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In preparing this Key two objects have been kept in view. It is intended first to save the time and lighten the work of teachers, and secondly to remove the difficulties of private students, leaving however sufficient demands upon their thought and intelligence to make the solutions in themselves a useful geometrical exercise. The Examples therefore have not in the majority of cases been worked out in detail, and the drawing of figures has been left to the reader.

The absence of figures may possibly give rise to some little difficulty in Examples which admit of a variety of cases, especially those in Book III. depending on angles in the same segment or on intersecting circles. It would of course be impossible within reasonable limits of space to deal separately with all the cases that may arise in every Example of this kind : we have therefore selected that case which we think would most naturally occur to a student in trying the problem for himself; and, when necessary, we have given some indication of the particular figure to which

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the proof refers. Other cases the student may easily, if he chooses, investigate for himself; the modifications which he will most frequently have to make will be the use of subtraction instead of addition of lines or angles, and the application of III. 22 instead of its kindred proposition III. 21.

As beginners are sometimes at a loss to know the form in which they may present the solution of an elementary geometrical question, the exercises occurring on pages 17— 17 B have been worked out fully and placed in an Introduction.

The Key is arranged for use with the Edition of our Euclid bearing the date 1892. For the convenience of those who use an earlier Edition, we here give a list of the few alterations which on careful revision we have thought well to make in the Examples of the book.

On Page 148 (Euclid), Ex. 40, read, “Produce a given straight line so that the rectangle contained by the whole line thus produced and the part produced, may be equal to the square on another given line.

Page 148, Ex. 41. Read, “Produce a given straight line so that the rectangle contained by the whole line thus produced and the given line shall be equal to the square on the part produced.

Page 217, Ex. 10. Read, “In any triangle, if a circle is described from the middle point of one side as centre and with a radius equal to half the sum of the other two sides, it will touch the circles described on these sides as diameters.”

Page 235, Ex. 18. Add, "and F, B, C, G are concyclic."

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