In two letters to Gerling, Gauss[5] expresses his regret that certain theorems of solid geometry depend upon the method of exhaustion, i. e. in modern phraseology, upon the axiom of continuity (or upon the axiom of Archimedes). Gauss mentions in particular the theorem of Euclid, that triangular pyramids of equal altitudes are to each other as their bases. Now the analogous problem in the plane has been solved.[6] Gerling also succeeded in proving the equality of volume of symmetrical polyhedra by dividing them into congruent parts. Nevertheless, it seems to me probable that a general proof of this kind for the theorem of Euclid just mentioned is impossible, and it should be our task to give a rigorous proof of its impossibility. This would be obtained, as soon as we succeeded in specifying two tetrahedra of equal bases and equal altitudes which can in no way be split up into congruent tetrahedra, and which cannot be combined with congruent tetrahedra to form two polyhedra which themselves could be split up into congruent tetrahedra.[7] [5] Werke, vol. 8, pp. 241 and 244. [6] Cf., beside earlier literature, Hilbert, Grundlagen der Geometric, Leipzig, 1899, ch. 4. [Translation by Townsend, Chicago, 1902.] [7] Since this was written Herr Dehn has succeeded in proving this impossibility. See his note: "Ueber raumgleiche Polyeder," in Nachrichten d. K. Gesellsch. d. Wiss. zu Göttingen, 1900, and a paper soon to appear in the Math. Annalen [vol. 55, pp. 405-478]. 4. PROBLEM OF THE STRAIGHT LINE AS THE SHORTEST DISTANCE BETWEEN TWO POINTS.