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Sup with the Gods


caledonia
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If thou wouldst follow Newton in his Book 1, Section 3, Proposition XI and deduce the inverse square law from Kepler's observations of elliptical planetary orbits, first know thy Conics.

In particular, the proof uses these properties of the ellipse :

1. The sum of the lines joining the foci to any point on the ellipse is constant

2. The lines joining the foci to a point on the ellipse make equal angles with the tangent there

3. The area of the parallelogram formed by the tangents at the ends of any diameter is constant

4 . The 'symptom' of the ellipse viz. for any diameter, the ratio of the square of an ordinate to the product of the abscissae is constant

 

Newton and his contemporary mathematicians were well versed in Apollonius where these and many other properties of the Conics are proved. But Apollonius is long and difficult : fortunately we have shorter methods :

Dandelin spheres easily prove 1.

It is easy to deduce 2 from 1.

Archimedes proved in Proposition 9 of his "Spheroids and Conoids" that we may regard the ellipse as a slant section of a right cylinder rather than a cone. Then for any diameter, the orthogonal projection of the parallelogram in 3 is a square describing the circular 'base'. All such squares are of course equal in area.

Again using the cylindrical definition of the ellipse, the abscissae / ordinate in 4 are projected onto a diameter and perpendicular ordinate of the circle. It is easy to prove the constancy of the ratio of (the square of) an ordinate and (the product of) its abscissae for the circle.

 

Now goest thou and follow Newton . . .

 

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Sorry. I accept that my posting should have been a blog. I would therefore like to remove my original posting (preaching?) but do not know how to do that. Feel free, if anyone does know . . .

 

 

 

 

I didn’t say you were preaching, I asked how it differs, but surely there’s an interesting question, embedded, that could be discussed.

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  • 1 year later...
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