8. Set up a one-to-one correspondence between the series of numbers _1_, _2_, _3_, _4_, ... and the series of even numbers _2_, _4_, _6_, _8_ .... Are we justified in saying that there are just as many even numbers as there are numbers altogether? 9. Is the axiom "The whole is greater than one of its parts" applicable to infinite assemblages? 10. Make out a classified list of all the infinitudes of the first, second, third, and fourth orders mentioned in this chapter. CHAPTER II - RELATIONS BETWEEN FUNDAMENTAL FORMS IN ONE-TO-ONE CORRESPONDENCE WITH EACH OTHER *23. Seven fundamental forms.* In the preceding chapter we have called attention to seven fundamental forms: the point-row, the pencil of rays, the axial pencil, the plane system, the point system, the space system, and the system of lines in space. These fundamental forms are the material which we intend to use in building up a general theory which will be found to include ordinary geometry as a special case. We shall be concerned, not with measurement of angles and areas or line segments as in the study of Euclid, but in combining and comparing these fundamental forms and in "generating" new forms by means of them. In problems of construction we shall make no use of measurement, either of angles or of segments, and except in certain special applications of the general theory we shall not find it necessary to require more of ourselves than the ability to draw the line joining two points, or to find the point of intersections of two lines, or the line of intersection of two planes, or, in general, the common elements of two fundamental forms.