INTRODUCTORY REMARKS.

The subject of Plane Geometry is here presented to the student arranged in six books, and each book is subdivided into propositions. The propositions are of two kinds, *problems* and *theorems.* In a problem something is required to be done; in a theorem some new principle is asserted to be true.

A proposition consists of various parts. We have first the general enunciation of the problem or theorem; as for example, *To describe an equilateral triangle on a given finite straight line,* or *Any two angles of a triangle are together less than two right angles.* After the general enunciation follows the discussion of the proposition. First, the enunciation is repeated and applied to the particular figure which is to be considered; as for example, *Let AB be the given straight line: it is required to describe an equilateral triangle on AB.* The *construction* then usually follows, which states the necessary straight lines and circles which must be drawn in order to constitute the *solution* of the problem, or to furnish assistance in the *demonstration* of the theorem. Lastly, we have the demonstration itself, which shews that the problem has been solved, or that the theorem is true.

Sometimes, however, no construction is required; and sometimes the construction and demonstration are combined.