In Fig. 25 the circle represents a meridian section of the earth; _P P´_ is the axis about which it rotates, and the dotted lines represent a beam of light coming from a star in the plane of the meridian, and so distant that the dotted lines are all practically parallel to each other. The several radii drawn through the points _1_, _2_, _3_, represent the direction of the vertical at these points, and the angles which these radii produced, make with the rays of starlight are each equal to the angular distance of the star from the zenith of the place at the moment the star crosses the meridian. We have already seen, in Chapter II, how these angles may be measured, and it is apparent from the figure that the difference between any two of these angles--e. g., the angles at _1_ and _2_--is equal to the angle at the center, _O_, between the points _1_ and _2_. By measuring these angular distances of the star from the zenith, the astronomer finds the angles at the center of the earth between the stations _1_, _2_, _3_, etc., at which his observations are made. If the meridian were a perfect circle the change of zenith distance of the star, as one traveled along a meridian from the equator to the pole, would be perfectly uniform--the same number of degrees for each hundred miles traveled--and observations made in many parts of the earth show that this is very nearly true, but that, on the whole, as we approach the pole it is necessary to travel a little greater distance than is required for a given change in the angle at the equator. The earth is, in fact, flattened at the poles to the amount of about 27 miles in the length of its diameter, and by this amount, as well as by smaller variations due to mountains and valleys, the shape of the earth differs from a perfect sphere. These astronomical measurements of the curvature of the earth's surface furnish by far the most satisfactory proof that it is very approximately a sphere, and furnish as its equatorial diameter 7,926 miles. Neglecting the _compression_, as it is called, i. e., the 27 miles by which the equatorial diameter exceeds the polar, the size of the earth may easily be found by measuring the distance _1_--_2_ along the surface and by combining with this the angle _1 O 2_ obtained through measuring the meridian altitudes of any star as seen from _1_ and _2_. Draw on paper an angle equal to the measured difference of altitude and find how far you must go from its vertex in order to have the distance between the sides, measured along an arc of a circle, equal to the measured distance between _1_ and _2_. This distance from the vertex will be the earth's radius.