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ELEMENTS OF GEOMETRY.

Definitions and Preliminary Remarks.

1. GEOMETRY is a science which has for its object the measure of extension.

Extension has three dimensions, length, breadth, and thick

ness.

2. A line is length without breadth.

The extremities of a line are called points. A point, therefore, has no extension.

3. A straight or right line is the shortest way from one point to another.

4. Every line, which is neither a straight line nor composed of straight lines, is a curved line.

Thus AB (fig. 1) is a straight line, ACDB is a broken line, or Fig. 1. one composed of straight lines, and AEB is a curved line.

5. A surface is that which has length and breadth, without thickness.

6. A plane is a surface, in which any two points being taken, the straight line joining those points lies wholly in that surface. 7. Every surface, which is neither a plane nor composed of planes, is a curved surface.

8. A solid is that which unites the three dimensions of ex

tension.

9. When two straight lines, AB, AC, (fig. 2), meet, the quan- Fig. 2. tity, whether greater or less, by which they depart from each other as to their position, is called an angle; the point of meeting or intersection A, is the vertex of the angle; the lines AB, AC, are its sides.

An angle is sometimes denoted simply by the letter at the vertex, as A; sometimes by three letters, as BAC, or CAB, the letter at the vertex always occupying the middle place.

Geom.

1

Angles, like other quantities, are susceptible of addition, subtraction, multiplication, and division; thus, the angle DCE Fig. 20. (fig. 20) is the sum of the two angles DCB, BCE, and the angle DCB is the difference between the two angles DCE, BCE.

Fig. 3.

Fig. 4.

Fig. 5.

10. When a straight line AB (fig. 3) meets another straight line CD in such a manner that the adjacent angles BAC, BAD, are equal, each of these angles is called a right angle, and the line AB is said to be perpendicular to CD.

11. Every angle BAC (fig. 4), less than a right angle, is an acute angle; and every angle, DEF, greater than a right angle is an obtuse angle.

12. Two lines are said to be parallel (fig. 5), when, being situated in the same plane and produced ever so far both ways, they do not meet.

13. A plane figure is a plane terminated on all sides by lines. If the lines are straight, the space which they contain is Fig. 6. called a rectilineal figure, or polygon (fig. 6), and the lines taken together make the perimeter of the polygon.

Fig. 7.

14. The polygon of three sides is the most simple of these figures, and is called a triangle; that of four sides is called a quadrilateral; that of five sides, a pentagon; that of six, a hexagon, &c.

15. A triangle is denominated equilateral (fig. 7), when the Fig, 8. three sides are equal, isosceles (fig. 8), when two only of its sides Fig. 9. are equal, and scalene (fig. 9), when no two of its sides are equal. 16. A right-angled triangle is that which has one right angle. The side opposite to the right angle is called the hypothenuse. Fig. 10. Thus ABC (fig. 10) is a triangle right-angled at A, and the side BC is the hypothenuse.

Fig. 11.

Fig. 12.

Fig. 13.

Fig. 14.

Fig. 15.

17. Among quadrilateral figures we distinguish ;

The square (fig. 11), which has its sides equal and its angles right angles, (See art. 80);

The rectangle (fig. 12), which has its angles right angles without having its sides equal (See art. above referred to);

The parallelogram (fig. 13), which has its opposite sides parallel ;

The rhombus or lozenge (fig. 14), which has its sides equal without having its angles right angles;

The trapezoid (fig. 15), which has two only of its sides parallel.

18. A diagonal is a line which joins the vertices of two angles not adjacent, as AC (fig. 42).

19. An equilateral polygon is one which has all its sides equal; an equiangular polygon is one which has all its angles equal.

20. Two polygons are equilateral with respect to each other, when they have their sides equal, each to each, and placed in the same order, that is, when by proceeding round in the same direction the first in the one is equal to the first in the other, the second in the one to the second in the other, and so on. In a similar sense are to be understood two polygons equiangular with respect to each other. The equal sides in the first case, and the equal angles in the second, are called homologous (A).

21. An Axiom is a proposition, the truth of which is self-evident.

A Theorem is a truth which becomes evident by a process of reasoning called a demonstration.

A Problem is a question proposed which requires a solution. A Lemma is a subsidiary truth employed in the demonstration of a theorem, or in the solution of a problem.

The common name of Proposition is given indifferently to theorems, problems, and lemmas.

A Corollary is a consequence which follows from one or several propositions.

A Scholium is a remark upon one or more propositions which have gone before, tending to show their connexion, their restriction, their extension, or the manner of their application.

A Hypothesis is a supposition made either in the enunciation of a proposition, or in the course of a demonstration.

Axioms.

22. Two quantities, each of which is equal to a third, are equal to one another.

23. The whole is greater than its part.

24. The whole is equal to the sum of all its parts.

25. Only one straight line can be drawn between two points. 26. Two magnitudes, whether they be lines, surfaces, or solids, are equal, when, being applied the one to the other, they coincide with each other entirely, that is, when they exactly fill the same space.

Fig. 42.

PART FIRST.

OF PLANE FIGURES.

SECTION FIRST.

First Principles, or the Properties of perpendicular, oblique, and

Demonstration. Let the straight line CD be perpendicular to Fig. 16. AB (fig. 16), and GH to EF, the angles ACD, EGH, will be equal.

Take the four distances CA, CB, GE, GF, equal to each other, the distance AB will be equal to the distance EF, and the line EF may be applied to AB, so that the point E will fall upon A, and the point F upon B. These two lines, thus placed, will coincide with each other throughout; otherwise there would be two straight lines between A and B, which is impossible (25). The point G therefore, the middle of EF, will fall upon the point C, the middle of AB. The side GE being thus applied to CA, the side GH will fall upon CD; for, let us suppose, if it be possible, that it falls upon a line CK, different from CD; since, by hypothesis (10), the angle EGH = HGF,

it follows that

Fig. 17.

But

and

besides, by hypothesis,

hence

and the line GH cannot fall upon a line CK different from CD ; consequently it falls upon CD, and the angle EGH upon ACD, and EGH is equal to ACD; therefore all right angles are equal.

THEOREM.

28. A straight line CD (fig. 17), which meets another straight line AB, makes with it two adjacent angles ACD, BCD, which, taken together, are equal to two right angles.