The first statement will account for the many, and perhaps seemingly needless, repetitions; and the second, for the _suggestive_ style of some of the questions in the lessons which _develop_ the matter contained in the review-lessons. Attention is respectfully directed to the following points:— First the particular, then the general. See page 25. Why is _m n g_ an acute angle? What is an acute angle? Here the attention is directed first to this particular angle; then this is taken as an example of its kind, and the idea generalized by describing the class. See also page 29. Why are the lines _e f_ and _g h_ said to be parallel? When are lines said to be parallel? Many of the questions are intended to test the vividness of the pupil’s conception. See page 29. Also page 78. If the circumference were divided into 360 equal parts, would each arc be large or small? Many of the questions are intended to test the attention of the pupil. The thing is not to be recognized by the definition; but the definition is to be a description of the thing, a description of the conception brought to the mind of the pupil by means of the name. CONTENTS. PART I. LINES 9 POINTS 9 CROOKED LINES 10 CURVED LINES 11 STRAIGHT LINES 11 OTHER LINES 11 POSITIONS OF LINES 14 ANGLES 17 RELATIONS OF ANGLES 20 ADJACENT ANGLES 20 VERTICAL ANGLES 21 KINDS OF ANGLES 23 RIGHT ANGLES 23 ACUTE ANGLES 24 OBTUSE ANGLES 24 RELATIONS OF LINES 27 PERPENDICULAR LINES 27 PARALLEL LINES 28 OBLIQUE LINES 28 INTERIOR ANGLES 30 EXTERIOR ANGLES 31 OPPOSITE ANGLES 32 ALTERNATE ANGLES 33 PROBLEMS RELATING TO ANGLES 38 POLYGONS 40 TRIANGLES 44 ISOSCELES TRIANGLES 48 PROBLEMS RELATING TO TRIANGLES 53 QUADRILATERALS 55 PARALLELOGRAMS 59 COMPARISON AND CONTRAST OF FIGURES 62