Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/276

252 THE FIRST BOOK.

Definitions. The first seven definitions have given rise to considerable discussion, on which however we do not propose to enter. Such a discussion would consist mainly of two subjects, both of which are unsuitable to an elementary work, namely, an examination of the origin and nature of some of our elementary ideas, and a comparison of the original text of Euclid with the substitutions for it proposed by Simson and other editors. For the former subject the student may hereafter consult Whewell's History of Scientific Ideas and Mill's Logic, and for the latter the notes in Camerer's edition of the Elements of Euclid.

We will only observe that the ideas which correspond to the words point, line, and surface, do not admit of such definitions as will really supply the ideas to a person who is destitute of them. The so-called definitions may be regarded as cautions or restrictions. Thus a point is not to be supposed to have any size, but only position; a line is not to be supposed to have any breadth or thickness, but only length; a surface is not to be supposed to have any thickness, but only length and breadth.

The eighth definition seems intended to include the cases in which an angle is formed by the meeting of two curved lines, or of a straight line and a curved line; this definition however is of no importance, as the only angles ever considered are such as are formed by straight lines. The definition of a plane rectilineal angle is important; the beginner must carefully observe that no change is made in an angle by prolonging the lines which form it, away from the angular point.

Some writers object to such definitions as those of an equilateral triangle, or of a square, in which the existence of the object defined is assumed when it ought to be demonstrated. They would present them in such a form as the following: if there be a triangle having three equal sides, let it be called an equilateral triangle.

Moreover, some of the definitions are introduced prematurely. Thus, for example, take the definitions of a right-angled triangle and an obtuse-angled triangle; it is not shewn until I. 1 7, that a triangle cannot have both a right angle and an obtuse angle, and so cannot be at the same time right-angled and obtuse-angled. And before Axiom 11 has been given, it is conceivable