TO THE TEACHER

THIS

HIS Syllabus is a collection of propositions in Elementary Geometry which may properly be regarded as forming the pupil's working outfit in this subject. They constitute the "book work" of the requirement in Geometry for admission to the Freshman Class of Harvard College and of the Lawrence Scientific School.*

Owing to the variety of ways in which the axioms and postulates may be assumed, it has seemed best not to include these in the Syllabus, but to leave this matter in the hands of the teacher. Attention should however be called to the desirability of introducing the Method of Limits before the propositions regarding the ratio and measure of angles and arcs (Book II) are taken up, and of using the method for the proof of these propositions.

So large a proportion of the present text-books agree in their division of the subject matter of Plane and Solid Geometry into books that their division has been here adopted. The order of propositions belonging to one and the same Book is not prescribed; but it is not expected that a proposition of a given Book shall be proved by the aid of propositions appearing in later Books. Should the candidate however have used a textbook in which the division into Books is inconsistent with the division of this Syllabus, and should he prefer to follow the order of propositions with which he is familiar, he will be allowed to do so on stating in his examination book the name of the text-book he has used.

The propositions of each Book have been arranged in one of the many possible logical orders. But no attempt has been

* A complete statement of the requirement, together with some suggestions to the teacher, will be found in the Harvard University Catalogue.

*

made to obtain a desirable teaching order. In fact, teachers may find even the separation of the Solid from the Plane Geometry an unfortunate one for purposes of instruction, when the excellent recommendations of the Committee of Ten regarding the early study of Concrete Geometry shall have been more generally adopted in the schools. These recommendations contemplate early training in the formation of space conceptions, so that the pupil will be in possession of a knowledge of the more important facts both of Plane and of Solid Geometry, which will have been studied together, before he begins the formal study of these subjects. It may then be desirable to use many of the simpler propositions of Solid Geometry as exercises in Plane Geometry, and to complete the earlier topics in Solid Geometry before the later ones in Plane Geometry have been taken up.

"the

For the sake of conciseness, such expressions as: product of two lines," "the product of a line and a surface," etc., have been used in the sense of: "the product of the lengths of two lines," "the product of the length of a line and the area of a surface," etc.

Candidates will be provided with a copy of the Syllabus for use at the admission examination in Geometry. They may refer to the propositions of the Syllabus by Book and number, instead of writing them out at length.

* Aids to such training are (1) the accurate drawing, by means of ruler and compass, of plane figures (e. g. the medial lines of a triangle, or a regular hexagon) and (2) the use of models and carefully drawn diagrams of solid figures. The pupil should provide himself with a sphere on which he can draw, and a hemispherical cup to fit the sphere, by means of which he can draw great circles. Models of the regular bodies are readily constructed out of card-board.

PLANE GEOMETRY

BOOKS I TO V.

BOOK I.

ANGLES, TRIANGLES, AND PERPENDICULARS.

THEOREM I.

If two triangles have two sides and the included angle of one respectively equal to two sides and the included angle of the other, the triangles are equal.

THEOREM II.

If two triangles have a side and the two adjacent angles of one respectively equal to a side and the two adjacent angles of the other, the triangles are equal.

THEOREM III.

In an isosceles triangle the angles opposite the equal sides. are equal.

Conversely, if two angles of a triangle are equal, the triangle is isosceles.

THEOREM IV.

If two angles of a triangle are unequal, the side opposite the greater angle is greater than the side opposite the less angle.

If two sides of a triangle are unequal, the angle opposite the greater side is greater than the angle opposite the less side.

THEOREM V.

If two triangles have two sides of one respectively equal to two sides of the other, and the included angles unequal, the triangle which has the greater included angle has the greater third side.

If two triangles have two sides of one respectively equal to two sides of the other, and the third sides unequal, the triangle which has the greater third side has the greater included angle.

THEOREM VI.

If two triangles have the three sides of one respectively equal to the three sides of the other, the triangles are equal.

THEOREM VII.

If straight lines are drawn from a point within a triangle to the extremities of a side, their sum is less than the sum of the other two sides of the triangle.

THEOREM VIII.

At a given point in a straight line but one perpendicular to the line can be drawn.

THEOREM IX.

The two adjacent angles which one straight line makes with another are together equal to two right angles.

Conversely, if the sum of two adjacent angles is two right angles, their exterior sides are in the same straight line

THEOREM X.

If two straight lines intersect each other, the opposite (or vertical) angles are equal.