106. All the theorems of "solid geometry" which relate merely to the surface of the sphere would be obtained in this way.

107. Some of the important differences between spherical geometry and plane geometry are that in spherical geometry

(a.) All theorems involving parallel lines are lacking.

(b.) The sum of the angles of a triangle is greater than two right angles.

(c.) Figures cannot be similar.

(d.) The area of a polygon is measured by the sum of its anglesthat is, by its spherical excess.

108. If the surface or covering of a pseudo-sphere be detached from the pseudo-sphere any surface into which it can be bent without stretching is a surface of constant negative curvature. geometry of such a surface is called pseudo-spherical geometry.

The

109. The straight-line axiom and the superposition axiom are true of the pseudo-spherical surface.

110. Through a given point of the pseudo-spherical surface two straight lines can be drawn to meet a given straight line at infinity, one meeting it at infinity in each direction. Consequently on this surface the following must be substituted for the parallel axiom :

Through a given point two straight lines can be drawn parallel to a given straight line.

On this surface two lines perpendicular to the same straight line diverge. The appearance of lines perpendicular to the same line on the plane, spherical, and pseudo-spherical surfaces respectively is shown in the above pictures.

111. Pseudo-spherical geometry can be built up from the axioms which are true on the pseudo-spherical surface. Some of the important differences between it and plane geometry are, that in pseudo-spherical geometry

(a.) Theorems which assume that non-parallel lines must meet are

not true.

(b.) Theorems involving parallelism must conform to the parallel axiom for a pseudo-spherical surface.

(c.) The sum of the angles of a triangle is less than two right angles. (d.) Figures cannot be similar.

(e.) The area of a triangle is measured by two right angles less the sum of its angles-that is, by its pseudo-spherical deficiency.

112. Remark.-The circumference of a circle on the plane surface =27; on the spherical surface the circumference is less than 2′′; on the pseudo-spherical surface the circumference is greater than The relation of the areas of circles on the three surfaces is the

277.

same.

113. There are many theorems which are true in the three kinds of geometry, such as

The sum of the two adjacent angles which one straight line makes with another straight line is equal to two right angles.

Every point in the perpendicular erected at the middle of a straight line is equally distant from the extremities of that line.

If two angles of a triangle are unequal the sides opposite to them are unequal and the greater side is opposite the greater angle.

The three bisectors of the angles of a triangle meet in a point.

NOTE.—The pseudo-sphere, page 527, is generated by revolving the curve whose equation is

about its y-axis. The radius of the base of the pseudo-sphere is a.