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GEOMETRY.

BOOK I.

DEFINITIONS.

1. GEOMETRY is the science which has for its object the measurement of magnitude.

2. Magnitude is that which is extended; or which has one or more of the three dimensions, length, breadth and thickness. Thus lines, surfaces and solids are magnitudes.

3. A line is length without breadth.

4. The extremities of a line, also the intersection of one line with another, are called points. A point, then, is that which has position, but occupies no space.

5. A straight line is the shortest distance from one point to another.

6. Every line, which is not straight, or composed of straight lines, is a curve line.

Thus, AB is a straight line; ACDB

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

A

E

D

B

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

8. A plane, or a plane surface is that in which, if any two points be assumed, and connected by a straight line, that line will lie wholly in the surface.

9. Every surface, which is not plane, or composed of plane surfaces, is a curve surface.

10. A solid is that which combines all the three dimensions of magnitude, viz., length, breadth and thickness.

11. When two straight lines AB, AC, meet or intersect each other, the opening between them is called an angle. The point of intersection A

is the vertex of the angle; and the lines A AB, AC, are its sides.

C

B

N. B. When there is but one angle at a point, it is usually designated by the letter placed at the vertex; as the angle A in the figure above. But if there are several angles at the same point, it is necessary to employ three letters to distinguish which angle is meant, viz. the letter at the vertex and the two letters at the extremities of the sides; always placing the one at the vertex in the middle. Thus the angle contained by the two sides CD, BC, in the next figure, is designated DCB, or BCD.

Angles, like all other quantities, are susceptible of addition, subtraction, multiplication and division. Thus the angle DCE is the sum of the two angles, DCB, BCE; and the angle DCB is the difference of the two angles DCE, BCE.

12. 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 those angles is called a right-angle; and the line AB C is said to be perpendicular to CD.

B

D

D

A

B

E

13. Every angle BAC, less D than a right angle, is an acute

angle; every angle DEF, great

A

B

F

E

er than a right-angle, is an obtuse angle.

14. Two lines are said to be parallel, when, being situated in the same plane, they cannot meet, how far soever both of them be produced.

15. A plane figure is a plane terminated on all sides by straight or curve lines.

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

16. Among polygons, we distinguish :
The triangle, which has three sides;
The quadrilateral, which has four sides ;
The pentagon, which has five sides;
The hexagon, which has six sides;
The heptagon, which has seven sides ;

The octagon, which has eight sides; and so on.

17. Among triangles we 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;

The acute-angled triangle, which has three acute gles;

an

gle;

The obtuse-angled triangle, which has an obtuse an

Δ Δ Δ

The right-angled triangle, which has a right-angle.
The side opposite the right-angle is

called the hypothenuse. Thus, the tri-
angle ABC is right-angled at A; and B

the side BC is its hypothenuse.

A

Note.-Acute and obtuse-angled triangles are sometimes called oblique-angled triangles, or simply oblique triangles.

18. Among quadrilaterals, we distinguish :

The square, which has its sides equal and its angles right-angles;

The rectangle, which has its angles rightangles without having its sides equal;

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

The lozenge or rhombus, which has its sides equal, without having its angles rightangles;

The trapezium, only two of whose sides are parallel.

A

19. A diagonal is a line which joins the vertices of two angles not adjacent to each other. Thus the line AC, in the figure above, is a diagonal.

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

21. 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. The phrase, mutually equiangular, has a corresponding signification.

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

22. An axiom is a self-evident proposition.

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

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

solution.

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

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

27. A corollary is an obvious consequence deduced from one or more propositions.

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

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