Concentric circles are parallel?

[phya]

You can see concentric circles as being parallel from the perspective of the centre of each circle (the distance from the first to the second circle is constant, as measured from the centre) but from another viewpoint (say, and observer looking towards both circles) then they are suddenly not following this rule.

From the centre of the circles viewpoint, this says in coordinate geometry that x^2+y^2=r^2 is "parallel" to x^2+y^2=R^2

but from the observer which is found situated far along the y-axis (as to observe both circles head on) then he will see two circles as "parallel" if they are of the form
x^2+y^2=r^2 and x^2+(y-c)^2=r^2

Now one property of parallel lines is that if line A is parallel to line B and line A is parallel to line C then B is parallel to C. Following this rule, this would mean that x^2+y^2=R^2 is "parallel" to x^2+(y-c)^2=r^2 ? This is why we give concentric circles a description other than "parallel". And we should keep it that way to avoid ambiguity.

In group of concentric circles, if A is parallel to B, B is parallel to C, then A is also parallel to C.

 

[Mentallic]

Your understanding is not necessarily correct

I beg to differ, notice that I wrote

but from the observer which is found situated far along the y-axis (as to observe both circles head on) then he will see two circles as "parallel" if they are of the form
x^2+y^2=r^2 and x^2+(y-c)^2=r^2

Your diagram doesn't express what I mentioned. So I took the liberty of explaining it to you in pretty pictures.

http://img691.imageshack.us/img691/9694/parallel.th.png
This diagram is supposed to be 2 dimensional.
In the first row of examples, the red and orange lines are of equal length. The circles are of the same size.

So when you say curves are parallel, we could be confused into thinking it's two circles above. How can we know you mean concentric circles? If you do mean concentric circles, then just say so. They already have a name for those so there's no need to try and label them as being parallel.
There are inconsistencies in trying to extend the meaning of parallel since it needs to hold for all cases. Clearly, from my diagram, if you consider it parallel to be the observer from a fixed point (as in the case of concentric circles) then why aren't two lines parallel if you observe them in the same way?

And to touch up on another point. When two parallel lines are transposed (moved) they are still parallel to each other. In your diagram, this rule breaks down. You move one of those concentric circles and suddenly they aren't "parallel" to each other anymore.

 

[G037H3]

I'm not sure that last point is exactly fair, because since a line is the same in homogeneous space, and distances between circles depend on their center and radius, it is much easier to claim that circles can't be parallel.

I stand by my previous post, concentric circles are parallel curves, but not parallel lines, because in mathematics lines are generally perfectly straight.

 

[Mentallic]

The point is, using the term "parallel curve" creates an ambiguity as I've already shown, unless it's specifically defined to be one or the other - which it isn't.

What is wrong with sticking to the term "concentric circles" anyway? I could just as easily argue that parallel lines are also concentric lines. It's just silly.

 

[G037H3]

The point is, using the term "parallel curve" creates an ambiguity as I've already shown, unless it's specifically defined to be one or the other - which it isn't.

What is wrong with sticking to the term "concentric circles" anyway? I could just as easily argue that parallel lines are also concentric lines. It's just silly.

I don't disagree, but I feel that the main point of the thread is to assert that concentric circles are parallel, which they are, because they're parallel curves. As long as the OP understands the difference between parallel lines (all the points that the line consists of can be rotated about themselves 360 degrees and the line remains parallel) and parallel curves, I don't really have a problem with the assertion.

 

[Mentallic]

Ok but if concentric circles are parallel, then surely circles of the same size that have been transposed aren't parallel. But with this same property of transposition for curves, phya claims they are parallel too.

 

[G037H3]

I'm sorry, but could you please elaborate on what you mean by transposition? :)

My knowledge in this kind of topic comes from studying Heath's version of Euclid's Elements, btw. I'm only about 10% through it, but it is a very beautiful book.

 

[Mentallic]

Tranpose or to move something. To slide in other words.

 

[G037H3]

Oh, I thought you meant to switch position, which makes no sense, because concentric circles have the same center, so they can't change places.

Circles of the same radius that have been moved aren't parallel in the usual sense, I agree. But if you take the curve of the half-circle on the same side of both circles, obviously they're the same, so they would be parallel curves, no? As in, there is a 1 to 1 match-up between points on the first curve and points on the last curve in that the value of the shortest distance between points that have the same value on the curve relative to the rest of the curve remains the same. Or, simply put, the curves fit perfectly with each other if the distance between the points are reduced to zero.

 

[Mentallic]

Yes, they're concentric circles. One property of concentric circles is that given some line that cuts through the centre of the circles, at the point where it intersects the circumference of each circle, the tangents made at those points will be parallel.

My parallel lines are concentric lines too in that case.

 

[G037H3]

"Concentric lines"...circles are concentric because of their uniform curve...two parallel lines cannot have any 'center', they can have a mid-point between them, wherein the group of all points that are halfway between the two lines create a new line that is parallel to both of the original parallel lines. Obviously.

I'm pretty sure that the strict mathematical definition of "line" always means a perfectly straight line, so I don't know what you could mean by 'concentric lines', as if this is true, only curves can be concentric from a particular point, in if the distance from the center is changed for one uniform curve to the distance (radii) of another curve, that both are exactly the same, in 2D space, with relation to the origin/center.

 

[Mentallic]

So then you agree that "concentric lines" is absurd, since there are many flaws and counter-examples to it. The same deal with parallel circles applies here.

 

[phya]

I beg to differ, notice that I wrote


Your diagram doesn't express what I mentioned. So I took the liberty of explaining it to you in pretty pictures.

http://img691.imageshack.us/img691/9694/parallel.th.png
This diagram is supposed to be 2 dimensional.
In the first row of examples, the red and orange lines are of equal length. The circles are of the same size.

So when you say curves are parallel, we could be confused into thinking it's two circles above. How can we know you mean concentric circles? If you do mean concentric circles, then just say so. They already have a name for those so there's no need to try and label them as being parallel.
There are inconsistencies in trying to extend the meaning of parallel since it needs to hold for all cases. Clearly, from my diagram, if you consider it parallel to be the observer from a fixed point (as in the case of concentric circles) then why aren't two lines parallel if you observe them in the same way?

And to touch up on another point. When two parallel lines are transposed (moved) they are still parallel to each other. In your diagram, this rule breaks down. You move one of those concentric circles and suddenly they aren't "parallel" to each other anymore.

You indeed have not understood my meaning. Must pay attention, we enter are not the familiar domain, therefore we are very easy to make a mistake. Must pay attention, we in exploration curve parallel, but is not straight line being parallel. In straight line parallel, after the straight line was parallel the migration, is still parallel, but in the curve - - circle was not that simple. We are conceivable, the straight line is the radius infinitely great circle, then what meaning the parallel motion straight line is? The parallel motion straight line is increasing or is reduced the circle (is certainly diameter infinitely great circle) the diameter. The concentric circle parallel migration is also so, is also changes the circle the diameter, but is not under the invariable diameter migration.

 

[Mentallic]

I can honestly say I have no idea what you just said.

 

[G037H3]

So then you agree that "concentric lines" is absurd, since there are many flaws and counter-examples to it. The same deal with parallel circles applies here.

Yes, concentric lines do not make sense, at least in Euclidean space, I agree.

I say that the argument with parallel circles is different than the obviously standard definition of parallelism (Playfair's axiom), because with parallel lines, the argument is meant for plane geometry, but can obviously be made to hold in 3D geometry. The same can be said for concentric circles -> concentric spheres, except that you must specify a 1 to 1 matching between points on each sphere, with the minimum distance value relationship that I mentioned above. With parallel lines, it is more of a conceptual consistency, with parallel curves it is always in relation to something else, if you ignore the superposition proof/argument.

 

[Mentallic]

Well there's no problem defining concentric circles to be parallel, but the only use I see this having is for it to be a shortcut way of describing the property that the tangents on each circle are parallel.

 

[phya]

I'm not sure that last point is exactly fair, because since a line is the same in homogeneous space, and distances between circles depend on their center and radius, it is much easier to claim that circles can't be parallel.

I stand by my previous post, concentric circles are parallel curves, but not parallel lines, because in mathematics lines are generally perfectly straight.

You in the past and in future border, therefore, on the one hand you acknowledge the curve parallel, on the other hand, you thought that the line is only a straight line. In fact, the line may be straight, may also be curving, the straight line is straight, but the line is not only straight.

 

[G037H3]

Well there's no problem defining concentric circles to be parallel, but the only use I see this having is for it to be a shortcut way of describing the property that the tangents on each circle are parallel.

Ah, wait a sec. Are you approaching this with a specific use in mind? You should just think about the objects and their relationship(s).

Also, I also have little idea what the hell phya is saying either. :/ idk if he's using translation software or just has a problem with vocabulary or grammar

 

[G037H3]

You in the past and in future border, therefore, on the one hand you acknowledge the curve parallel, on the other hand, you thought that the line is only a straight line. In fact, the line may be straight, may also be curving, the straight line is straight, but the line is not only straight.

Well, I personally use Archimedes' definition that a line is the shortest distance between two points, but here is evidence that the mathematical term 'line' refers to a straight one: http://mathworld.wolfram.com/Line.html

 

[Mentallic]

Ah, wait a sec. Are you approaching this with a specific use in mind?

Oh no not at all. Phya wanted to extend the definition of parallel to describe curves too, and I've given examples that have shown that for circles (which can be for curves too) you can think of parallel in different ways, depending on how you view it. This leaves room for confusion which is why one would need to define what exactly parallel for curves is and the properties that accompany it.

You should just think about the objects and their relationship(s).

But I have thought about them. Their relationship is that they are concentric and one of their properties is that given a line cutting through the centre of the circles, at the points where it intersects the circumferences, tangential lines at those points will be parallel.

This doesn't mean I'm ready to accept a flawed definition of "concentric circles are parallel curves".

 

[phya]

The point is, using the term "parallel curve" creates an ambiguity as I've already shown, unless it's specifically defined to be one or the other - which it isn't.

What is wrong with sticking to the term "concentric circles" anyway? I could just as easily argue that parallel lines are also concentric lines. It's just silly.

The concentric circle is only an example, this is only to explain the curve the parallel phenomenon. Certainly, curve parallel is not the limitation in the concentric circle. For example, the parabola may also have the parallel line. The curve parallel is the curve to the curve distance constant invariable. This is the curve parallel essence. Certainly is also the surface parallel essence. This is an axiom.

 

[G037H3]

Oh no not at all. Phya wanted to extend the definition of parallel to describe curves too, and I've given examples that have shown that for circles (which can be for curves too) you can think of parallel in different ways, depending on how you view it. This leaves room for confusion which is why one would need to define what exactly parallel for curves is and the properties that accompany it.


But I have thought about them. Their relationship is that they are concentric and one of their properties is that given a line cutting through the centre of the circles, at the points where it intersects the circumferences, tangential lines at those points will be parallel.

This doesn't mean I'm ready to accept a flawed definition of "concentric circles are parallel curves".

Well a line is a straight curve. Hum.

A transposition of two parallel lines would do nothing to change the values of any additional figures or forms created by any other lines. You are undoubtedly correct in your assertion about tangent lines to two points on the circumference of both circles being tangent, but what I was trying to communicate is that the straight line connecting the two is essential because of the 1 to 1 mapping, that is, the larger circle simple has a larger radius. Both circles have the same equation. Now, how is this really different than saying that two lines with y intercept 1 and y intercept 2 have the same equation if they are parallel? The shortest possible distance between any two points on the parallel lines is going to be 1. Obviously. Now, curves are not really any different, except for two things. The length of the curve, if it were flattened out to a straight line, and the uniformity of the curve. A line can be perfectly superimposed on any other line, but a curve can only be if they share a common value for the radii, so my contention is that concentric circles are an example of parallel lines, but if you want to transform the curves in any way, it changes things and they aren't always parallel if you transform/flip a curve.

 

[phya]

I don't disagree, but I feel that the main point of the thread is to assert that concentric circles are parallel, which they are, because they're parallel curves. As long as the OP understands the difference between parallel lines (all the points that the line consists of can be rotated about themselves 360 degrees and the line remains parallel) and parallel curves, I don't really have a problem with the assertion.

I was not only saying that the concentric circle parallel, I was also saying the curve parallel, moreover was also saying the non-Euclid geometry exists the question, sees #43 to paste:https://www.physicsforums.com/showpost.php?p=2919121&postcount=43

 

[phya]

So then you agree that "concentric lines" is absurd, since there are many flaws and counter-examples to it. The same deal with parallel circles applies here.

https://www.physicsforums.com/showpost.php?p=2921163&postcount=63

 

[phya]

I'm sorry, but could you please elaborate on what you mean by transposition? :)

My knowledge in this kind of topic comes from studying Heath's version of Euclid's Elements, btw. I'm only about 10% through it, but it is a very beautiful book.

：）
His meaning is the parallel migration. I already replied him.

 

[Mentallic]

If I were to think of two parabolas being parallel, then I would instantly think of some parabola y=ax^2+bx+c and then one that is parallel to it would be of the form y=ax^2+bx+k.

But apparently to have the parabolas follow this same rule as the concentric circles do, if say the "centre" of the parabola y=x²-1 is (0,0) then for another parabola that is a ratio of m:1 distance further from this parabola, it must be of the form y=x²/m-m

So which do you consider to be parallel parabolas? May I remind you this will be a definition that you create and which is not already widely accepted. I hope you can see why by now.

 

[phya]

Well, I personally use Archimedes' definition that a line is the shortest distance between two points, but here is evidence that the mathematical term 'line' refers to a straight one: http://mathworld.wolfram.com/Line.html

I understand your idea, but, the present mathematics is also in historical mathematics, but has not been separated from the historical mathematics. In the past, the humanity did not have the negative number concept, but afterward had. At present the humanity did not think that the curve is parallel, but the future will think not like this? Is indefinite. In the spherical surface between two spot most short distances are the straight line? On ellipsoid surface? On column surface? In paraboloid? In random surface? The line is the broad concept, but the straight line is the relative narrow concept.

 

[G037H3]

I understand your idea, but, the present mathematics is also in historical mathematics, but has not been separated from the historical mathematics. In the past, the humanity did not have the negative number concept, but afterward had. At present the humanity did not think that the curve is parallel, but the future will think not like this? Is indefinite. In the spherical surface between two spot most short distances are the straight line? On ellipsoid surface? On column surface? In paraboloid? In random surface? The line is the broad concept, but the straight line is the relative narrow concept.

Negative numbers were known. The thing is that the Greeks treated mathematics in a very pure manner, so that things like the square root of 2 really bothered them because of their desire for clean numerical relationships.

After that, negative numbers were mostly ignored because mathematics was used mainly for physics and physical applications, so negative results were thrown out as invalid. Which they are, if they're describing something physical on their own, with nothing to compare them to.

Concentric circles are parallel curves, but not parallel lines, because you can't perfectly place a section of the smaller circle onto a section of the larger circle and have them fit.

The shortest distance between two points is always a straight line. Non-Euclidean geometries only change this by changing the rules of the space so that it is not homogeneous in all directions as Euclidean space is. For Riemannian (elliptic) geometry, an example of the planet Earth is given, that two points on the surface are connected by a great circle, but still the shortest distance between those two points is a straight line going through the Earth, but in Riemannian geometry this sort of thing isn't allowed...basically, its still Euclidean geometry, just with somewhat different rules. Both models can describe space adequately.

Again, I think that you are mistaking the difference between a curve and a line. A line is always a straight line, a line is a straight curve. Get it?

 

[phya]

Oh no not at all. Phya wanted to extend the definition of parallel to describe curves too, and I've given examples that have shown that for circles (which can be for curves too) you can think of parallel in different ways, depending on how you view it. This leaves room for confusion which is why one would need to define what exactly parallel for curves is and the properties that accompany it.

But I have thought about them. Their relationship is that they are concentric and one of their properties is that given a line cutting through the centre of the circles, at the points where it intersects the circumferences, tangential lines at those points will be parallel.

This doesn't mean I'm ready to accept a flawed definition of "concentric circles are parallel curves".

Certainly, if two curves are parallel, then their corresponding points of curvature circle must concentric, this is certain. The curve parallel defined me already to say. The concentric circle is only a curve parallel phenomenon, but is not all.

 

[phya]

Well a line is a straight curve. Hum.

A transposition of two parallel lines would do nothing to change the values of any additional figures or forms created by any other lines. You are undoubtedly correct in your assertion about tangent lines to two points on the circumference of both circles being tangent, but what I was trying to communicate is that the straight line connecting the two is essential because of the 1 to 1 mapping, that is, the larger circle simple has a larger radius. Both circles have the same equation. Now, how is this really different than saying that two lines with y intercept 1 and y intercept 2 have the same equation if they are parallel? The shortest possible distance between any two points on the parallel lines is going to be 1. Obviously. Now, curves are not really any different, except for two things. The length of the curve, if it were flattened out to a straight line, and the uniformity of the curve. A line can be perfectly superimposed on any other line, but a curve can only be if they share a common value for the radii, so my contention is that concentric circles are an example of parallel lines, but if you want to transform the curves in any way, it changes things and they aren't always parallel if you transform/flip a curve.

You know what I mean?
https://www.physicsforums.com/showpost.php?p=2921163&postcount=63

 

[phya]

If I were to think of two parabolas being parallel, then I would instantly think of some parabola y=ax^2+bx+c and then one that is parallel to it would be of the form y=ax^2+bx+k.

But apparently to have the parabolas follow this same rule as the concentric circles do, if say the "centre" of the parabola y=x²-1 is (0,0) then for another parabola that is a ratio of m:1 distance further from this parabola, it must be of the form y=x²/m-m

So which do you consider to be parallel parabolas? May I remind you this will be a definition that you create and which is not already widely accepted. I hope you can see why by now.

https://www.physicsforums.com/showpost.php?p=2921406&postcount=79

 

[Mentallic]

So then the second column of curves in this picture aren't supposedly parallel?
http://img691.imageshack.us/i/parallel.png/"

 

[phya]

Negative numbers were known. The thing is that the Greeks treated mathematics in a very pure manner, so that things like the square root of 2 really bothered them because of their desire for clean numerical relationships.

After that, negative numbers were mostly ignored because mathematics was used mainly for physics and physical applications, so negative results were thrown out as invalid. Which they are, if they're describing something physical on their own, with nothing to compare them to.

Concentric circles are parallel curves, but not parallel lines, because you can't perfectly place a section of the smaller circle onto a section of the larger circle and have them fit.

The shortest distance between two points is always a straight line. Non-Euclidean geometries only change this by changing the rules of the space so that it is not homogeneous in all directions as Euclidean space is. For Riemannian (elliptic) geometry, an example of the planet Earth is given, that two points on the surface are connected by a great circle, but still the shortest distance between those two points is a straight line going through the Earth, but in Riemannian geometry this sort of thing isn't allowed...basically, its still Euclidean geometry, just with somewhat different rules. Both models can describe space adequately.

Again, I think that you are mistaking the difference between a curve and a line. A line is always a straight line, a line is a straight curve. Get it?

Why wants online front to add on " Straight " ? Therefore so the rhetoric has the reason, because the line is containing the curve. Your line's concept is forms in the history, therefore has the limitation. In the early history, we speak of the human often to understand the man, actually person this concept contains the woman, also contains the man.

 

[phya]

So then the second column of curves in this picture aren't supposedly parallel?
http://img691.imageshack.us/i/parallel.png/"

You give the link is unable to open.

 

[Mentallic]

It's just this one.

http://img691.imageshack.us/img691/9694/parallel.th.png

 

[phya]

What do you mean by 'parallel'? There are some definitions of "parallel" in "concentric circles are parallel" is true and some in which it is not. The "usual" definition of parallel in Euclidean geometry specifically defines only "parallel lines" and so, with that definition, it is not true.

Euclid's parallel is narrow parallel, my parallel is generalized being parallel.

 

[phya]

In the spherical surface does not have the parallel line, this viewpoint is not correct. Should say that in the spherical surface does not have the straight line parallel line, but has the curve parallel line.

 

[HallsofIvy]

Euclid's parallel is narrow parallel, my parallel is generalized being parallel.

So, once again, you state that you are not using the "standard" definition but refuse to state what definition you are using.

 

[Mentallic]

Well, his own definition. It's generalized, and it's flawed. There's nothing else to it really.

 

[G037H3]

generalized parallelism as a definition doesn't work because you can just flip one of the figures over and they aren't parallel anymore

 

[phya]

So, once again, you state that you are not using the "standard" definition but refuse to state what definition you are using.

Actually parallel had reflected a kind of natural phenomenon, this is the distance maintains the constant invariable phenomenon. This kind of phenomenon has between the straight line, also has between the curve. Therefore not only has straight line parallel, moreover also has curve being parallel. Its essence is the distance constant invariable.

 

[phya]

generalized parallelism as a definition doesn't work because you can just flip one of the figures over and they aren't parallel anymore

What meaning?

 

[G037H3]

here you go phya, pages 191-194 of Heath's Euclid's Elements, vol 1

"DEFINITION 23.
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.
IIapa.AATJAo~ (alongside one another) written in one word does not appear
in Plato; but with Aristotle it was already a familiar term.
Ei~ U1rEtpOV cannot be translated "to infinity" because these words might
seem to suggest a region or place infinitely distant, whereas El~ a1rEtpov, which
seems to be used indifferently with E1r' <'f1rEtpOV, is adverbial, meaning "without
limit," i.e. "indefinitely." Thus the expression is used of a magnitude being
"infinitely divisible," or of a series of terms extending without limit.
In both directions, Ecf>' £KaTEpa TO. fJ-EpTJ, literally "towards both the parts"
where "parts" must be used in the sense of "regions" (cf Thuc. II. 96).
It is clear that with Aristotle the general notion of parallels was that of
straight lines which do not meet, as in Euclid: thus Aristotle discusses the
question whether to think that parallels do meet should be called a
geometrical or an ungeometrical error (Anal. post. 1. 12, 77 b 22), and (more
interesting still in relation to Euclid) he observes that there is nothing
surprising .in different hypotheses leading to the same error, as one might
conclude that parallels meet by starting from the assumption, either (a) that
the interior (angle) is greater than the exterior, or (b) that the angles of a
triangle make up more than two right angles (Anal. prior. n. 17, 66 a II).
Another definition is attributed by Proclus to Posidonius, who said that
"parallel lines are those which, (being) in one plane, neither converge nor diverge,
but have all the perpendiculars equal which are drawn from the points 0/ one
line to the other, while such (straight lines) as make the perpendiculars less and
less continually do converge to one another; for the perpendicular is enough
to define (opi'Etv OVVaTaL) the heights of areas and the distances between lines.
For this reason, when the perpendiculars are equal, the distances between the
straight lines are equal, but when they become greater and less, the interval is
lessened, and the straight lines converge to one another in the direction in
which the less perpendiculars are" (Proclus, p. 176, 6- 17).
Posidonius' definition, with the explanation as to distances between straight
lines, their convergence and divergence, amounts to the definition quoted by
Simplicius (an-Nairizi, p. 25, ed. Curtze) which described straight lines as
parallel if,'when they are produced indefinitely both ways, the distance between
them, or the perpendicular drawn from either of them to the other, is always
equal and not different. To the objection that it should be proved that the
distance between two parallel lines is the perpendicular to them Simplicius
I. DEF. 23] NOTES ON DEFINITIONS 22, ~3
replies that the definition will do equally well if all mention of the perpendicular
be omitted and it be merely stated that the distance remains equal,
although" for proving the matter in 'question it is necessary to say that one
straight line is perpendicular to both" (an-Nairizi, ed. Besthorn-Heiberg, p. 9)'
He then quotes the definition of "the philosopher Aganis": ," Parallel
straight lines are straight lines, situated in the same plane, the distance between
which, if they are produced indefinitely in both directions at the same time, is
everywhere the same." (This definition forms the basis of the attempt of
"Aganis" to prove the Postulate of Parallels.) On the definition Simplicius
remarks that the words "situated in the same plane" are perhaps unnecessary,
since, if the distance between the lines is everywhere the same, and one does
not incline at all towards the other, they must for that reason be in the same
plane. He adds that the "distance" referred to in the definition is the
shortest line which joins things disjoined. Thus, between .point and point,
the distance is the straight line joining them; between a point and a straight
line or between a point and a plane it is the perpendicular drawn from the point
to the line or plane; "as regards the distance between two lines, that distance
is, if the lines are parallel, one and the same, equal to itself at all places on
the lines, it is the shortest distance and, at all places on the lines, perpendicular
to both" (ibid. p. 10).
Tb-e same idea occurs in a quotation by Proclus (p. 177, II) from
Geminus. As part of a classification of lines which do not meet he observes:
" Of lines which do not meet, some are in one plane with one another, others
not. Of those which meet and are in one plane, some are always the same
distance from one another, others lessen the distance continually, as the hyperbola
(approaches) the straight line, and the conchoid the straight line (i.e. the
asymptote in each case). For these, while the distance is being continually
lessened, are continually (in the position of) not meeting, though they converge
to one another; they never converge entirely, and this is the most paradoxical
theorem in geometry, since it shows that the convergence of some lines is nonconvergent.
But of lines which are always an equal distance apart, those
which are straight and never make the (distance) between them smaller, and
which are in one plane, are parallel."
Thus the equidistance-theory of parallels (to which we shall return) is very
fully represented in antiquity. I seem also to see traces in Greek writers of a
conception equivalent to the vicious direction-theory which has been adopted
in so many modern text-books. Aristotle has an interesting, though obscure,
.allusion in Anal. prior. II. 16, 65 a 4 to a petitio principii committed by "those
who think that they draw parallels" (or "establish the theory of parallels,"
which is a possible translation of TO.'; 1rapaAA~AOV<; ypo.<!ml'): "for they unconsciously
assume such things as it is not possible to demonstrate if parallels
do not exist." It is clear from this that there was a vicious circle in the then
current theory of parallels; something which depended for its truth on the
properties of parallels was assumed in the actual proof of those properties,
e.g. that the three angles of a triangle make up two right angles. This is not
the case in Euclid, and the passage makes it clear that it was Euclid himself
who got rid of the petilio principii in earlier text-books by formulating and
premising before I. 29 the famous Postulate 5, which must ever be regarded
as among the most epoch-making achievements in the domain of geometry.
But one of the commentators on Aristotle, Philoponus, has a note on the
above passage purporting to give the specific character of the petitio principii
alluded to; and it is here that a direction-theory of parallels may be hinted at,
whether Philoponus is or is not right in supposing that this was what Aristotle
had in mind. Philoponus says: "The same thing is done by those who draw
parallels, namely begging the original question; for they will have it that it is
possible to draw parallel straight lines from the meridian circle, and they
assume a point, so to say, falling on the plane of that circle and thus they
draw the straight lines. And what was sought is thereby assumed; for he
who does not admit the genesis of the parallels will not admit the point
referred to either." What is meant is, I think, somewhat as follows. Given
a straight line and a point through which a parallel to it is to be drawn, we
are to suppose the given straight line placed in the plane of the meridian.
Then we are told to draw through the given point another straight line in the
plane of the meridian (strictly speaking it should be drawn in a plane parallel
to the plane of the meridian, but the idea is that, compared with the size of
the meridian circle, the distance between the point and the straight line is
negligible); and this, as I read Philoponus, is supposed to be equivalent to
assuming a very distant point in the meridian plane and joining ~he given
point to it. But obviously no ruler would stretch to such a point, and the
objector would say that we cannot really direct a straight line to the assumed
distant point except by drawing it, without more ado, parallel to the given
straight line. And herein is the petitio principii. I am confirmed in seeing
in Philoponus an allusion to a direction-theory by a,·remark of Schottel',) on a
similar reference to the meridian plane supposed to be used by advocates of
that theory. Schotten is arguing that direction is not in itself a conception
such that you can predicate one direction of two different lines. " If anyone
should reply that nevertheless many lines can be conceived which all have the
direction from north to south," he replies that this represents only a nominal,
not a real, identity of direction.
Coming now to modern times, we may classify under three groups
practically. all the different definitions that have been given of parallels
(Schotten, op. cit. II. p. 188 sqq.).
(I) Parallel straight lines have no point common, under which general
conception the following varieties of statement may be included:
(a) they do not cut another,
(b) they meet at infinity, or
(c) they have a common point at infinity.
(2) Parallel straight lines have the same, or like, direction or directions,
under which class of definitions must be included all those which introduce
transversals and say that the parallels make t'qual angles with a transversal.
(3) Parallel straight lines have the distance between them constant;
with which group we may connect the attempt to explain a parallel as the
geometrical locus of all poil/ts 10hich are equidistant from a straight line.
But the three points of view have a good deal in common; some of them
lead easily to the others. Thus the idea of the lines having no point common
led to the notion of their having a common point at infinity, through the
influence of modern geometry seeking to embrace different cases under one
conception; and then again the idea of the lines having a common point at
infinity might suggest their having the same direction. The" non-secant"
idea. would also naturally lead to that of equidistance (3), since our
observation shows that it is things which come nearer to one another that
tend to meet, and hence, if lines are not to meet, the obvious thing is to see
that they shall not come nearer, i.e. shall remain the same distance apart.
We will now take the three groups in order.
(I) The first observation of Schotten is that the varieties of this group
which regard parallels as (a) meeting at infinity or (b) having a common
point at infinity (first mentioned apparently by Kepler, 16°4, as a "fac;on de
parler" and then used by Desargues, 1639) are at least unsuitable definitions
for elementary text-books. How do we know that the lines cut or meet at
infinity? We are not entitled to assume either that they do or that they do
not, because "infinity" is outside our field of observation and we cannot verify
either. As Gauss says (letter to Schumacher), "Finite man cannot claim to
be able to regard the infinite as something to be grasped by means of ordinary
methods of observation." Steiner, in speaking of the rays passing through a
point and successive points of a straight line, observes that as the point of
intersection gets further away the ray moves continually in one and the same
direction (" nach einer und derselben Richtung hin"); only in one position,
that in which it is parallel to the straight line, "there is no real cutting"
between the ray and the straight line; what we have to say is that the ray is
"directed towards the infinitely distant point on the straight line." It is true
that higher geometry has to assume that the lines do meet at infinity: whether
such lines exist in nature or not does not matter (just as we deal with "straight
lines" although there is no such thing as a straight line). But if two lines do
not cut at any finite distance, may not the same thing be true at infinity also?
Are lines conceivable which would not cut even at infinity but always remain
at the same distance from one another even there? Take the case of a line
of railway. Must the two rails meet at infinity so that a train could not stand
on them there (whether we could see it or not makes no difference)? It
. seems best therefore to leave to higher geometry the conception of infinitely
distant points on a line and of two straight lines meeting at infinity, like
imaginary points of intersection, and, for the purposes of elementary geometry,
to rely on the plain distinction between "parallel" and "cutting" which
average human intelligence can readily grasp. This is the method adopted
by Euclid in his definition, which of course belongs to the group (I) of
definitions regarding parallels as non-secant.
It is significant, I think, that such authorities as Ingrami (Elementi di
geometria, 1904) and Enriques and Amaldi (Elementi di geometria, 1905),
after all the discussion of principles that has taken place of late years, give
definitions of parallels equivalent to Euclid's: "those straight lines in a plane
which have not any point in common are called parallels." Hilbert adopts
the same point of view. Veronese, it is true, takes a different line. In his
great work Fondamenti di geometn"a, 1891, he had taken a ray to be parallel to
another when a point at infinity on the second is situated on the first; but he
appears to have come to the conclusion that this definition was unsuitable for
his Elementi. He avoids however giving the Euclidean definition of parallels
as "straight lines in a plane which, though produced indefinitely, never meet,"
because" no one has ever seen two straight lines of this sort," and because
the postulate generally used in connexion with this definition is not evident in
the way that, in the field of our experience, it is evident that only one straight
line can pass through two points. Hence he gives a different definition, for
which he claims the advantage that it is independent of the plane. It is
based on a definition of figures "opposite to one another with respect to a
point " (or reflex figures). "Two figures are opposite to one another with
respect to a point 0, e.g. the figures ABC ... and A'B' C' ..., if to every point
of the one there corresponds one sole point of the other, and if the segments
H. E. I',) OA, OB, OC, ... joining the points of one figure to 0 are respectively equal
and opposite to the segments OA', OB', OC', ... joining to 0 the corresponding
points of the second": then, a transversal of two straight lines being any
segment having as its extremities one point of one line and one point of the
other, "two straight lines are called parallel if one of them contains two points
opposite to two points of the other with respect to the middle point of a common
transversal." It is true, as Veronese says, that the parallels so defined and the
parallels of Euclid are in substance the same; but it can hardly be said that
the definition gives as good an idea of the essential nature of parallels as does
Euclid's. Veronese has to prove, of course, that his parallels have no point in
common, and his "Postulate of Parallels" can hardly be called more evident
than Euclid's: "If two straight lines are parallel, they are figures opposite to
one another with respect to the middle points of all their transversal segments."
(2) The direction-theory.
The fallacy of this theory has nowhere been more completely exposed
than by C. L. Dodgson (Euclid and his modern Rivals, 1879). According to
Killing (Einfiihrung in die Grundlagtn der Geomelrie, I. p. 5) it would appear
to have originated with no less a person than Leibniz. In the text-books
which employ this method the notion of direction appears to be regarded as a
primary, not a derivative notion, since no definition is given. But we ought
at least to know how the same direction or like directions can be recognised
when two different straight lines are in question. But no answer to this
question is forthcoming. The fact is that the whole idea as applied to noncoincident
straight lines is derived from knowledge of the properties of
parallels; it is a case of explaining a thing by itself. The idea of parallels
being in the same direction perhaps arose from the conception of an angle as
a difference of direction (the hollowness of which has already been exposed) ;
sameness of direction for parallels follows from the same "difference of
direction" which both exhibit relatively to a third line. But this is not
enough. As Gauss said (Werke, IV. p. 365), "If it [identity of direction] is
recognised by the equality of the angles formed with one third straight line,
we do not yet know without an antecedent proof whether this same equality
,will also be found in the angles formed with a fourth straight line" (and any
number of other transversals); and in order to make this theory of parallels
valid, so far from getting rid of axioms such as Euclid's, you would have to
assume as an axiom what is much less axiomatic, namely that "straight lines
which make equal corresponding angles with a certain transversal do so with
any transversal" (Dodgson, p. 101). .
(3) In modern times the conception of parallels as equidistant straight
lines was practically adopted by Clavius (the editor of Euclid, born at
Bamberg, 1537) and (according to Saccheri) by Borelli (Euclides restitu/us,
1658) although they do not seem to have defined parallels in this way.
Saccheri points out that, before such a definition can be used, it has to
be proved that "the geometrical locus of points equidistant from a straight
line is a straight line." To do him justice, Clavius saw this and tried to
prove it: he makes out that the locus is a straight line according to the
definition of Euclid, because "it lies evenly with respect to all the points
on it"; but there is a confusion here, because such "evenness" as the locus
has is with respect to the straight line from which its points are equidistant,
and there is nothing to show that it possesses this property with respect
to itself. In fact the theorem cannot be proved without a postulate."

the jumbled up spammish words are Greek...i'm not going to go find the words and write them out, sorry, though i can link a pdf to the whole book if anyone is interested

 

[phya]

Actually so long as we acknowledged that own and oneself is parallel, then we can not but acknowledge that the curve and the curve were also parallel. For instance a straight line, it own and oneself is parallel. Similarly, a circle own and oneself is also parallel. Therefore the curve and the curve are may mutually parallel.

 

[G037H3]

Actually so long as we acknowledged that own and oneself is parallel, then we can not but acknowledge that the curve and the curve were also parallel. For instance a straight line, it own and oneself is parallel. Similarly, a circle own and oneself is also parallel. Therefore the curve and the curve are may mutually parallel.

concentric circles are parallel curves

i don't know what other assertion you're making other than this

parallel curves are possible with a general definition of parallelism, but with a strict definition, only parallel lines on a plane, and maybe geodesics, are parallel.

 

[phya]

Möbius strip is parallel.

 

[phya]

The curve of the two edges are parallel.

 

[G037H3]

The curve of the two edges are parallel.

err, what?...the top and bottom edges of the line? that's a pretty loose definition of parallelism, and even if they were parallel, how would that help you?

if you flip a curve (a reflex transform), it is no longer parallel. a line is.

 

[Upisoft]

Concentric circles are not parallel, they are equidistant. "Parallel" is defined only for straight lines.

 

[G037H3]

Concentric circles are not parallel, they are equidistant. "Parallel" is defined only for straight lines.

parallel curves exist, it's just that phya is using too loose of a definition for parallel :/

http://mathworld.wolfram.com/ParallelCurves.html