same way; namely, by working them out. It is necessary at first to give easy problems; but the doing of easy problems prepares the way for harder ones and still harder.

The exercises here given consist of a great number of easy problems for beginners, and enough harder ones for more advanced scholars. The exercises in each section are carefully graded, and some of the more difficult sections can be omitted without destroying the unity of the work. The book can be used in connection with any text-book on Geometry, as soon as the geometrical processes of reasoning are well understood.

A Syllabus of Geometry is given, not only for reference, but with a view of making the book by itself convenient for reviewing the study of Geometry. Lessons can be assigned consisting partly of book-work taken from the Syllabus, and partly of original work, and the two parts can be so fitted to each other that a thorough knowledge of the book-work will be necessary in order to do the original work, and the doing of the original work will firmly fix in the mind the principles involved in the book-work.

Any corrections or any suggestions relating to the work will be thankfully received.

PHILLIPS EXETER ACADEMY,

September, 1884.

G. A. WENTWORTH.
G. A. HILL.

SYLLABUS

OF

PLANE GEOMETRY.

AXIOMS.

1. Magnitudes which can be made to coincide are equal. 2. Two magnitudes, each equal to a third, are equal to each other.

3. If equals are added to equals, the sums are equal. 4. If equals are taken from equals, the remainders are equal.

5. If equals are added to unequals, the sums are unequal in the same sense.

6. If equals are taken from unequals, the remainders are unequal in the same sense.

7. If unequals are taken from equals, the remainders are unequal in the opposite sense.

8. The whole is equal to the sum of its parts.

9. Through two points only one straight line can be drawn.

10. A straight line is the shortest line between two points.

11. Through a point not in a straight line only one parallel to the line can be drawn.

BOOK I.

THE STRAIGHT LINE.

DEFINITIONS.

12. Body, surface, line, point, straight line, curved line, broken line, plane surface or plane, curved surface, figure, plane figure, similar figures, equivalent figures, equal figures, test of the equality of geometrical figures, method of superposition.

Object of Geometry, Plane Geometry, Solid Geometry, axiom, theorem, corollary, scholium, problem, postulate, proposition, hypothesis, conclusion, proof, converse theorem, contrary theorem.

Comparison of lines as regards magnitude, linear units, length of a line, addition, subtraction, multiplication, and division of lines.

Angle, its sides, its vertex, naming of an angle, straight angle, right angle, acute angle, obtuse angle, generation of an angle, comparison of angular magnitudes, division of the right angle into degrees, minutes, and seconds, adjacent angles, vertical angles, supplementary angles, complementary angles, bisector of an angle, perpendicular lines, oblique lines.

Parallel lines, secant, alternate-interior angles, alternateexterior angles, exterior-interior angles, interior and exterior angles on the same side of the secant.

Polygon, its sides, its angles, its vertices, its parts, triangle, quadrilateral, pentagon, hexagon, octagon, decagon, dodecagon, perimeter, diagonal, exterior angles, convex