Apropos of the definitions of geometry, the great French philosopher and mathematician, Pascal, set forth certain rules relating to this subject, as also to the axioms employed, and these may properly sum up this chapter. 1. Do not attempt to define terms so well known in themselves that there are no simpler terms by which to express them. 2. Admit no obscure or equivocal terms without defining them. 3. Use in the definitions only terms that are perfectly understood or are there explained. 4. Omit no necessary principles without general agreement, however clear and evident they may be. 5. Set forth in the axioms only those things that are in themselves perfectly evident. 6. Do not attempt to demonstrate anything that is so evident in itself that there is nothing more simple by which to prove it. 7. Prove whatever is in the least obscure, using in the demonstration only axioms that are perfectly evident in themselves, or propositions already demonstrated or allowed. 8. In case of any uncertainty arising from a term employed, always substitute mentally the definition for the term itself. =Bibliography.= Heath, Euclid, as cited; Frankland, The First Book of Euclid, as cited; Smith, Teaching of Elementary Mathematics, p. 257, New York, 1900; Young, Teaching of Mathematics, p. 189, New York, 1907; Veblen, On Definitions, in the _Monist_, 1903, p. 303. FOOTNOTES: [53] Free use has been made of W. B. Frankland, "The First Book of Euclid's 'Elements,'" Cambridge, 1905; T. L. Heath, "The Thirteen Books of Euclid's 'Elements,'" Cambridge, 1908; H. Schotten, "Inhalt und Methode des planimetrischen Unterrichts," Leipzig, 1893; M. Simon, "Euclid und die sechs planimetrischen Buecher," Leipzig, 1901. [54] For a facsimile of a thirteenth-century MS. containing this definition, see the author's "Rara Arithmetica," Plate IV, Boston, 1909. [55] Our slang expression "The cart before the horse" is suggestive of this procedure. [56] Loc. cit., Vol. II, p. 94. CHAPTER XIII HOW TO ATTACK THE EXERCISES The old geometry, say of a century ago, usually consisted, as has been stated, of a series of theorems fully proved and of problems fully solved. During the nineteenth century exercises were gradually introduced, thus developing geometry from a science in which one learned by seeing things done, into one in which he gained power by actually doing things. Of the nature of these exercises ("originals," "riders"), and of their gradual change in the past few years, mention has been made in Chapter VII. It now remains to consider the methods of attacking these exercises.