PLANE GEOMETRY.

BOOK I.

FUNDAMENTAL PRINCIPLES.

DEFINITIONS.

I. GEOMETRY is the science which has for its object the measurement of extension.

Extension has three dimensions, length, breadth, and height.

2. A line is length without breadth. The extremities of a line are called points; a point, therefore, has no extension.

3. A straight line is the shortest distance from one point to another. 4. Every line which is neither a straight line, nor composed of straight lines, is a curve line.

Thus, AB is a straight line.

ACDB a broken

E

line, or a line composed of straight lines, and AEB is a curve line.

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

D

B

6. A plane is a surface in which any two points being taken, the straight line joining them lies wholly in the surface.

7. Every surface, which is neither a plane surface nor composed of plane surfaces, is a curved surface.

8. A solid, or body, is that which combines all the three dimensions of extension.

Note.—The word volume is often used to designate a solid.

9.

When two straight lines, AB, AC, meet each other, the quantity, greater or less, by which they are separated from each other, in regard to their position, is called an angle. The point of meeting or intersection, A, is the vertex of the angle, and the lines AB, AC are its sides.

The angle is sometimes designated simply by the letter at the vertex, sometimes by three letters, BAC or CAB, care being taken to place the letter at the vertex in the middle.

Angles, like all other quantities, are susceptible of addition, subtraction, multiplication, and division: thus, the angle A DCE is the sum of the two angles DCB,

BCE; and the angle DCB is the differ

ence of the two angles DCE, BCE.

-B

NOTE.-Two angles, DCB and BCE, which have the same vertex, C, and a common side, CB, are called adjacent angles.

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

II.

B

C

A

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.

B

D

E

12. Parallel straight lines are such as are in A

the same plane, and which, being produced C

ever so far both ways, 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 enclose is called a rectilineal figure, or polygon. The lines themselves are called the sides; and taken together they form the contour, or perimeter, of the polygon.

-B

-D

14. The polygon of three sides, the simplest of all, is called a triangle; the polygon of four sides is called a quadrilateral; that of five sides, a pentagon; that of six, a hexagon, etc.

15. An equilateral triangle is one which has three equal sides; an

isosceles triangle is that which has two sides equal; and a scalene triangle is one which has three unequal sides.

Δ Δ

16. A right-angled triangle is one which has a right angle. The side opposite the right angle is called the hypothenuse: thus, ABC is a right-angled triangle, with the right angle at A; the side, BC, being the hypothenuse.

17. Among the quadrilaterals we distinguish : The square, which has its sides equal, and its angles right angles. (See Prop. xxx., Book I.)

The rectangle, which has its angles right angles, but not all its sides equal. (See the same Prop.)

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

B

The lozenge, or rhombus, which has all its sides equal, but its angles are not right angles.

Lastly, the trapezoid, of which only two sides are parallel.

18. A diagonal of a polygon is a line which joins two vertices, not adjacent to one another. Thus, AC, AD, AE, AF, are diagonals.

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

20. Two polygons are mutually equilateral, when they have their sides equal, each to each, and placed in the same order; that is to say, when following their perimeters in the same direction, the first side of the one is equal to the first side of the other, the second of the one to the second of the other, the third to the third, and so on. In the same way with respect to their angles, two polygons are said to be mutually equiangular.

In both cases, the equal sides, or the equal angles, are called homologous sides or angles.

N. B. In the four first books, it is only plane figures, or figures traced on a plane surface, which will come under consideration.

EXPLANATION OF TERMS AND SIGNS.

An axiom is a self-evident proposition.

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

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

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

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

A corollary is an obvious consequence which flows from one or more propositions.

A scholium is a remark on one or more preceding propositions, tending to point out their connection, their use, their restriction, or their extension.

An hypothesis is a supposition made either in the enunciation of a proposition, or in the course of a demonstration.

AXIOMS.

1. Things which are equal to the same thing are equal to one another.

2. The whole is greater than its part.

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

4. From drawn.

one point to another only one straight line can be

5. Two magnitudes, lines, surfaces, or solids, etc., which coincide throughout their whole extent, are equal to one another.

PROPOSITION I.

THEOREM.

Two straight lines which have two points, A and B, common, coincide throughout their whole extent.

In the first place, the two lines coincide between A and B ; for, otherwise, there would be two straight lines from A to B, which is impossible (Ax. 4). Suppose, however, that they separate from each other at B, the one becoming BC, the other, BE. Turn the line

ABE about the point A until one of the points, E, of the line falls on one of the points, F, of the line ABC. In this movement the point B will fall on some point above, as G, and the line ABED will take the position AGFH. Whence, it would follow, that from the point A to the point F there could be two straight lines, which is impossible (Ax. 4). Hence, the lines cannot separate, and, therefore, they coincide throughout their whole extent.

PROPOSITION II.

THEOREM.

Through a point, C, on a straight line, AB, only one perpendicular, CD, can be drawn to that line.

D K

For, through the point C draw any other straight line, CK. Then we shall have the angle ACK greater than ACD (Ax. 2). But ACD is equal to BCD (Def. 10). Therefore, ACK is greater than BCD; but BCD is greater than BCK (Ax. 2); much more, then, is ACK, greater than BCK. Hence, the straight line CK makes with AB angles which are not equal, and is, therefore, not perpendicular to this line (Def. 10). Therefore, etc.

A

B