in the Elements of Euclid, that one example prepares the way for a set of others which are much easier than itself. It should be observed that the exercises relate to pure Geometry; all examples which would find a more suitable place in works on Trigonometry or Algebraical Geometry have been carefully rejected. It only remains to advert to the mechanical execution of the volume, to which great attention has been devoted. The figures will be found to be unusually large and distinct, and they have been repeated when necessary, so that they always occur in immediate connexion with the corresponding text. The type and paper have been chosen so as to render the volume as clear and attractive as possible. The design of the editor and of the publishers has been to produce a practically useful edition of the Elements of Euclid, at a moderate cost; and they trust that the design has been fairly realised. Any uggestions or corrections relating to the work will be most thankfully received. I. TODHUNTER. St John's COLLEGE, October 1862. 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. |