CHAPTER XXXIV 

ON GEOMETRY
At the same time it will not be forgotten that the physical reality of geometry can not be put in evidence with full clarity unless there is an abstract theory also. . . . Thus, for example, while the term electron may have more than one physical meaning, it is by no means such a protean object as a point or a triangle. (259) Oswald veblen 



Euclidean space is simply a group. (4in 
HENRI POINCARE 



It is only in Euclidean "gravitationless" geometry that integrability obtains. (55d Hermann weyl
The fundamental fact of Euclidean geometry is that the square of the distance between two points is a quadratic form of the relative coordinates of the two points (Pythagoras' Theorem). But if we look upon this law as being strictly valid only for the case when these two points are infinitely near, we enter the domain of Riemann's geometry. (347) Hermann weyl
... parallel displacement of a vector must leave unchanged the distance which it determines. Thus, the principle of transference of distances or lengths which is the basis of metrical geometry, carries with it a principle of transference of direction; in other words, an amne relationship is inherent in metrical
Space. (547) HERMANN WEYL
But before dealing with the brain, it is well to distinguish a second characteristic of nervous organization which renders it an organization in
levels. 1411) HENRY PIERON 

Section A. Introductory.
The main metrical rule in geometry is the familiar Pythagorean theorem. In 1933 this rule is no longer considered as generally valid outside of the euclidean system, as its proof depends on the doubtful postulate of parallels. It is considered as an empirical generalization in which the relative error decreases when the distances become smaller. Indeed the small element of length, ds, given by the pythagorean rule is considered convenient and reliable in our exploration of the world.
The pythagorean rule states that in any right triangle, ABC, the square of the side opposite the right angle (the hypotenuse) is equal to the sum of the squares of the two other sides (the legs). In symbols,. If we build squares
on all three sides of the triangle ABC and denote the areas of the squares by C, A', and B' then we have
The above rule is also the main metrical rule for coordinate geometry, which gives us the length of the line segment joining any two points. Consider, for example, two points in two dimensions, Pi and Pj, whose
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