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fig. 15.

25. Or acute angled, having all the angles acute, as F. fig. 15.

F

· 26. Acute and obtuse angled triangles are in general called oblique angled triangles, in all which any side may be called the base, and the other two the sides.

fig. 16.

A

В.

27. The perpendicular height of a triangle is a line drawn from the vertex to the base perpendicularly: thus if the triangle ABC, be proposed, and BC be made its base, then if from the vertex A the perpendicular AD be drawn to BC, the line AD will be the height of the triangle ABC, standing on BC as its base, fig. 16.

D

Hence all triangles between the same parallels have the same height, since all the perpendiculars are equal from the nature of parallel lines.

28. Any figure of four sides is called a quadrilate

ral figure.

29. Quadrilateral figures whose opposite sides are parallel, are called [see fig. 3. parallelograms: thus ABCD is a

page 25.] parallelogram. fig. 3. 17. and fig. 18. 19.

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30. A parallelogram whose sides are all equal and angels right, is called a square, as ABCD. fig. 17.

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31. A parallelogram whose opposite sides are equal and angles right, is called a rectangle or an oblong, as ABCD, fig. 3.

fig. 18.

32. A rhombus is a parallelogram of equal sides, and has its angles oblique, as fig. 18.

A

Ag. 19.

33. A rhomboides is a parallelogram whose opposite sides are equal and angles oblique; as B fig. 19.

B

fig. 100.

B

34. Any quadrilateral figure that is not a parallelogram, is called a trapezium. fig. 100.

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35. Figures which consist of more than four sides are called polygons: if the sides are all equal to each other, they are called regular polygons. They sometimes are named from the number of their sides, as a five-sided figure is called a pentagon, one of six sides a hexagon, &c. but if their sides are not equal to each other, then they are called irregular polygons ;

36. Four quantities are said to be in proportion when the product of the extremes is equal to that of means : thus if A multiplied by D, be equal to B multiplied by C, then A is said to be to B as C is to D.

POSTULATES OR PETITIONS.

1. That a right line may be drawn from any one given point to another.

2. That a right line may be produced or continued at pleasure.

3. That from any centre and with any radius, the circumference of a circle may be described.

4. It is also required that the equality of lines and angles to others given, be granted as possible: that it is possible for one right line to be perpendicular to another, at a given point or distance; and that every magnitude has its half, third, fourth, &c. part.

Note. Though these postulates are not always quoted, the reader will easily perceive where, and in wbat sense they are to be understood.

AXIOMS OR SELF-EVIDENT TRUTHS.

1. Things that are equal to one and the same thing, are equal to each other.

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3. The whole is equal to all its parts.

4. If to equal things, equal things be added, the wholes are equal.

5. If from equal things, equal things be deducted, the remainders are equal.

6. If to or from unequal things, equal things be added or taken, the sums or remainders are unequal.

7. All right angles are equal to one another.

8. If two right lines not parallel, be produced towards their nearest distance, they will intersect each other.

9. Things which mutually agree with each other, are equal.

NOTES.

A theorem is a proposition, wherein something is proposed to be demonstrated.

A problem is a proposition, wherein something is to be done or effected."

A lemma is some demonstration, previous and necessary, to render what follows the more easy.

A corollary is a consequent truth, deduced from a foregoing demonstration.

A scholium, is a remark or observation made upon something going before.

SIGNIFICATION OF SIGNS.

The sign=, denotes the quantities between which it stands to be equal.

The sign +, denotes the quantity it precedes to be added.

The sign , denotes the quantity which it preGedes to be subtracted.

The sign x, denotes the quantities, between them to be multiplied into each other.

To denote that four quantities, A, B, C, D, are proportional, they are usually written thus, A:B:: C:D; and read thus, as A is to B, so is C to D; but when three quantities A, B, C, are proportional, the middle quantity is repeated, and they are written A:B::B:C.

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