all rather a waste of time and a useless exercise,
development since Euclid made most of the theorems rather common
knowledge. Now trig was of interest because you could actually use it
Exactly. The advancement in tools makes most of geometry rather
retrograde. I've learned how do to square roots by division several
times and then promptly forgot how, and learned to do square roots by
subtration of 9s or something like that on old electric calculators
before computers, but I certainly don't wouldn't do that anymore.
It's as quaint as building a canoe by hollowing a log.
Lots of math teachers don't make things clear. The silly word
problems were a constant problem for most students, in simple math,
algebra, chemistry, and physics. They new that you did something with
A, B, an C, but weren't sure whether it was A/B xC or A/C x B. Like
most students I hated them. It wasn't until I started teaching
chemistry and making up my own word problems that they became rather
fun, but I was always clear with my students that word problems were
puzzles that required a thought and not mechanically adding, dividing,
etc. I say silly word problems because most contained the exact
number of data to find a solution. When you introduce needed and
useless data into a problem, and tell the student that a graph or some
sort of depiction may help, the student can start understanding.
Not hardly from the same teacher. And my point was precisely that
many terms are archaeic and inappropriate for the present day.
On Mon, 02 Aug 2004 21:34:53 GMT, "George E. Cawthon"
The formal part provided a basis for understanding the informal
results. It also taught to not rely on assumption, but granted that
was an exercise in logic rather than the properties of shapes.
I use geometric constructions when I want dead-on accuracy, or want to
form a shape not doable otherwise nearly as efficiently. If I want
dead-on 90 degrees, 60 degrees or other variations I use compasses and
straightedge [the 180 kind ... :-) ]. Lots of other similar uses
also. Perhaps I should write the book using the KISS principle?
I can't tell you the number of times friends who are pro carpenters
tell me they really shoulda listened! Now they are using geometry
[including Pythagora theorem!] every day. if you want a REALLY good
book, get "the Carpenter's Square" and learn how to do things
efficiently, including building a spiral staircase.
Just on that point, it's tough enough to teach youngsters what is to
them complicated algebra. Those "word problems" are the simplest
applications dreamt up. If you can make up a textbook full of simpler
applications not using "word problems" I'd have been the first to buy
it. The only other alternative is some "real world" problems, usually
well beyond the young and inexperienced. We see it better only in
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