Focus of parabola

[caledonia]

In Lemma XIII of the Principia, Newton says "The latus rectum of a parabola belonging to any vertex is quadruple the distance of that vertex from the focus of the figure. And this is demonstrated by the writers on the conic sections".

 

Well, Apollonius does not state this proposition. And I can't find any 'writer' who does.

 

Hence, can anyone help ?

 

Thanks.

 

 

[studiot]

I do believe that this relationship was known to the ancientGreeks.

 

Pappus wrote

 

The four books of Euclid's 'Conics' were completed by Appolonius, who added four more and gave us eight books of Conics.

 

However Euclid's books were lost.

 

Aristaeus wrote five further books of the Conics.

 

These books contained all the knowledge to prove geometrically the relationship ie equivalent statements, so perhaps it was never explicity stated, it was used however.

[studiot]

I do not have access to this book now, but try here

 

Charles Taylor : Ancient and Modern Geometry of Conics (Cambridge 1881) contains much history of the subject.

 

The original book was by John Wallis in about 1661

[EdEarl]

I do not have access to this book now, but try here

 

Charles Taylor : Ancient and Modern Geometry of Conics (Cambridge 1881) contains much history of the subject.

 

The original book was by John Wallis in about 1661

I did not find this book, but did find this from Cyclopaedia, volume 1. And, another person on another forum showed me the focal point in this forum: http://www.openscad.org/community.html in this thread: Parabolic Trough, by Don Bright on Aug 31, 2013; 8:16pm. (Ref: Dandelin spheres)

[studiot]

Thank you for your references, Ed Earl.

 

However the thread you originally replied to was this one, not here.

 

http://www.scienceforums.net/topic/36783-conics-general-question/

 

[EdEarl]

Thank you for your references, Ed Earl.

 

However the thread you originally replied to was this one, not here.

 

http://www.scienceforums.net/topic/36783-conics-general-question/

 

LOL, that's right. That was a senior moment.

[studiot]

Since there is continued interest in this subject, the proof of the proposition is remarkably simple.

 

The Greeks knew the parabola as a curve such that any point on the curve is equidistant from a given point (S in my sketch) and a given straight line (MN in my sketch). The distance to the line being the distance along a perpendicular to the line. LR is the latus rectum.

 

The proof follows immediately from the definition.

 

 

Edited September 9, 2013 by studiot

[EdEarl]

tyvm