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12. A figure is that which is inclosed by one or more boundaries.

13. A circle is a plane figure, contained by one line, ... called the circumference, which is every where equally distant from a point within the figure, called the centre.

14. Rectilineal figures are those which are contained by right lines.

15. All plane figures, bounded by three right lines, are called triangles.

16. An equilateral triangle, is that which has all its fides equal to each other.

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17. An ifofceles triangle, is that which has only twe of its fides equal to each other.

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18. A right-angled triangle, is that which has one right angle; the fide which is opposite to the right angle being called the hypothenuse.

19. An obtufe-angled triangle, is that which has one obtufe angle.

20. Parallel right lines are fuch as are in the same plane, and which, being produced ever fo far both ways, will

never meet.

21. Every plane figure, bounded by four right lines, is called a quadrangle, or quadrilateral.

22. A parallelogram, is a quadrangle whofe oppofite fides are parallel.

23. The diagonal of a quadrangle, is a right line join- / oing any two of its oppofite angles,:

24. The base of any figure is that fide upon which it is supposed to stand; and the vertical angle is that which is oppofite to the bafe.

NOTE, When an angle is expreffed by means of three letters, the one which ftands at the angular point, must always be placed in the middle.

POSTULATES.

1. Let it be granted that a right line may be drawn from any one given point to another.

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2. That a terminated right line, may be produced to any length in a right line.

3. That a circle may be described from any point as a centre, at any distance from that centre.

4. And that a right line, which meets one of two parallel right lines, may be produced till it meets the other.

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AXIOM S.

1. Things which are equal to the fame thing are equal to each other.

2. If equals be added to equals the wholes will be equal.

3. If equals be taken from equals the remainders will be equal.

4. If equals be added to unequals the wholes will be unequal.

5. If equals be taken from unequals the remainders will be unequal.

6. Things which are double of the fame thing are equal to each other.

7. Things which are halves of the fame thing are equal to each other..

8. The whole is equal to all its parts taken together.

9. Magnitudes which coincide, or fill the fame space, are equal to each other.

REMARKS.

A PROPOSITION, is something which is either proposed to be done, or to be demonstrated.

A PROBLEM, is something which is proposed to be done.

A THEOREM, is something which is proposed to be demonftrated.

A LEMMA, is fomething which is previoully demon, strated, in order to render what follows more easy.

A COROLLARY, is a confequent truth, gained from fome preceding truth, or demonftration.

A SCHOLIUM, is a remark or observation made upon fomething going before it.

PROPOSITION I. PROBLEM.

UPON a given finite right line to describe an equilateral triangle.

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Let AB be the given right line; it is required to defcribe an equilateral triangle upon it.

From the point A, at the distance AB, describe the circle BCD (Pof. 3.)

And from the point B, at the distance BA, describe the circle ACE (Pof. 3.)

Then, because the two circles pass through each other's centres, they will cut each other.

And, if the right lines CA, CB be drawn from the point of intersection c, ABC will be the equilateral triangle required.

For, fince A is the centre of the circle BCD, AC is equal to AB (Def. 13.)

And, because B is the centre of the circle ACE, BC is alfo equal to AB (Def. 13.)

But things which are equal to the same thing are equal to each other (4x. 1); therefore AC is equal to CB.

And, fince AC, CB are equal to each other, as well as to AB, the triangle, ABC is equilateral; and it is defcribed upon the right line AB, as was to be done.'