28.

AN ACUTE-ANGLED TRIANGLE has three acute angles.

29.

A SQUARE is a Quadrilateral having all its sides equal, and all its angles right angles.

30.

AN OBLONG is a Quadrilateral which has not all its sides equal, but all its angles are right angles.

31.

A RHOMBUS is a Quadrilateral having all its sides equal, but its angles are not right angles.

32.

A RHOMBOID is a Quadrilateral having its opposite sides equal to one another, but all its sides are not equal, nor its angles right angles.

Some writers on Mensuration of Surfaces speak of a Trapezoid as a Quadrilateral having one pair of opposite sides parallel, and consider the Trapezium as having neither pair of opposite sides parallel. This is not in accordance with Euclid's language in Book I. Prop. 35, where a Quadrilateral with one pair of opposite sides parallel is called a Trapezium.

This thirty-third def. is limited by Def. 34, which is usually appended as a Note to the Enunciation of Prop. 34, Book I.

34.

A PARALLELOGRAM is a Quadrilateral of which the opposite sides are parallel; and the diagonal, or diameter, is the straight line joining two of its opposite angles.

The Square, Oblong, Rhombus, and Rhomboid are each of them Parallelograms, as this definition shows. For the terms Rhombus and Rhomboid that of Parallelogram is often used; and for Oblong the term Rectangle.

35.

PARALLEL STRAIGHT LINES are such as are in the same plane, and which, being continually produced, never meet.

POSTULATES.

1.

Let it be granted that a straight line may be drawn from any one 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, at any distance from that centre.

The Postulates are 'Requests' that Euclid makes for certain things to be allowed as permissible in the study of Geometry. They are but three: 1. The drawing of a straight line from any one point to any other. 2. The producing, to any length, of a straight line already drawn. 3. The describing of a circle from any centre with any radius.

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 taken from equals, the remainders are equal.

4.

If equals be added to unequals, the wholes are unequal.

5.

If equals be taken from unequals, the remainders are unequal.

6.

Things which are double of the same are equal to one another.

7.

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

8.

Magnitudes which coincide with one another—that is, which fill exactly the same space-are equal to one another.

9.

The whole is greater than its part.

10.

Two straight lines cannot enclose space.

11.

All right angles are equal to one another.

12.

If a straight line meets two straight lines so as to make the two interior angles on the same side of it, taken together, less than two right angles, these straight lines being continually produced shall at length meet upon that side on which are the angles which are less than two right angles.

The Axioms are

Common Notions,' or self-evident Truths. To them Euclid, on this ground, claims assent. Axioms 10, 11, and 12 are considered by some to be of the nature of Postulates rather than Axioms. This, however, is a distinction the consideration of which the beginner in Euclid may postpone.

HINTS TO THE LEARNER.

1. Make the figures of a good size, and as accurately as possible. A well-drawn figure is of great value towards the understanding of the Proposition.

2. Do not copy the figures of any Proposition, but draw them, step by step, as directed in the Construction.'

3. Remember that in the 'Proof' of any Proposition Euclid employs those Propositions only which are previous to the one under consideration. He never expects you to have a knowledge beyond what you ought thus to have already acquired.

EXPLANATION OF TERMS.

A COROLLARY is a Theorem, or Problem, which arises easily and directly from the Proposition to which it is attached.

HYPOTHESIS is a supposition assumed, for the time, to be true. Q. E. F. stand for Quod erat faciendum, meaning which was to be done. They stand at the end of Problems.

Q. E. D. stand for Quod erat demonstrandum, meaning which was to be demonstrated or proved. They stand at the end of Theorems. The following abbreviations are used: