Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/278

254 the original Greek. They are objectionable, because lines and angles are magnitudes to which the axiom may be applied, but they cannot be said to fill space.

On the method of superposition we may refer to papers by Professor Kelland in the Transactions of the Royal Society of Edinburgh, Vols. , and.

The eleventh axiom is not required before I. 14, and the twelfth axiom is not required before I. 29; we shall not consider these axioms until we arrive at the propositions in which they are respectively required for the first time.

The first book is chiefly devoted to the properties of triangles and parallelograms.

We may observe that Euclid himself does not distinguish between problems and theorems except by using at the end of the investigation phrases which correspond to and respectively.

I. 2. This problem admits of eight cases in its figure. For it will be found that the given point may be joined with either end of the given straight line, then the equilateral triangle may be described on either side of the straight line which is drawn, and the sides of the equilateral triangle which are produced may be produced through either extremity. These various cases may be left for the exercise of the student, as they present no difficulty.

There will not however always be eight different straight fines obtained which solve the problem. For example, if the point A falls on BC produced, some of the solutions obtained coincide; this depends on the fact which follows from I. 32, that the angles of all equilateral triangles are equal.

I. 5. "Join FC" Custom seems to allow this singular expression as an abbreviation for "draw the straight line FC," or for "Join F to C by the straight line FC"

In comparing the triangles BFC, CGB, the words " and the base BC is common to the two triangles BFC, CGB" are usually inserted, with the authority of the original. As however these words are of no use, and tend to perplex a beginner, we have followed the example of some editors and omitted them.

A corollary to a proposition is an inference which may be deduced immediately from that proposition. Many of the corollaries in the Elements are not in the original text, but introduced by the editors.