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Abraham Kaestner and Euclid's Fifth (Parallel) Postulate

Date: 12/02/96 at 16:53:11
From: Anonymous

I am a seventh grade student doing a research paper on a mathematician 
named Abraham Kaestner.  I cannot find any information on him and was 
wondering if you could help me.  

Thanks, Kris

Date: 12/02/96 at 17:35:13
From: Doctor Sarah

Hi Kris -

There's not a whole lot of information about Kaestner on the Web, but 
I did find some in the St. Andrews history archive:   

               Abraham Gotthelf Kaestner

           Born: 27 Sept 1719 in Leipzig, Germany
           Died: 20 June 1800 in Gottingen, Germany

Kaestner taught at Leipzig, then from 1746 at Gottingen where he was 
to succeed to Segner's chair.  He was an excellent expositor of 
mathematics, although it is reported that Gauss didn't bother to go to 
his lectures as they were too elementary.  However, he did influence 
Gauss, in particular with his interest in Euclid's parallel postulate. 

In fact, Kaestner's interest in the parallel postulate indirectly 
influenced Bolyai and Lobachevsky too (Kaestner taught Bolyai's father 
and Bartels, one of Kaestner's students, taught Lobachevsky). 


    1. _Dictionary of Scientific Biography_. 
    2. G Goe, Kaestner, Forerunner of Gauss, Pash, Hilbert, 
       _Proceedings 10th International Congress of the History 
       of Science II_ (Paris, 1964), 659-661.   

Here's the name of a 4-volume book by Kaestner:

Kaestner, Abraham Gotthelf  (1719-1800).  _Geschichte der Mathematik 
seif der Wiederherstellung der Wissenschaften bis an das Ende des 
achtzehnten Jahrhunderts_.  Four volumes.  Rosenbusch, Gottingen, 

This isn't really enough for a research paper, but you could follow 
the leads from the St. Andrews archive to Bolyai and Lobachevsky and 
also talk about Euclid's Parallel Postulate.  There's a nice paper 
about it with illustrations at   

It begins "On Euclid's Fifth Postulate: al-Haytham's Innovative 
"Proof" and Omar Khayyam's Response":

One of the most fascinating aspects of mathematics is that there exist
statements that are both true and false.  Perhaps the most famous of 
these is Euclid's controversial fifth postulate.  Throughout history, 
almost from the postulate's conception, mathematicians have tried, in 
vain, to prove or disprove it.  It seems that Euclid himself did not 
entirely trust the postulate, for he avoided using it as long as he 
could in his great work, _The Elements_, by proving his first 28 
propositions without it.  This paper will present a brief history of 
the postulate, a particularly inventive proof by al-Haytham, and the 
great Persian mathematician Omar Khayamm's comments upon its validity. 

From the beginning, Euclid's fifth postulate, also called the parallel
postulate, stood out from among its brethren.  The first four 
postulates are short, brief, and to the point, whereas the fifth is 
longer and rather strange sounding. The postulates are listed in _The 
Elements_ as such: 

1. To draw a straight line from any point to another. 
2. To produce a finite straight line continuosly in a straight line. 
3. To describe a circle with any centre and distance. 
4. That all right angles are equal to each other. 
5. That, if a straight line falling on two straight lines make the
   interior angles on the same side less than two right angles, if
   produced infinitely, meet on that side on which are the angles 
   less than the two right angles (Rosenfield 35).

The paper goes on from there. If you follow the arguments laid out in 
it, it's really pretty interesting.

-Doctor Sarah,  The Math Forum
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