may be looked on as an example of this. Those Examples (pages 130, 131) which depend chiefly on Euclid (1. 4; 1.8; I. 26) had better be solved before the Exercises are begun.

The Author begs to thank his friends Messrs Nicolls, B.A., Senior Moderator, Trinity College, Dublin, and L. J. Ryan, Head Master, Central Model Schools, Dublin, for their kindness in correcting the proof-sheets.

19, ST ANDREW STREET, DUBLIN.

ERRATA.

Page 16, sixth and seventh lines from the bottom, for AHG read AHC.

Page 54, last line, for O read O1.

Page 59, line 13, for AE read BE, for AF read BF.

GEOMETRICAL EXERCISES

FOR BEGINNERS.

FUNDAMENTAL PROPOSITIONS.

PROP. 1. The straight line joining the middle points of the sides of a triangle is parallel to the third side and equal to half of it.

Let D, E, F, be the middle points of the sides BC, CA, AB, of any triangle ABC. Join FE, BE, CF, and

ED.

C. G.

1

As the triangles AFE, BFE stand upon equal bases AF, BF, and are between the same parallels, since they have a common vertex, E, they are equal (I. 38). For the same reason the triangles AFE, CFE, are equal; therefore the triangles BFE, CFE are equal. But they stand upon the same base, FE, and upon the same side of it; therefore they are between the same parallels (I. 39). Hence FE is parallel to the side BC. Similarly the line ED is parallel to the side AB; therefore BFED is a parallelogram, and therefore (I. 34) FE, BD are equal: that is, FE is equal to half the base.

Note. The converse of this very important theorem may be stated thus:-If, through the middle point of one side of a triangle, a parallel be drawn to the base, it bisects the other side.

PROP. 2. Half the hypotenuse of a right-angled triangle is equal to the straight line joining its middle point to the vertex.

E

A

B

Let ABC be a right-angled triangle, of which AB is the hypotenuse. Bisect AB in O. Join CO: the straight lines AO, BO, CO are equal.

Bisect AC in E; and join OE.