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

### Indhold

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### Andre udgaver - Se alle

### Almindelige termer og sætninger

A₁ absolute polar antipodal points axiom axis B₁ B₂ Bolyai bundle of lines C₁ centre Chap circle circumcircles coincide common perpendicular conic conjugate constant coordinates cosh cross-ratio curve dihedral angle distance draw elliptic geometry equation equidistant equidistant-curve Euclid euclidean geometry exterior angle fixed line fixed point formulae Gauss given line Hence homographic horocycle horosphere hyperbolic geometry hypothesis ideal points imaginary inversion 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 right-angled triangle segments sides sinh space sphere straight line tangents tanh theorem theory of parallels transformation triangle ABC unit of length vertex

### Populære passager

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 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 25 - ... It determines not how, nor where anything must be brought to pass. It limits not the Holy One, and leaves all things possible with God, to be executed according to the good pleasure of his will. The laws of nature are uniform, not by any compulsory necessity that they should be so, as that the sum of the angles of a triangle must be equal to two right angles; not because God could not, for cause, wholly change their working, as he has been pleased to do in the case of miracles ; but because,...

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

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 - If there exists a single triangle in which the sum of the angles is equal to two right angles, then in every triangle the sum of the angles must likewise be equal to two right angles.

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 80 - Ufcits of area in the area of a rectangle is equal to the product °f the numbers of units of length in its sides. It would take us too far out of our way to examine completely the notion of area. We shall simply take advantage of the fact, that when we are dealing with a very small region of the plane we can apply euclidean geometry. Thus, while there exists no such thing as a euclidean square in non-euclidean geometry, if we take a regular quadrilateral 1 with all its sides very small we may take...