By Otto Schreier, Emanuel Sperner

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**Additional resources for Projective Geometry of N Dimensions (Volume Two of Introduction to Modern Algebra and Matrix Theory)**

**Example text**

Alten] First, he mentions the art of counting and calculating; we will skip its justiﬁcation here. Then, the dialogue partners agree on geometry as the second science. Again, its use for commanders of war is cited ﬁrst: “Clearly, he said, we are concerned with that part of geometry which relates to war; for in pitching a camp, or taking up a position, or closing or extending the lines of an army, or any other military manoeuvre, whether in actual battle or on a march, it will make all the diﬀerence whether a general is or is not a geometrician.

This supposedly means that geometry was studied for its own sake, in contrast to acting according to rules given by necessities, practical life or sacred purposes. Hence, here lies the beginning of the development of pure mathematics. The number was at the core of Pythagoras’s religious-philosophical teaching. , the Pythagoreans developed the ﬁrst foundations of number theory as well as a musical theory based on simple numeric ratios. When dealing with mathematical questions, they had the devastating cognition that irrational ratios exist.

Each of the four corner squares has the outer half (along a diagonal) cut oﬀ, which gives you an octagon approximated to the circle. Calculate its area and approximate it to the most proximate square root, from which you extract the root! Accordingly, this root is the side length of a square, which together with the circle of diameter d = 9 are almost of equal area. To what extent does the Egyptian formula result from that? b) In order to judge the quality of this approximation formula, calculate the resulting value for n with two digits after the comma!

### Projective Geometry of N Dimensions (Volume Two of Introduction to Modern Algebra and Matrix Theory) by Otto Schreier, Emanuel Sperner

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