The thirteen books of Euclid's Elements Volume 1

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1908 Excerpt: ..."parallel" and "cutting" which average human intelligence can readily grasp. This is the method adopted by Euclid in his definition, which of course belongs to the group (1) of definitions regarding parallels as non-secant. (It is significant, I think, that such authorities as Ingrami (Elementi di geometria, 1904) and Enriques and Amaldi (Elementi di geometria, 1905), after all the discussion of principles that has taken place of late years, give I definitions of parallels equivalent to Euclid's: "those straight lines in a plane which have not any point in common are called parallels." Hilbert adopts the same point of view. Veronese, it is true, takes a different line. In his 1 great work Fondamenti di geometria, 1891, he had taken a ray to be parallel to another when a point at infinity on the second is situated on the first; but he I appears to have come to the conclusion that this definition was unsuitable for his Elementi. He avoids however giving the Euclidean definition of parallels I as "straight lines in a plane which, though produced indefinitely, never meet," 'because "no one has ever seen two straight lines of this sort," and because the postulate generally used in connexion with this definition is not evident in the way that, in the field of our experience, it is evident that only one straight line can pass through two points. Hence he gives a different definition, for which he claims the advantage that it is independent of the plane. It is based on a definition of figures "opposite to one another with respect to a point" (or reflex figures). "Two figures are opposite to one another with respect to a point O, e.g. the figures ABC... and A'B'C..., if to every point of the one there ...