11. If two lines meet each other in a point, the inclinanation or opening between them is called an angle. Thus if the line AB meets the line CB in the point B, the inclination of these lines to each other determines the angle at B.

B

An angle is designated by a letter at the vertex or point where the lines meet, as B; but if two or more angles have the same vertex, then each angle is designated by naming the sides about the angle; naming the letter at the vertex always in the middle.

Thus the angle CBA, or ABC, designates the angle formed by the intersection of the lines CB and AB at the vertex B; and by the angle DBC or CBD, the angle formed by the meeting of

C

the lines CB and DB is in like manner designated.

B

Angles are greater or less, as the lines forming the angles are more or less inclined or opened, and like other magnitudes or quantities are susceptible of addition, subtraction, multiplication and division.

Thus the angle ABD is the sum of the two angles ABC and CBD, and the angle CBD is the difference of the two angles ABD and ABC.

Hence the quantity of an angle depends not upon the length of the lines forming

B

D

A

that angle, but altogether upon their relative position or inclination.

13. One right line is said to be perpendicular to another, when it makes with

it equal adjacent angles A perpendicular at the extremity of a line is that which makes an angle with it equal to the adja

cent angle which would be formed by producing the line beyond that extremity.

13. A right angle is the angle formed by a

right line and a perpendicular to it.

14. An acute angle is less than a right angle.

15. An obtuse angle is greater than a right angle.

16. By the distance of a point from a right line is meant the perpendicular from that point to the line; and one line is said to be equidistant from another, when every point in the one is equidistant from the other.

A

a

b

с

17. Right lines are parallel, when they are at the same perpendicular distance from each other in all their parts. Thus, if the two right lines AB and CD are so posited, that perpendiculars, ad, be, cf, &c., from the several points a, b, c, &c., in one line, drawn to the other, are C equal each to each, wherever posited, the lines AB and CD are parallel.

-D

f

18. A surface is that which has two dimensions of extension, viz: length and breadth without thickness.

19. A plane is a surface in which if two points be assumed at pleasure and connected by a right line, that line will lie wholly in the surface.

20. Every surface which is not a plane surface, or composed of plane surfaces, is a curved surface.

21. A plane figure is an enclosed plane surface.

2. If it be bounded by right lines only it is

called a rectilineal figure.

23. A polygon is a name used to comprehend every species of rectilineal figures, without regard to the number of sides; but figures are more particularly distinguished as follows, viz:

A figure of three sides is called a triangle.

The quadrilateral has four sides.

The pentagon has five sides.

The hexagon has six sides.

The heptagon has seven sides.

The octagon has eight sides; and so on.

24. All the sides of the polygon taken together form the contour or perimeter of the polygon.

25. Among triangles we may distinguish

The equilateral triangle, which has its three sides equal; The isosceles triangle which has two of its sides equal; The scalene triangle, which has its three sides unequal.

Δ Δ

The acute angled triangle, which has three acute angles.

The obtuse angled triangle which

has one obtuse angle.

26. A right angled triangle is one which has a right angle. 27. In a right angled triangle, the side opposite the right angle is called the hypothenuse. If, for example, the angle A is right, the side BC is the hypothenuse. Any side of a triangle may be considered as its base, but it is usual in the case of

B

A

the isoceles triangle to confine this term to that side that is not equal to either of the others.

28. A rhomboid or parallelogram is a quadrilateral whose opposite sides are parallel.

29. If only two of the opposite sides are parallel, the figure is a trapezium.

30. A rhombus is a rhomboid, the adjacent sides of which are equal.

31. A rectangle is a right angled homboid.

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32. A square is a right angled rhombus.

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C

33. The right line which joins the vertices of two opposite angles of a quadrilateral is called a diagonal. Thus the line AC joining the vertices of the opposite angles DAB, DCB, of the quadrilateral ABCD, is a diagonal.

B.

34. Plane figures are equal, when, by supposing them to be applied to each other, they would coincide throughout; and they are said to be equivalent, when they enclose equal portions of space, and are at the same time incapable of such coincidence.

35. An equilateral polygon is one which has all its sides equal. An equiangular polygon is one which has all its angles equal.

36. Two polygons are mutually equilateral, when they have their sides equal each to each and placed in the same order, that is, when following their perimeters in the same direction, the first side of one is equal to the first side of the other, the second of one to the second of the other, the third to the third, and so on. The phrase, mutually equiangular, has a corresponding signification with respect to the angles. In both cases, the equal sides, or the equal angles, are called homologous sides or angles.

POSTULATES.

1. Grant that a right line may be drawn from one point to another.

2. And that it may be either increased till it is equal to a greater right line, or diminished till it be equal to a less.

3. Grant also, that an angle may be increased till it is equal to a greater angle, or diminished till it is equal to a less.

4. And lastly, that from a point either within or without a right line a perpendicular thereto may be drawn.

PROPOSITION I. THEOREM.

From the same point in a given right line, more than one perpendicular thereto cannot be drawn.

Let CE be perpendicular to the right line AC or AB, CB being the production of AC; and if the proposition be denied, let some other line, as CD, drawn from the same point C, be also perpendicular to AC.

B

Then because the angles ACE and BCE are equal (Def. 12.), the angle ACE must be greater than the angle DCB, since the sum of the two angles DCB and DCE is only equal to the angle BCE or ACE. But CD is perpendicular to AB by hypothesis, therefore the angle DCB must be equal to the angle BCE or ACE (Def. 12.), which is a manifest absurdity; therefore CD cannot be perpendicular to AC or АВ.

PROPOSITION II. THEOREM.

All right angles are equal to each other.

Let ABC be a right angle, and DEF any other right angle, then, if it be denied that these two angles are equal to each other, one of them, as ABC, must be supposed greater than the other, so that DEF must be equal to some portion of ABC.

Let ABƒ represent that portion; then, because ABf is a right angle, Bf is perpendicular to AB (Def. 12.); but ABC is also a right angle, therefore BC is likewise perpendicular to AB,

Α

C

BD

F

that is, from the same point B in the right line AB, two perpendiculars thereto are drawn, which is impossible. (Prop. I.) Therefore ABC cannot be greater than DEF, and in a similar manner it may be proved that DEF cannot be greater than ABC; the two angles are therefore equal.

PROPOSITION III. THEOREM.

The adjacent angles which one right line makes with another which it meets, are together equal to two right angles.

Let AB and CD be the right lines, meeting each other at C, then will the angle ACD+the angle DCB, be equal to two right angles.

At the point C, erect CE perpendicular to AB. The angle ACD is the sum of the angles ACE, ECD; therefore ACD+BCD is the sum of the three angles ACE, ECD,

E

D

A

C B