INTRODUCTION In 1827 Gauss presented to the Royal Society of Goettingen his important paper on the theory of surfaces, which seventy-three years afterward the eminent French geometer, who has done more than any one else to propagate these principles, characterizes as one of Gauss's chief titles to fame, and as still the most finished andusefulintroductiontothestudyofinfinitesimalgeometry.? Thismemoirmay be called: General Investigations of Curved Surfaces, or the Paper of 1827, to distinguish it from the original draft written out in 1825, but not published until 1900. A list of the editions and translations of the Paper of 1827 follows. There are three editions in Latin, two translations into French, and two into German. The paper was originally published in Latin under the title: Ia. Disquisitiones generales circa superficies curvas auctore Carolo Friderico Gauss. Societati regiae oblatae D. 8. Octob. 1827, and was printed in: Commentationes societatis regiae scientiarum Gottingensis recentiores, Commentationes classis mathematicae. Tom. VI. (ad a. 1823-1827). Gottingae, 1828, pages 99-146. This sixth volume is rare; so much so, indeed, that the British Museum Catalogue indicates that it is missing in that collection. With the signatures changed, and the paging changed to pages 1-50, Ia also appears with the title page added: Ib. Disquisitiones generales circa superficies curvas auctore Carolo Friderico Gauss. Gottingae. Typis Dieterichianis. 1828. II. In Monge's Application de l'analyse a la geometrie, fifth edition, edited by Liouville, Paris, 1850, on pages 505-546, is a reprint, added by the Editor, in Latin under the title: Recherches sur la theorie generale des surfaces courbes; Par M. C.-F. Gauss. IIIa. A third Latin edition of this paper stands in: Gauss, Werke, Her- ausgegeben von der Koeniglichen Gesellschaft der Wissenschaften zu Goettingen, Vol. 4, Goettingen, 1873, pages 217-258, without change of the title of the original paper (Ia). IIIb. The same, witho
