Anwendung der Grassmann'schen Ausdehnungslehre auf die by Rudolf Mehmke PDF

By Rudolf Mehmke

This ebook was once initially released sooner than 1923, and represents a replica of an incredible old paintings, keeping an analogous structure because the unique paintings. whereas a few publishers have opted to follow OCR (optical personality popularity) expertise to the method, we think this results in sub-optimal effects (frequent typographical error, unusual characters and complicated formatting) and doesn't safely protect the ancient personality of the unique artifact. We think this paintings is culturally vital in its unique archival shape. whereas we attempt to thoroughly fresh and digitally improve the unique paintings, there are sometimes cases the place imperfections corresponding to blurred or lacking pages, bad photos or errant marks could have been brought as a result of both the standard of the unique paintings or the scanning strategy itself. regardless of those occasional imperfections, we've introduced it again into print as a part of our ongoing worldwide ebook maintenance dedication, delivering buyers with entry to the absolute best ancient reprints. We delight in your knowing of those occasional imperfections, and essentially wish you take pleasure in seeing the booklet in a structure as shut as attainable to that meant through the unique writer.

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This booklet used to be initially released sooner than 1923, and represents a duplicate of a tremendous historic paintings, conserving an analogous structure because the unique paintings. whereas a few publishers have opted to observe OCR (optical personality popularity) expertise to the method, we think this ends up in sub-optimal effects (frequent typographical blunders, unusual characters and complicated formatting) and doesn't safely shield the ancient personality of the unique artifact.

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Additional info for Anwendung der Grassmann'schen Ausdehnungslehre auf die Geometrie der Kreise in der Ebene

Example text

2 we find a point E not lying on the line AD (see Fig. 1). Then we apply the axiom A10 to the points C and E. As a result on the line CE we find a point F such that the point E lies in the interior of the segment [CF ]. Let’s draw the lines AE and F B, then consider the triangle F BC. The line AE crosses the line F C at the point E which is an interior point for the segment [F C]. The intersection of the lines AE and BC coincides with the point A which is outside the segment [BC]. Therefore, according to Pasch’s axiom A12, the line AE should cross the side [F B] of the triangle F BC at some interior point G.

But the point Ai+1 does not belong to the segment [Aj , Aj+1 ]. 3). , 1998, 2007. 36 CHAPTER II. AXIOMS OF INCIDENCE AND ORDER. 1. Let A1 , . . , An be a monotonic sequence of points on a straight line and let B be some point of this line coinciding with none of the points A1 , . . , An . Then one can join the point B to the points A1 , . . , An and enumerate the resulting set of points so that the monotonic sequence of points A1 , . . , An+1 will be formed. Proof. Let’s consider the three points A1 , An , and B.

Proof. Assume that pAB and pCD are the mappings of congruent translation on a straight line a and assume that pAB = pCD = p. Let’s define the positive direction on the −→ line a by means of the vector AC and thus define a relation of precedence for the points of this line a. Then A ≺ C. 2: p(A) ≺ p(C), [p(A)p(C)] ∼ = [AC]. But p(A) = pAB (A) = B and p(C) = pCD (C) = D. Hence, −→ −→ A ≺ C and B ≺ D, which means that the vectors AC and BD −→ −→ are codirected. Moreover, [BD] ∼ = [AC], therefore, AC = BD.

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Anwendung der Grassmann'schen Ausdehnungslehre auf die Geometrie der Kreise in der Ebene by Rudolf Mehmke


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