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B i, • • • ~r Bk = C = Cx + . . + Ct\ and suppose further that the formulas A\ = /i (Bf), • • Ak = f h (Bk)y Cx = gi (B{), . . • . , C i = gi (Bt) hold, where f x, . f 1n gly . g t are certain motions (so that A x = B l9 Cx = B X9 and so on). These formulas just mean that each of the figures A and C is equidecomposable with B . Let F tj = B t p|B j , where i =■ 1 , . . , 7c; j = 1 , . ,Z (note that some 7The father of the famous J. Bolyai, who arrived at the non-Euclidean geometry independently o f N.

This gives a rectangle aefb which is equidecomposable with the triangle abc. In fact, the triangles marked 1 in Fig. 28 are congruent, and so are those marked 2. But each of the figures abc and aefb consists of the shaded trapezoid and a pair of triangles marked 1 and 2 . Lemma 3. Any two rectangles with the same area are equi decomposable. In fact, we arrange the rectangles of equal area (the rectangles oabc and omnp in Fig. 30) in such a way that they have a common right angle. , FIG. 28 = -47- , which implies FIG.

We choose polygons G and H such that G CL F d II and s (H) — s (G) < e (Theorem3). H ') — s (£') = 5 (if) — 5 (G) < 8. It follows that the figure i*' = g (F) is measurable. The validity of property (7 *) for arbitrary measur able figures is now proved in just the same way as for rectangles (with a reference to Theorem 5 rather than to Theorem 2). The statement (7 *) is a more general property of area than axiom (7 ), and it is sometimes chosen as an axiom instead of (7 ), leading to the construction of a theory of area based on the properties (a), (P)> (Y*)> (6)* regarded as axioms.