A rational integral function or form in any number of variables with real coefficients such that it becomes negative for no real values of these variables, is said to be definite. The system of all definite forms is invariant with respect to the operations of addition and multiplication, but the quotient of two definite forms—in case it should be an integral function of the variables—is also a definite form. The square of any form is evidently always a definite form. But since, as I have shown,[38] not every definite form can be compounded by addition from squares of forms, the question arises—which I have answered affirmatively for ternary forms[39]—whether every definite form may not be expressed as a quotient of sums of squares of forms. At the same time it is desirable, for certain questions as to the possibility of certain geometrical constructions, to know whether the coefficients of the forms to be used in the expression may always be taken from the realm of rationality given by the coefficients of the form represented.[40]
I mention one more geometrical problem:
[38] Math. Annalen, vol. 32.
[39] Acta Mathematica, vol. 17.
[40] Cf. Hilbert: Grunglagen der Geometrie, Leipzig, 1899, Chap. 7 and in particular § 38.
18. BUILDING UP OF SPACE FROM CONGRUENT POLYHEDRA.
If we enquire for those groups of motions in the plane for which a fundamental region exists, we obtain various answers, according as the plane considered is Riemann's (elliptic), Euclid's, or Lobachevsky's (hyperbolic). In the case of the elliptic plane there is a finite number of essentially different kinds of fundamental regions, and a finite number of congruent regions suffices for a complete covering of the whole plane; the group consists indeed of a finite number of motions only. In the case of the hyperbolic plane there is an infinite number of essentially different kinds of fundamental regions, namely, the well-known Poincaré polygons. For the complete covering of the plane an infinite number of congruent regions is necessary. The case of Euclid's plane stands between these; for in this case there is only a finite number of essentially different kinds of groups of motions with fundamental regions, but for a complete covering of the whole plane an infinite number of congruent regions is necessary.
Exactly the corresponding facts are found in space of three dimensions. The fact of the finiteness of the groups of motions in elliptic space is an immediate consequence of a fundamental theorem of C. Jordan,[41] whereby the number of essentially different kinds of finite groups of linear substitutions in
variables does not surpass a certain finite limit dependent upon