By Ferenc Kárteszi (Eds.)
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Extra info for Appendix: The Theory of Space
DX quod (nonnisi ab 9 dependens) iam primurn per y exprimerr. durn est; unde II intrgrando prodit. ) ad=p, ac=q, c t c d z r , ntqiie cab& dS =s sit; poterit (uti in 11;) ostenrli, esse m p , d cdn (uti hf d h Cot m dg 1 pi -9(e c --L ). Potest hoc absque integratione 8 s. , quoque deduci. Aequatione e. g, circuli (ex 31 111), rectae (ex 31, 111, sectionis c o d (per praec) expressis; poterunt areae quoque his lineis cJau0 sae exprimi. Palam est, superficiem z ad figuram planam p (in distantia q ) Itlam, esse ad p i n ratione potentiarurn 2darum linearum homologarum, sive uti 9 - 9 l ( e T t e 8 4 Porro-computllm soliditatis pari rnorio tracratnm.
Plane? In other words, he asked if there exist plane geometries, Euclidean in the small, which are different from the geometries of EUCLIDand of BOLYAI-LOBACHEVSKY. He failed to notice, however, that another, stronger assumption had crept into his starting hypothesis: according to this tacit assumption, all motions in the plane taking a point into any other point and all rotations about any point are possible. For brevity, call the explicit and the tacit assumption the local and the kinematic one, respectively.
Et r in vel extra fg cadit (si cd non =fg, ubi res jam patet). 1. In casu primo estfrs non (2R- r f m = f g n ) , quia r s j ~ ~ f mast ; cur11 TsIllgn s i t , est etiarn frs non f g n i adeoque f r s = f g n , et rfm-fft-s= g p ~ t fgn = 2 8 . Itaque e t dcpj-cdg t2R. II. Si T extra fg cadat ;tunc n g c m f r , sitque m f g n ~ n g h t s l h b aet ita porro, usquequo f k = v e l prima vice f r fiat. ). ); adeoque rfmt-frs = kfmj-fdo = 2R; si veto r in Irl cadat, turn (per z2R rfmvrs = dcp-fcdq.