Il6	SPACE   AND   GEOMETRY
saccheri's theory of parallels.
Among the inquirers notable for their contributions to the theory of parallels are the Italian Sac-cheri and the German mathematician Lambert. In order to render their mode of attack intelligible, we will remark first that the existence of rectangles and squares, which we fancy we constantly observe, cannot be demonstrated without the aid of the Fifth Postulate. Let us consider, for example, two congruent isosceles triangles ABC, DEC, having right angles at A and D (Fig. 16), and let them be laid . together at their hypothenuses BC so as to form the

Fig. 16.
equilateral quadrilateral ABCD; the first twenty-seven propositions of Euclid do not suffice to determine the character and magnitude of the two equal (right) angles at B and C. For measure of length and measure of angle are fundamentally different and directly not comparable; hence the first propositions regarding the connection of sides and angles are qualitative only, and hence the imperative necessity of a quantitative theorem regarding angles, like that of the angle-sum. Be it further remarked that theorems analogous to the twenty-seven planimetric propositions of Euclid may be set up for the surface