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

20. Parallel right lines are fuch as are in the fame 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.

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23. The diagonal of a quadrangle, is a right line joining any two of its oppofite angles.

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

NOTE, When an angle is expreffed by means of three letters, the one which ftands at the angular point, muft 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.

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

3. That a circle may be defcribed 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.

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.

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

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5. If equals be taken from unequals the remainders will be unequa!.

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.

REMARK S.

A PROPOSITION, is fomething which is either pro'pofed to be done, or to be demonftrated.

A PROBLEM, is fomething which is propofed to be done.

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

A LEMMA, is fomething which is previously demonftrated, 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 obfervation made upon fomething going before it.

PROPOSITION I. PROBLEM.

UPON a given finite right line to defcribe 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, defcribe the circle BCD (Pof. 3.)

And from the point в, at the distance BA, defcribe the circle ACE (Pof. 3.)

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

And, if the right lines CA, CB be drawn from the point of interfection 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 fame thing are equal to each other (Ax. 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.

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PROP. II. PROBLEM.

From a given point to draw a right line equal to a given finite right line.

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Let A be the given point, and BC the given right line; it is required to draw a right line from the point A, that fhall be equal to BC.

Join the points A, B, (Pos. 1.); and upon BA describe the equilateral triangle BAD (Prop. 1.)

From the point в, at the distance BC, defcribe the circle CEF (Pof. 3.) cutting DB produced in F.

And from the point D, at the distance DF, defcribe the circle FHG (Pof. 3.), cutting DA produced in G, and AG will be equal to BC, as was required.

For, fince B is the centre of the circle CEF, BC is equal to BF (Def. 13.)

And, because D is the centre of the circle FHG, DG is equal to DF (Def. 13.)

But the part DA is also equal to the part DB (Def. 16.), whence the remainder AG will be equal to the remainder BF (Ax. 3.)

And fince AG, BC have been each proved to be equal to BF, AG will alfo be equal to BC (Ax. 1.)

A right line AG, has, therefore, been drawn from the point A, equal to the right line BC, as was to be done.

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