At 12 minutes 42 seconds, after showing how rational parameterizations for an ellipse can be constructed from one rational point on the curve, and a straight line, he notes that this can be done for any conic.
I have a feeling that it could be done more generally, for any curves that can be defined as a plane cut by a ruled surface. It seems to me that the only surfaces Euclid considered serously were ruled surfaces: a sphere is not a surface in Euclid's view. This would explain his definition of the circle and semicircle: from Fitzpatrick's edition of Euclid's Elements. Now look at Eucld's definition of a right-lne "A straight line is any line which lies evenly with the points on it", and his definition of a right-angle as "If a right-line is stood upon its end on another right-line, such the angles on either side of the vertical are equal, then those angles are right-angles". Then in a later book, "All right angles are equal. Now think about a right-line AB from point A to point B, and imagine it does not lie evenly with the points on it. Draw a straight line from A to C, and another from C to B. This is a three-sided figure. Now look at Euclid's definition of a triangle: it is a proposition. A triangle has internal angles which sum to two right angles. Now look at Euclid's definition of a circle and a semi-circle.
Now think about the nature of error: Aristotle on The Continuum.
A ruled surface is the boundary of a cone of light from a point source, so the study of intersections of ruled surfaces is one which can be done just by looking at the action of shadows cast by straight edges onto a screen. Thinking about Plato's "Myth of the Cave" in The Republic, in this context, should give a fruitful way to interpret the notion of Platonic reality. Then read the first chapter of Aristotle's Physics, to see how Aristotle's and Plato's philosophies relate to each other: Aristotle on The Continuum , ... See Rafael's fresco The School of Athens at the Vatican:
Now, by systematically exploring these relationships, one may develop quite a useful body of theory which makes a fairly sound basis for observational astronomy and geodesy. In the latter, you would be interested in ways to locate, say the center of a circle based on observations of an eliptical projection of the circle. Think about relating the readings of sundials and solar and lunar eclipses, at different places, such as at Athens and at Alexandria, ... and collecting the data over several years, then bringing it all together and turning it into a theory about something or other, such as the actual shape of the earth?
This is a reconstruction of Feynman's derivation of Kepler's law from Newton's law. It might be interesting to think about how Kepler's law could have been used to derive Newton's laws, ... with a little help from Euclid, maybe, see All About e and Quaternions Visualised
Now listen to this lecture on Electrical Engineering, by Eric Laithwaite, ...
Now before the people at Cambridge go nuts, watch this beautiful talk, which may sound tongue-in cheek, but isn't. It's totally genuine, and utterly brilliant!
With that in mind, go back and watch the above videos again, and then watch this one:
Some people 'round here need to belt up!
... and study Architecture:
Apparently you don't have to believe in shared simultaneous spaces in a rotating reference frame to be able to do astrophysics! See There's Trouble Down't Dark Satanic Mill!
I have a feeling that it could be done more generally, for any curves that can be defined as a plane cut by a ruled surface. It seems to me that the only surfaces Euclid considered serously were ruled surfaces: a sphere is not a surface in Euclid's view. This would explain his definition of the circle and semicircle: from Fitzpatrick's edition of Euclid's Elements. Now look at Eucld's definition of a right-lne "A straight line is any line which lies evenly with the points on it", and his definition of a right-angle as "If a right-line is stood upon its end on another right-line, such the angles on either side of the vertical are equal, then those angles are right-angles". Then in a later book, "All right angles are equal. Now think about a right-line AB from point A to point B, and imagine it does not lie evenly with the points on it. Draw a straight line from A to C, and another from C to B. This is a three-sided figure. Now look at Euclid's definition of a triangle: it is a proposition. A triangle has internal angles which sum to two right angles. Now look at Euclid's definition of a circle and a semi-circle.
Now think about the nature of error: Aristotle on The Continuum.
A ruled surface is the boundary of a cone of light from a point source, so the study of intersections of ruled surfaces is one which can be done just by looking at the action of shadows cast by straight edges onto a screen. Thinking about Plato's "Myth of the Cave" in The Republic, in this context, should give a fruitful way to interpret the notion of Platonic reality. Then read the first chapter of Aristotle's Physics, to see how Aristotle's and Plato's philosophies relate to each other: Aristotle on The Continuum , ... See Rafael's fresco The School of Athens at the Vatican:
Now, by systematically exploring these relationships, one may develop quite a useful body of theory which makes a fairly sound basis for observational astronomy and geodesy. In the latter, you would be interested in ways to locate, say the center of a circle based on observations of an eliptical projection of the circle. Think about relating the readings of sundials and solar and lunar eclipses, at different places, such as at Athens and at Alexandria, ... and collecting the data over several years, then bringing it all together and turning it into a theory about something or other, such as the actual shape of the earth?
This is a reconstruction of Feynman's derivation of Kepler's law from Newton's law. It might be interesting to think about how Kepler's law could have been used to derive Newton's laws, ... with a little help from Euclid, maybe, see All About e and Quaternions Visualised
Now listen to this lecture on Electrical Engineering, by Eric Laithwaite, ...
Now before the people at Cambridge go nuts, watch this beautiful talk, which may sound tongue-in cheek, but isn't. It's totally genuine, and utterly brilliant!
Some people 'round here need to belt up!
... and study Architecture:
Apparently you don't have to believe in shared simultaneous spaces in a rotating reference frame to be able to do astrophysics! See There's Trouble Down't Dark Satanic Mill!
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