...and results of Lobatchevsky. Lobatchevsky begins his original matter with his Theorem 16 as follows : "All straight lines which in a plane go out from a...those lines will be called parallel to the given line. "From the point A (Fig. i) let fall upon the line BC the perpendicular AD, to which again draw the...

...or a side and the adjacent angles equal. From here follow the other theorems wilh their eaplanations and proofs. 16. All straight lines which in a plane...those lines will be called parallel to the given line. From the point A (Fig. 1 ) let fall upon the line BC the perpendicular AD, to which again draw the...

...hypothenuse is greater than either of the other sides, and the two angles adjacent to it are acute. From here follow the other theorems with their explanations...those lines will be called parallel to the given line. From the point A (Fig. 1.11) let fall upon the line BC the perpendicular AD, to which again draw the...

...all (Prop. 20). The new definition of parallelism (Prop. 16) introduces the nonEuclidean atmosphere: All straight lines which in a plane go out from a...given straight line in the same plane, be divided in two classes - into cutting and not-cutting. The boundary lines of the one and the other class of...

...necessary outcome of human reason? In 1840, Nikolai Ivanovich Lobachevsky (1792-1856) argued that: All straight lines which in a plane go out from a...plane, be divided into two classes into cutting and non-cutting. The boundary lines of the one and the other class of those lines will be called parallel...

...statement above. Properly analysed, he claimed, it must go as follows: All straight lines in a plane which go out from a point can, with reference to a given...those lines will be called parallel to the given line. From the point A let fall upon the line BC the perpendicular AD, to which again draw the perpendicular...

...of all, that there is a group of theorems which are independent of the fifth postulate. He considers "all straight lines which in a plane go out from a...line in the same plane, be divided into two classes" [Lobachevsky, §16, Bonola (1955), p. 13]: the "cutting" and "non-cutting". And "the boundary lines...

...a straight line having two ends be infinite in length ? VI. In his theorem 16, Lobatschewsky says: "All straight lines which in a plane go out from a...lines will be called parallel to the given line." Does Lobatschewsky regard the boundary line as a cutting or a not-cutting line, or neither ? In proposition...