Bractals
2011-12-13 06:43:47 |
Tangent line
How would the ancient greek mathematicians define a tangent line to a curve?
The curve might be a circle, ellipse, hyperbola, or parabola.
How would this definition apply to y = x^3 - x ?
Can a curve have more than one tangent line at a point on the curve? |
brianjn
2011-12-14 06:22:50 |
Re: Tangent line
Bractals? I realise that with your acquaintance and standing with Perplexus that you are not placing a problem here for solution as so many novices have done.
What is this about?
I am recounting a documentary whereby a Greek archaeological team were reconstructing columns of the Parthenon. The columns were actually disks of stone and they were uniquely fashioned; no "disk" could be placed correctly on top of another so was the inter-facial surface matching. It would appear that the vertical profile of each column was also true linear but that was not the case. The Greek woman heading that team was to find that the profile was a circular curve which was over 1 km in radius and yet that profile was re-markedly accurate. (The Greeks somehow realised that standing at the base of a column and looking upwards, a straight line profile was visually unaesthetic).
On site on one of the Grecian islands someone had found structures built using columns of the same construction. In the ruins was a flooring, maybe a wall, which had something as I recall being like a clinometer. Using this, and some extrapolation of shadows cast by the Sun they were able to arrive at the desired vertical profile.
How truly I have remembered and recounted those memories I am at a loss, but, is something like this at the root of your question?
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Jer
2011-12-14 09:11:03 |
Re: Tangent line
I wouldn't be surprised if Euclid uses the term, so I would suggest searching there. The definition is probably a line that touches the curve at a single point (as opposed to a secant) and they surely knew the tangent line is perpendicular to the curve.
An algebraic curve would have been meaningless. It was Descartes that invented the coordinate plane and graphing equations.
Generally a curve cannot have more than one tangent point but I suppose you could consider a sort of cusp to have more than one. |
Bractals
2011-12-14 16:44:20 |
Re: Tangent line
How do you define "touches"? Does it differ from intersect? |
brianjn
2011-12-19 01:48:37 |
Re: Tangent line
I did not really understand the focus of the original post. I now realise that we have a philosophical point before us.
Yes, "touch", "intersect", "cross" ...
What are those shades of meaning? In standard English we'd have huge arguments.
Equilibrium? As we consider gravity and a uniform beam at rest over a fulcrum point, might they have had a similar concept but in relationship to a very specific point, the "gravitational attraction" being perpendicular to the curve. But now there seems to be a problem. Would "perpendicular to a point on a curve" have been within their lexicon?
Where did "tangent" first appear in Greek mathematics? That might provide the clue to how the concept was understood and then defined.
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Bractals
2011-12-19 14:30:54 |
Re: Tangent line
The reason that I brought up the topic was a proof I read in a pdf on the internet.
The author gave the following definition for a tangent line:
A tangent line intersects a curve at one point and all other points of the curve lie on one side of the line.
This would mean that the curve y = |x| could have more than one tangent line at the point x = 0.
Also, this definition would exclude the tangent lines to the curve y = x^3 - x when y' = 0 (since they intersect the curve at more than one point).
So it seems that we would need to "localize" the definition. |
Jer
2011-12-27 01:49:05 |
Re: Tangent line
The above definition would have to specify that
1. The curve be smooth (differentiable) at the point
2. It is confined to a local neighborhood
Certainly http://mathworld.wolfram.com/TangentLine.html gives a definition that does not have these problems, but would not be understood by the ancient Greeks. Another possibility is they assumed they know what they meant by the term and didn't give a very good (from our perspective) definition either.
In other words the definition my not be sufficient. Or, since the term has been defined, the author _wants_ your examples to be as you mention them.
I would like to see this pdf if you don't mind sharing.
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Bractals
2011-12-27 15:46:05 |
Re: Tangent line
Jer - Checkout www.amotlpaa.org/math/parabtan.pdf |
