23. A square is one which has all its sides equal, and all its angles right angles.

24. A rectangle' is one which has its opposite sides equal and all its angles right angles.

25. A rhombus is one which has all its sides equal, but its angles not right angles.

26. A rhomboid is one which has its opposite sides equal, but all its sides are not equal, nor its angles right angles.

27. A trapezoid is one which has one pair of opposite sides parallel.

28. A trapezium is one which has no two sides parallel.

29. The diagonal' of a quadrilateral is a straight line joining its opposite angles.

Circle.

30. A circle is a plane figure, enclosed by one line, which is called the circumference, and is such that all lines drawn from a certain point within the figure to the circumference are equal to one another.

1 Rectus, right; angulus, an angle. 2 Dia, through; gōnia, an angle.

31. These lines are called radii, and this point is the centre of the circle.

32. A diameter of a circle is any straight line drawn through the centre, and terminated both ways by the circumference.

33. A semicircle is the figure contained by the diameter and the part of the circumference cut off by it.

34. The perimeter of a polygon is the sum of the lines which bound it. The perimeter of a circle is its circumference.

Postulates.

1. Let it be granted that a straight line may be drawn from any point to any other point;

2. That a terminated straight line may be produced to any length in a straight line;

3. That a circle may be described from any centre, with a radius equal to any given line.

Axioms.

1. Things which are equal to the same thing are equal to one another.

2. If equals be added to equals, the wholes are equal.

3. If equals be subtracted from equals, the remainders are equal.

4. If equals be added to unequals, the wholes are unequal. 5. If equals be subtracted from unequals, the remainders are unequal.

6. Things which are doubles of the same thing are equal to one another.

7. Things which are halves of the same thing are equal to one another.

8. Magnitudes which coincide with one another are equal to one another.

9. The whole is greater than its part.

10. All right angles are equal to one another.

11. Two straight lines which intersect each other cannot both be parallel to the same straight line.

12. The shortest distance between two points is the straight line which joins them.

NOTE.-The student should carefully construct all the figures he uses in the following propositions, drawing perpendiculars, equal angles, squares, etc. accurately with ruler and compass, and by the methods described in their proper places. He may also impress upon his mind the truth of some of the theorems by actual experiment. To illustrate: The 30th Prop. of Book I. proves that the three angles of any triangle are equal to two right angles. Let him draw, on paper, triangles of various shapes, and cut them out; then cut off two of the angles, and place them contiguous to the third.

Analyses are given to some of the propositions of Book I. As the exercise is very useful, the student is advised to continue it through the work. After the examples given, he need have but little difficulty in applying analyses to most propositions; indeed, he will find it to lessen the labor of understanding and remembering the proofs, by giving him a correct and comprehensive idea of the method.

He should be careful, in studying a demonstration, to understand the reason of every step. Nothing is assumed except the definitions, postulates and axioms; and every statement may be referred to these, or to some previous proposition which itself is founded upon them. Taking them for granted, the whole geometry follows as a logical necessity. A thorough, honest mastery of the first half of the first book will make the remainder of geometry, and higher mathematics founded on geometry interesting and comparatively easy.

The method usually followed in the propositions is

1. A general statement or enunciation of the theorem to be proved or problem to be solved.

2. A restatement of it as applied to the particular figure on the paper. 3. The construction of any additional lines which may be needed to prove the proposition.

4. The proof, which terminates with the statement which we wished to prove.

5. Any corollaries or scholiums which may attach to the proposition.

Proposition 1.

Problem. From the greater of two given straight lines, to

Proof.

AD is a radius of the circle, and is therefore equal (Def. 30) to C.

NOTE.-A small arc cutting AB in D is sufficient in practice.

Proposition 2.

Theorem.-Two sides of a triangle are together greater

than the third side.

Restatement.

Let ABC be a triangle; any two sides are together greater

than the third.

Proof.

Because (Ax. 12) the shortest distance between two points is

B

C

the straight line which joins them, the line BC is the shortest distance from B to C, and is therefore less than BA and AC. Therefore BA and AC are together greater than BC. In the same way, any other two sides may be shown to be together greater than the third.

Proposition 3.

Problem. To construct a triangle, having given the three

sides.

Let A, B, C be the three sides of a triangle: Restatement. it is required to construct it.

It is a necessary condition that any two of the sides be greater than the third.

1) the lines GD, GF. The triangle GDF is the required

triangle.

Proof.

Because DG is a radius of the circle GHK, it

is equal (Def. 30) to B; and because FG is a radius of GKL, it is equal to C; and DF was made equal to A; therefore GDF is a triangle having its three sides respectively equal to A, B, C.

Scholium.-In practice we need not describe the whole circle. Two small arcs intersecting at G will be sufficient.