[FOM] non-Euclidean geometry and FOM programs

Antonino Drago drago at unina.it
Tue Nov 20 17:57:13 EST 2007

> andre rodin"  wrote:
>>> All his [Lobachevsky] reasoning involves explicit geometrical 
>>> constructions
>>> with multiple
>>> parallels to any given straight line.
>>> These explicit constructions are certainly NOT  Euclidean models of
>>> Lobachevsky's geometry like Beltrami's model.
> Antonio Drago wrote:
>> This idea does not appear in Lobachevsky writings. Only in Geoemetrische
>> Untersuchungen... (1840) he devotes the proposition no. 23 to a 
>> "geometrical
>> construction" for finding out the parallel line to a given line 
>> (actually,
>> its proof is incorrect, because it relies upon a wishfull sentence: "it 
>> must
>> exist such a parallel line FG").
> andre rodin:
> I refer to "New Foundations of Geometry with the Complete theory of
> Parallels" first published in Russian in Kazan in 1835-1838 (then
> republished in 1949 as a part of Complete Works but for the best of my
> knowledge never fully translated in another language). The "New 
> Foundations"
> is the most systematic presentation of Lobachevsky's work. Unfortunately I
> don't have this text at hand and cannot check. But do you take it into
> consideration when you claim that "the idea doesn't appear in L. 
> writings"?

This book was translated in English in 1897, in German by Engel in 1899, in 
French in 1901, in Italian in 1974. Surely, French and Italian translations 
suppressed some chapters, but not the more relevant ones for presenting the 
foundations of Lobachevsky's geometry.
In the book you refer to the kind of foundation was the same of the booklet 
I quoted, i.e. an intuitive arguing about the possibility of more than one 
parallel line and then some theorems about the sum of the angles inside a 
triangle. A different kind of foundation, essentially relying upon 
trigonometry,  was given in Foundations of Geometry (1829/30), Gèométrie 
Imaginaire (1835) and Pangeometry (1855-6).
Let me quote the paper by  S. Cicenia and myself The organizational 
structures of geometry in Euclid, L. Carnot and Lobachevsky. An analysis of 
Lobachevsky' s works, ). In Memoriam N. I. Lobachevskii, 3, pt. 2 (1995) 
ibidem, 116-124 .
Best regards
Antonino Drago

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