be drawn to the other two sides, their sum is constant, and equal to the perpendicular from one of the equal angles to the opposite side.

Suggestion.-Construct an isosceles triangle having such parts as the conditions require, namely, two perpendiculars from a point in the base to the sides, and a third perpendicular from either of the equal angles to the opposite side. State the theorem with reference to the diagram. To the third perpendicular draw, from the common point in the base, a fourth. The demonstration is based mainly upon the equality of triangles.

13. All the equal chords in a circle may be touched by another circle. (See Th. III.)

14. If two opposite sides of an inscribed quadrilateral are equal, the other two sides are parallel. (See Ths. IV and V.)

15. If the diameter of a circle be made one of the equal sides of an isosceles triangle, the base will be bisected by the circumference.

16. Two chords of a circle which cut a diameter in the same point and make equal angles with it, are equal.

17. An inscribed angle is acute or obtuse, according as it is inscribed in

a segment greater or less than a semicircle.

CHAPTER V.

CONSTRUCTIONS.

OBSERVATIONS.

The student may be surprised that methods for constructing diagrams have not hitherto been given. But, first, it was unnecessary that our diagrams should be rigorously exact, since they were employed merely to aid our mental conceptions in reasoning about principles; and, secondly, their precise construction involves the principles previously discussed.

The constructions of Elementary Geometry are effected by the straight line and circumference, with the aid of a ruler and compass.

The object of these constructions is twofold,—to afford an excellent review of geometrical facts, and to impart skill in the application of knowledge.

While they may be understood by simple inspection, the student is warned against a method of study so partial and unsatisfactory, unattended by either the profit or the delight of art. They will be impressed upon the mind with far greater ease and force, if, in every instance, they are actually drawn. Science is acquired by study-art by practice.

It will be observed that the solution of a problem consists of three parts:

1. To draw the parts given;

2. To draw the parts requisite for construction;

3. To give the reasons of the process, or, the proof that the construction is correct.

done.

Note.-Q. E. F.' stands for quod erat faciendum, which is read, which was to be

be drawn to the other two sides, their sum is constant, and equal to the perpendicular from one of the equal angles to the opposite side.

Suggestion.-Construct an isosceles triangle having such parts as the conditions require, namely, two perpendiculars from a point in the base to the sides, and a third perpendicular from either of the equal angles to the opposite side. State the theorem with reference to the diagram. To the third perpendicular draw, from the common point in the base, a fourth. The demonstration is based mainly upon the equality of triangles.

13. All the equal chords in a circle may be touched by another circle. (See Th. III.)

14. If two opposite sides of an inscribed quadrilateral are equal, the other two sides are parallel. (See Ths. IV and V.)

15. If the diameter of a circle be made one of the equal sides of an isosceles triangle, the base will be bisected by the circumference.

16. Two chords of a circle which cut a diameter in the same point and make equal angles with it, are equal.

17. An inscribed angle is acute or obtuse, according as it is inscribed in a segment greater or less than a semicircle.

CHAPTER V.

CONSTRUCTIONS.

OBSERVATIONS.

The student may be surprised that methods for constructing diagrams have not hitherto been given. But, first, it was unnecessary that our diagrams should be rigorously exact, since they were employed merely to aid our mental conceptions in reasoning about principles; and, secondly, their precise construction involves the principles previously discussed.

The constructions of Elementary Geometry are effected by the straight line and circumference, with the aid of a ruler and compass.

The object of these constructions is twofold,—to afford an excellent review of geometrical facts, and to impart skill in the application of knowledge.

While they may be understood by simple inspection, the student is warned against a method of study so partial and unsatisfactory, unattended by either the profit or the delight of art. They will be impressed upon the mind with far greater ease and force, if, in every instance, they are actually drawn. Science is acquired by study-art by practice.

It will be observed that the solution of a problem consists of three parts:

1. To draw the parts given;

2. To draw the parts requisite for construction;

3. To give the reasons of the process, or, the proof that the construction is correct.

done.

Note.-Q. E. F.' stands for quod erat faciendum, which is read, which was to be

PROBLEM I.

At a given point within a line, to erect a perpendicular to that line.

G

D

A

B

E

C

F

Let A B be the given line, and C the given point within it.

From C as a centre, with any radius, describe an arc intersecting A B in the points E and F; from E and F as centres, with a radius greater than one

half of E F, describe arcs intersecting in G. Through G draw DC, which will be the required perpendicular.

For, by construction, C is one point equally distant from E and F, and G is another; hence D C is perpendicular to E F, and therefore to A B. (Ch. II, Th. XX, Ex. 2.)

Q. E. F.

Note.-Suppose a carpenter wished to mortise a 'king-post' into a 'purline' at a given point B, and at right angles to it, without the use of a set-square. Having laid off, on the 'purline,' B A B C, he might take two poles of equal length, and, fasttening their lower extremities at A and C, place their upper extremities against the 'post'; then move the post, till the ends of the poles are just opposite, or in the same straight line, and the 'kingpost will be perpendicular to the 'purline.'

What principle guarantees the accuracy of this method?

PROBLEM II.

From a given point without a line, to draw a perpendicular