=Bibliography.= Smith, Teaching of Elementary Mathematics, New York, 1900; Young, The Teaching of Mathematics, New York, 1901; Moore, On the Foundations of Mathematics, _Bulletin of the American Mathematical Society_, 1903, p. 402; Betz, Intuition and Logic in Geometry, _The Mathematics Teacher_, Vol. II, p. 3; Hilbert, The Foundations of Geometry, Chicago, 1902; Veblen, A System of Axioms for Geometry, _Transactions of the American Mathematical Society_, 1904, p. 343. FOOTNOTES: [42] From the Greek [Greek: ge], _ge_ (earth), + [Greek: metrein], _metrein_ (to measure), although the science has not had to do directly with the measure of the earth for over two thousand years. [43] From the Arabic _al_ (the) + _jabr_ (restoration), referring to taking a quantity from one side of an equation and then restoring the balance by taking it from the other side (see page 37). [44] One of the clearest discussions of the subject is in W. B. Frankland, "The First Book of Euclid's 'Elements,'" p. 26, Cambridge, 1905. [45] "Grundlagen der Geometrie," Leipzig, 1899. See Heath's "Euclid," Vol. I, p. 229, for an English version; also D. E. Smith, "Teaching of Elementary Mathematics," p. 266, New York, 1900. [46] We need frequently to recall the fact that Euclid's "Elements" was intended for advanced students who went to Alexandria as young men now go to college, and that the book was used only in university instruction in the Middle Ages and indeed until recent times. [47] For example, he moves figures without deformation, but states no postulate on the subject; and he proves that one side of a triangle is less than the sum of the other two sides, when he might have postulated that a straight line is the shortest path between two points. Indeed, his followers were laughed at for proving a fact so obvious as this one concerning the triangle. [48] T. L. Heath, "Euclid," Vol. I, p. 200. [49] For a resume of the best known attempts to prove this postulate, see Heath, "Euclid," Vol. I, p. 202; W. B. Frankland, "Theories of Parallelism," Cambridge, 1910. [50] For the early history of this movement see Engel and Staeckel, "Die Theorie der Parallellinien von Euklid bis auf Gauss," Leipzig, 1895; Bonola, Sulla teoria delle parallele e sulle geometrie non-euclidee, in his "Questioni riguardanti la geometria elementare," 1900; Karagiannides, "Die nichteuklidische Geometrie vom Alterthum bis zur Gegenwart," Berlin, 1893. [51] This limitation upon elementary geometry was placed by Plato (died 347 B.C.), as already stated. [52] Book I, Proposition 20. CHAPTER XII THE DEFINITIONS OF GEOMETRY