## The Elements of Non-Euclidean GeometryG. Bell, 1914 - 274 sider |

### Indhold

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### Andre udgaver - Se alle

### Almindelige termer og sætninger

absolute polar antipodal points axioms axis B₁ Bolyai bundle of lines C₁ called centre Chap circle circumcircles coincide common perpendicular conic conjugate constant coordinates cosh coth cross-ratio curve determine dihedral angle distance draw elliptic geometry equal equation equidistant equidistant-curve Euclid euclidean geometry fixed point formulae Gauss given line Hence homographic horocycle horosphere hyperbolic geometry ideal points imaginary infinity or ideal involution line at infinity lines and planes Lobachevsky locus marginal images non-euclidean geometry non-intersecting opposite orthogonal pairs parallel lines paratactic passes pencil of lines point of intersection points at infinity projective geometry prove quadric quadrilateral radius ratios represented right angles second degree segment sides Similarly sinh space sphere straight line surface tangents tanh theorem three dimensions transformation triangle ABC unit of length vertex y₁

### Populære passager

Side 21 - Euclid, but in which the famous postulate is assumed false, and in which the sum of the angles of a triangle is always less than two right angles.

Side 85 - The diagonals of a quadrilateral intersect at right angles. Prove that the sum of the squares on one pair of opposite sides is equal to the sum of the squares on the other pair.

Side 12 - PROPOSITION 17. The sum of any two angles of a triangle is less than two right angles.

Side 3 - That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

Side 34 - Euclid, eg first asserts and proves, that the exterior angle of a triangle is greater than either of the interior opposite angles...

Side 23 - I have made such wonderful discoveries that I am myself lost in astonishment, and it would be an irreparable loss if they remained unknown. When you read them, dear Father, you too will acknowledge it. I cannot say more now except that out of nothing I have created a new and another world. All that I have sent you hitherto is as a house of cards compared to a tower.

Side 220 - Two circles touch at A ; T is any point on the tangent at A; from T are drawn tangents TP, TQ to the two circles. Prove that TP = TQ. What is the locus of points from which equal tangents can be drawn to two circles in contact ? tEx.

Side 18 - C'. We thus obtain a curvilinear triangle in which the sum of the angles is less than two right angles ; and since the angles in this triangle are equal to those in the nominal triangle, our result is proved.

Side 1 - It is impossible to say when electricity was first discovered. Records show that as early as 600 BC the attractive properties of amber were known. Thales of Miletus (640-546 BC), one of the "seven wise men...

Side 154 - The fact that a straight line can be represented by an equation of the first degree enables us to represent noneuclidean straight lines by straight lines on the euclidean plane.