German translation of final sentences of a paper by Hilbert

Question

I am translating a paper by Hilbert into English. I am finished except for the last few sentences, which are confusing me. If anyone can give me a rough/quick translation it would be greatly appreciated.

Here are the sentences:

Zum Schluß erlaube ich mir darauf hinzuweisen, daß ich bei der vorstehenden Entwickelung stets den nirgends konkaven Körper als ganz im Endlichen gelegen angenommen habe. Wenn jedoch in der durch die ursprünglichen Axiome definierten Geometrie eine Gerade und ein Punkt vorhanden ist von der Eigenschaft, daß durch diesen Punkt zu der Geraden nur eine einzige Parallele möglich ist, so ist jene Annahme nicht gerechtfertigt. Es wird leicht erkannt, welche Abänderungen meine Betrachtung dann zu erfahren hat.

Kleinteich bei Ostseebad Rauschen, den 14. August 1894.

Here is my poor, certainly very wrong, translation of the above:

Finally, I take the liberty to point out that I, in the above development, have always assumed that the the nowhere concave body [convex set] is very finitely located [bounded?]. However, if in the original axioms that defined the geometry a straight line and a point have only one parallel possible [through that point], then that assumption is not justified [here]. It is easily recognized then, that amendments have to be found.

Thank you in advance!

For Hilbert's paper, see page 88 of Grundlagen der Geometrie.

Answer 1

Your translation is not ''poor'', though indeed it is ''wrong'' in several respects. I will run through your proposed translation systematically. For easier comparison, let me first recall that your proposed translation was

and let me also add that the original is

(source: David Hilbert: ``Grundlagen der Geometrie''. Zweite, durch Zusätze vermehrte und mit fünf Anhängen versehene Auflage. Leipzig. Druck und Verlag von B. G. Teubner. 1903.)

Now the systematic discussion of your proposed translation: your translation is fine up until the first 'the' in

"Finally, I take the liberty to point out that I, in the above development, have always assumed that the the nowhere concave body [convex set] is very finitely located [bounded?].

but the second 'the' is an instance of

'dittography' (cf., e.g., p. 48 of e.g. P. D. Wagner: A student's guide to textual criticism of the bible)

and then you begin to go wrong when writing ''is very finitely located''. It is hopefully needless to say that this is not even correct English, in and of itself, and moreover it is not a translation of the German. In the context of Hilbert's words ``als ganz im Endlichen gelegen'', the 'ganz' is not an adverb modifying the participle 'gelegen' (in principle, you have a point in that in German 'ganz' can, sometimes, mean 'very'); rather, Hilbert means that the entirety of the body lies in a bounded region of the plane (which is why your comment in the square brackets is correct).

Then you go wrong again by letting the 'in' refer to 'the original axioms': Hilbert lets the 'in' refer to the 'geometry which is defined by the axioms'. In modern parlance, Hilbert lets the 'in' refer to a model, not to the formulas.

What is more, your last sentence ''It is easily recognized then, that amendments have to be found.'', while more or less contentually correct, sounds akward (to me).

For completeness, here is a correct translation:

Finally, I permit myself to point out that in the foregoing developments I always assumed the nowhere-concave body to lie entirely within a finite region. If, however, the geometry defined by the original axioms contains a line and a point of such a kind that there is a unique parallel to the line through this point, then this assumption is not justified. It is easily seen what modifications have to be made to my discussion then.

Kleinteich, by the Baltic resort Rauschen, August 14, 1894

Detail.

The 'Kleinteich' in the last line of Hilbert's above-cited appendix seems to be the name of a no longer existing district of the already small town 'Rauschen': in the old book

Rudolf Friedrich Reusch, ``Sagen des preußischen Samlandes aus dem Munde des Volks erzählt'' Königsberg 1838. Druck und Verlag der Hartungschen Hofbuchdruckerei

one finds:

which means

The castle-mountain. The castle-mountain is situated near Kleinteich, which is a part of the village Rauschen. This mountain is essentially nothing but a foothill of the elevations which encircle Rauschen on its seaside, half-round, almost without shrubbery, and covered by ling.