Geometrical deductions are problems which are intended to be solved by the application of recognised geometrical methods and propositions. They are divided into several classes. A geometrical deduction is termed a rider when it is given as an exercise on a particular proposition. It generally happens that the difficulty of a deduction is greatly diminished when it is given in this way, for we know in what direction to seek for a solution. When a deduction is presented as a rider, it is, of course, expected that the proposition to which the deduction is appended shall be made use of in the solution. It will occasionally happen, with carelessly constructed riders, that a simpler solution, not in volving this proposition, is available; but generally there can be no difficulty in so arranging the proof as to introduce the proposition on which the deduction is supposed to be founded.