37. Or obtuse angled, having one obtuse angle, as E. fig. 14.

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

39. 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.

40. 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.

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

41. Any figure of four sides is called a quadrilateral figure.

42. Quadrilateral figures, whose opposite sides are parallel, are called parallelograms: thus ABCD is a parallelogram, Fig. 3. 17, and AB, fig. 18 and 19.

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

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

45. A rhombus is a parallelogram of equal

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sides, and has its angles oblique, as A. fig. 18. and is an inclined square.

46. A rhomboides is a parallelogram whose opposite sides are equal and angles oblique; as B. fig. 19, and may be conceived as an inclined rectangle.

47. Any quadrilateral figure that is not a parallelogram, is called a trapezium. Plate 7. fig. 3.

48. 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, as an irregular pentagon, hexagon, &c.

49. Four quantities are said to be in proportion when the product of the extremes is equal to that of the 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.

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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 what 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.

2. Every whole is greater than its part.

3. Every whole is equal to all its parts taken together.

4. If to equal things, equal things be added, the whole will be equal.

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

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

GEOMETRICAL THEOREMS.

THEOREM I.

PL. 1. fig. 20.

IF a right line falls on another, as AB, or EB, does on CD, it either makes with it two right angles, or two angles equal to two right angles.

1. If AB be perpendicular to CD, then (by def. 10.) the angles CBA, and ABD, will be each a right angle.

2. But if the line fall slantwise, as EB, and let AB be perpendicular to CD; then the 2 DBA= DBE+EBA, add ABC to each; then, DBA+ ABC=DBE+EBA+ABC; but CBE EBA +ABC, therefore the angles DBE+EBC= DBA+ABC, or two right angles. Q. E. D.

Q

Corollary 1. Whence if any numbers of right lines were drawn from one point, on the same side of a right line; all the angles made by these lines will be equal to two right angles.

2. And all the angles which can be made about a point, will be equal to four right angles.

THEO. II.

PL. 1. fig. 21.

AEB to

If one right line cross another, (as AC does BD,) the opposite angles made by those lines will be equal to each other: that is, CED, and BEC to AED.

By theorem 1. BEC+ CED= 2 right angles. and CED+DEA= 2 right angles.

Therefore (by axiom 1.) BEC+CED=CED+ DEA: take CED from both, and there remains BEC=DEA. (by axiom 5.) Q. E. D.

After the same manner CED+AED=2 right angles; and AED +AEB = two right angles; wherefore taking AED from both, there remains CED AEB. Q. E. D.

THEO. III.

PL. 1. fig. 22.

If a right line cross two parallels, as GH does AB and CD, then,

1. Their external angles are equal to each other, that is, GEB = CFH.

2. The alternate angles will be equal, that is, AEF = EFD and BEFCFE.

3. The external angle will be equal_to_the_internal and opposite one on the same side, that is, GEB EFD and AEG CFE.

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4. And the sum of the internal angles on the same side, are equal to tivo right angles; that is, BEF + DFE are equal to two right angles, and AEF + CFE are equal to two right angles.

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