ELEMENTS OF PLANE GEOMETRY.

FIRST BOOK.

Mathematics is that branch of science which treats of Measurable Quantity.

Geometry is a branch of mathematics, which treats of that species of quantity called Magnitude; it is either Theoretical or Practical.

Theoretical Geometry investigates the relations and properties of magnitudes.

Magnitudes are of one, two, or three dimensions; as lines, surfaces, and solids. They have no material existence, but they may be represented by diagrams.

That branch of Geometry which refers to magnitudes described upon a plane, is called Plane Geometry.

DEFINITIONS OF MAGNITUDES.

1. A point is that which has position, but not magnitude. 2. A line is length without breadth.

COROLLARY.-The extremities of a line are points; and the intersections of one line with another are also points.

3. Lines which cannot coincide in two points, without coinciding altogether, are called straight or right lines.

COR.-Hence two straight lines cannot inclose a space. Neither can two straight lines have a common segment; that is, they cannot coincide in part, without coinciding altogether.

4. A crooked or broken line is composed of

two or more straight lines.

5. A line, of which no part is a straight line, is called a curved line, a curve line, or curve.

6. A convex or concave line is such that it cannot be cut by a straight line in more than

two points; the concavity of the intercepted portion is turned towards the straight line, and the convexity from it.

7. A superficies is that which hath only length and breadth. COR. The extremities of a superficies are lines; and

the intersections of one superficies with another are also lines.

8. A plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies.

9. A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line.

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When several angles are at one point B, any one of them is expressed by three letters, of which the letter that is at the vertex of the angle—that is, at the point in which the straight lines that contain the angle meet one another-is put between the other two letters, and one of these two is somewhere upon one of those straight lines, and the other. upon the other line: Thus the angle which is contained by the straight lines AB, CB, is named the angle ABC, or CBA; that which is contained by AB, BD, is named the angle ABD, or DBA; and that which is contained by BD, CB, is called the angle DBC, or CBD; but if there be only one angle at a point, it may be expressed by a letter placed at that point; as the angle at E. It is sometimes convenient to name an angle by a small letter placed within the angle; thus, angle DBC may be called angle n; angle ABD, m; and angle ABC, m n.

10. When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle; and the straight line which stands on the other is called a perpendicular to it.

11. An obtuse angle is that which is greater than a right angle.

12. An acute angle is that

which is less than a right angle..

13. A figure is that which is inclosed by one or more boundaries. The space contained within the boundary of a plane figure is called its surface; and its surface in reference to that of another figure, with which it is compared, is called its area.

14. A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another.

15. This point is called the centre of the circle.

16. A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference. 17. A line drawn from the centre to the circumference of a circle is called a radius.

18. A semicircle is the figure contained by a diameter and the part of the circumference cut off by the diameter. 19. Rectilineal figures are those which are contained by straight lines.

20. Trilateral figures, or triangles, by three straight lines. 21. Quadrilateral, by four straight lines.

22. Multilateral figures, or polygons, by more than four straight lines.

23. Of three-sided figures, an equilateral triangle is that which has three equal sides, as E. 24. An isosceles triangle is that which has only two sides equal, as I.

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25. A scalene triangle is that which has three unequal sides, as S.

26. A right-angled triangle is that which has a right angle.

27. An obtuse-angled tri

angle is that which has an obtuse angle.

28. An acute-angled triangle is that which has three acute angles.

29. Of four-sided figures, a square is that which has all its sides equal, and all its angles right angles.

30. An oblong or rectangle is that which has all its angles right angles, but has not all its sides equal.

31. A rhombus is that which has all

its sides equal, but its angles are not right angles.

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32. A parallelogram is a quadrilateral, of which the opposite sides are parallel.

33. All other four-sided figures besides these, are called Trapeziums.

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34. When an angle of a rectilineal figure is less than two right angles, it is called a salient angle; A and when greater than two right angles, it is said to be re-entrant, as B.

35. Any side of a rectilineal figure may be called the base. In a right-angled triangle, the side opposite to the right angle is called the hypotenuse; either of the other two sides, the base; and the other, the perpendicular. In an isosceles triangle, the side which is neither of the equal sides is called the base. The point at which an angle is formed, is called the angular point. The angular point opposite to the base of a triangle is called the vertex; and the angle at the vertex, the vertical angle.

36. The altitude of a triangle, or a parallelogram, is a perpendicular drawn from the opposite angle or side upon

the base.

37. A straight line joining two of the opposite angular points of a quadrilateral, is called a diagonal.

38. Parallel straight lines are such as are in the same plane, and which, being produced ever so far both ways, do not 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. And that a circle may be described from any centre, and 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 exactly fill the same space-are equal to one another. 9. The whole is greater than its part.

10. All right angles are equal to one another.

11. Two straight lines cannot be drawn through the same point, parallel to the same straight line, without coinciding with one another.

12. It is possible for another figure to exist, equal in every respect to any given figure.

DEFINITIONS OF TERMS.

1. A proposition is a portion of science, and is a theorem, a problem, or a lemma.

2. A theorem is a truth which is established by a demonstration.

3. A problem either proposes something to be effected, as the construction of a figure; or it is a question that requires a solution.

4. A lemma is a subordinate truth previously established, to be employed in the demonstration of a theorem, or the solution of a problem.

5. A hypothesis is a fact assumed without proof, either in the enunciation of a proposition, or in the course of a demonstration.