10. When a straight line AB meets another straight line CD, so as to make the adjacent angles BAC, BAD, equal to each other, each of these angles is called a right angle; and the line AB is said to be C perpendicular to CD.

11. Every angle BAC, 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, when being situated in the same plane, they cannot meet, how far soever, either way, both of them be produced

13. A plane figure is a plane terminated on all sides by lines, either straight or curved.

If the lines are straight, the space they enclose is called a rectilineal figure, or polygon, and the lines themselves, taken together, form the contour, or perimeter of the polygon.

A

-B

14. The polygon of three sides, the simplest of all, is called a triangle; that of four sides, a quadrilateral; that of five, a pentagon; that of six, a hexagon; that of seven, a heptagon; that of eight, an octagon; that of nine a nonagon; that of ten, a decagon; that of twelve, a dodecagon.

Δ.Δ.

15. An equilateral triangle is one which has its three sides equal; an isosceles triangle, one which has two of its sides equal; a scalene triangle, one which has its three sides unequal. 16. A right-angled triangle is one which

has a right angle. The side opposite the
right angle is called the hypothenuse. Thus,
in the triangle ABC, right-angled at A, B

17. Among the quadrilaterals, we distinguish :

The square, which has its sides equal, and its angles right angles.

The rectangle, which has its angles right angles, without having its sides equal.

The parallelogram, or rhomboid, which has its opposite sides parallel.

The rhombus, or lozenge, which has its sides equal, without having its angles right angles.

And lastly, the trapezoid, only two of whose sides are parallel.

18. A diagonal is a line which joins the vertices of two angles not adjacent to each other. Thus, AF, AE, AD, AC, are diagonals.

19. An axiom is a self-evident proposition.

20. A theorem is a truth, which becomes evident by means of a train of reasoning called a demonstration.

21. A problem is a question proposed, which requires a solution.

22. A lemma is a subsidiary truth, employed for the demonstration of a theorem, or the solution of a problem.

23. The common name, proposition, is applied indifferently, to theorems, problems, and lemmas.

24. A corollary is an obvious consequence, deduced from one or several propositions.

25. A scholium is a remark on one or several preceding propositions, which tends to point out their connexion, their use, their restriction, or their extension.

26. A hypothesis is a supposition, made either in the enun

Axioms.

1. Things which are equal to the same 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.

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

5. If equals be taken from unequals, the remainders will be unequal.

6. Things which are double of the same thing, are equal to each other.

7. Things which are halves of the same thing, are equal to each other.

8. The whole is greater than any of its parts.

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

10. All right angles are equal to each other.

11. From one point to another, only one straight line can be drawn.

12. Through the same point, only one straight line can be drawn which shall be parallel to a given line.

13. Magnitudes, which being applied to each other, coincide throughout their whole extent, are equal.

CHAPTER III.

Description of the Instruments used for Delineating or Drawing Lines and Angles on paper. Construction of Problems.

18. Drawings, or delineations on paper, are the copies of things which they are intended to represent.

In order that these copies may be exact, their different parts must bear the same proportion to each other that exists between the corresponding parts of the things themselves.

To enable us to delineate lines and angles correctly, upon paper, certain instruments are necessary; these we will now

DIVIDERS.

19. The dividers is the most simple and useful of the instruments used for drawing. It consists of two legs ba, bc, which may be easily turned around a joint at b.

One of the principal uses of this instrument is to lay off on a line, a distance equal to a given line.

For example, to lay off on CD a dis

tance equal to AB.

For this purpose, place the forefinger on the joint of the dividers, and set one

А

B

AT

E D

foot at A: then extend, with the thumb and other fingers, the other leg of the dividers, until its foot reaches the point B. Then raise the dividers, place one foot at C, and mark with the other the distance CE: this will evidently be equal to AB.

RULER AND TRIANGLE.

20. A Ruler of a convenient size, is about twenty inches in length, two inches wide, and a fifth of an inch in thickness. It should be made of a hard material, perfectly straight and smooth.

The hypothenuse of the right-angled triangle, which is

length, and it is most convenient to have one of the sides considerably longer than the other. We can solve, with the ruler and triangle, the two following problems.

I. To draw through a given point a line which shall be parallel to a given line.

Let C be the given point, and AB the given line.

Place the hypothenuse of the triangle against the edge of the ruler, and then

C

A

B

place the ruler and triangle on the paper, so that one of the sides of the triangle shall coincide exactly with AB: the triangle being below the line.

Then placing the thumb and fingers of the left hand firmly on the ruler, slide the triangle with the other hand along the ruler until the side which coincided with AB reaches the point C. Leaving the thumb of the left hand on the ruler, extend the fingers upon the triangle and hold it firmly, and with the right hand, mark with a pen or pencil, a line through C: this line will be parallel to AB.

II. To draw through a given point a line which shall be perpendicular to a given line.

Let AB be the given line, and D the given point.

A

D

B

Place the hypothenuse of the triangle against the edge of the ruler, as before. Then place the ruler and triangle so that one of the sides of the triangle shall coincide exactly with the line AB. Then slide the triangle along the ruler until the other side reaches the point D: draw through D a right line, and it will be perpendicular to AB.

SCALE OF EQUAL PARTS.

.2 .3.4.5 .6.7.8.910

a

2

1

21. A scale of equal parts is formed by dividing a line of a given length into equal portions.

If, for example, the line ab of a given length, say one inch, be divided into any number of equal parts, as 10, the scale thus