The geometry of such a surface (of positive curvature) is called Riemannian or Gaussian.

The plane satisfies hypothesis III if it be assumed that no other such line may be drawn. The geometry of the plane is termed Euclidean, and IIb is true on the pseudosphere. Through a point outside a given line two parallels to the line may be drawn. The appearance of the parallels is indicated by the figure. The geometry of a surface of constant negative curvature is called Lobatschewskian.

A

B

Fig. 51 PARALLELS ON A PSEUDO-SPHERICAL SURFACE.

The curvature of a point (regarded as a sphere of zero radius) is infinite. Starting with a point, let the radius. increase and curvature decrease. As the curvature runs continuously through the values from +∞ down to zero the surface has a Riemannian geometry of no parallels. When curvature passes through zero, for an instant the surface is a plane with the property of one parallel, the curvature becoming negative. The Lobatschewskian geometry applies and there are two parallels. Continuing the curvature becomes larger and larger negatively, with radius becoming smaller until finally the surface closes up again into a point and the complete course has been run. Paralleling the case with the conic section, the parabola was seen to be the boundary between the ellipse and the

hyperbola. So the Riemannian geometry is said to be elliptic, the plane parabolic and the pseudosphere hyperbolic; these terms come, however, from a different property of the spaces.

It is a curious fact that in the simple Riemannian plane the straight line cuts through the plane without cutting it in two. This cut cannot well be pictured, but an idea of its meaning may be got by thinking of the surface of a ring with a cut extending around the outside of it.

In Lobatchevskian space the unit of measure is a continuously decreasing length, while in Riemannian space it is continuously increasing.

Riemann, in his celebrated paper on 'The Hypotheses which Lie at the Basis of Geometry,' first advanced the theory that space might be unbounded without being infinite, in these words: In the extension of space-construction to the infinitely great, one must distinguish between unboundedness and infinite extent. That space is an unbounded three-fold manifoldness is an assumption which is developed by every conception of the outer world. The unboundedness of space possesses a greater empirical certainty than any external appearance. But its infinite extent by no means follows. On the other hand, if we assume independence of bodies from position, and therefore ascribe to space constant curvature, it must necessarily be finite, provided this curvature has ever so small a positive value. If we prolong all the geodesics from one point in a surface of constant curvature, this surface would take the form of a sphere.

The question as to whether the space of experience is Euclidean, Lobatchevskian or Riemannian is one which can never be determined. Are there two parallels, one or none? could only be settled in one of two ways, by reason or by measurement. A better form for the question is as to whether the sum of the angles of a triangle is less than, equal to, or greater than two right angles. As to reason, the geometry of one hypothesis is just as consistent as

that of another. As to measurement, it is conceivable that an error in the measurement of the three angles of a triangle which may be drawn on this page would not show an error which would easily be detected if the triangle were drawn with sides 10 miles in length.

The largest triangles ever possible to measure have as a side the diameter of the earth's orbit, the opposite vertex being a celestial body. That no deviation from two right angles in the sum for this triangle is found is no evidence that if it were a million times as great the deviation would not be appreciable. The most that can be said is that if space is curved, the curvature is slight.

The study of non-Euclidean spaces enables one better to appreciate the insight of the old Greek geometer who 2,000 years ago realized that the proof of his fifth postulate was beyond his powers.

All measurement in mathematics is concerned either with that of lines or of angles. Euclid developed a complete theory of measurement of lines, but aside from the right angle and several of its exact divisors—as % of a right angle, etc.—the only relations which he determined were those of greater and less; thus, if the sides of a triangle are 3, 4, 5, it is known by geometry that the angle opposite the side 5 is a right angle, and further that the angle opposite the side 4 is greater than that opposite 3, but exactly how much Euclid gives us no means of determining.

CHAPTER V

TRIGONOMETRY

TRIGONOMETRY is the science of the triangle with reference to the particular problem of finding the value of the unknown parts when three independent parts are given, as finding the angles when the three sides are given, etc. In a right triangle, ABC, lettered as in Fig. 49, six ratios are involved, which remain the same so long as the angles are not changed, the size of the triangle changing at will but preserving its shape. These six ratios are functions of the angles; that is, they depend for their value upon the values of the angles. They are named in the table below, with the abbreviations usually assigned to them given last.

In the first column the functions are arranged in pairs, the second of the pair having its name from the first, with the prefix co. The origin of this prefix is from the rela

A

B

C

Fig. 52 RATIOS OF A TRIANGLE.

tion which exists between A and B, the sum of which is I right angle. It therefore takes B to fill a right angle together with A, or B is said to be the complement of A or co-A. Looking at the second column, one sees that b

the ratio is the sin B or sin of co-A or cosine A.

C

These six ratios were originally used in connection with a right triangle alone. When it became desirable to consider angles greater than I right angle, such angles not being found in a right triangle, the definitions for sine,