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EUCLID was a famous Greek mathematician. He was born about 330 B.C., and taught at Alexandria during the reign of Ptolemy I., King of Egypt.

The • Elements' of Euclid is a work on elementary mathematics, and is divided into thirteen books. Books I., II., III., IV., and VI. treat of Plane Geometry; Book V. treats of Proportion; Books VII., VIII., and IX. treat of the properties of numbers; Book X. treats of surd quantities; Books XI. and XII. treat of Solid Geometry; and Book XIII. contains miscellaneous Propositions in Plane and Solid Geometry. Modern editions of Euclid's · Elements' usually include Books I.-IV., part of Book V., Books VI. and XI., and sometimes one or two propositions from Book XII.

Geometry (Gk, yî, earth; métpov, measure) treats of the measurement of lines, surfaces, and solids, with their various properties and relations.

ECC.

B

EUCLID'S ELEMENTS

BOOK I.

DEFINITIONS.

1. A point is that which has position, but no magnitude.
2. A line is that which has length without breadth.
3. The extremities of a line are points.

4. A straight line is a line which lies evenly between its extreme points.

5. A surface is that which has length and breadth, but no thickness.

6. The extremities of a surface are lines.

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

8. A plane angle is the inclination of two lines to one another in a plane, which meet together, but are not in the same direction.

9. A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line.

NOTES ON THE DEFINITIONS. Each Bouk of Euclid begins with Definitions. A Definition is an explanation of the exact meaning of some important word or term used in the book.

1 and 2. In Euclid a point is supposed to have no size, and a line is supposed to have no breadth. But we cannot actually mark a point without marking some small area; nor can we draw a line without giving it some breadth.

3. A line may be considered as being formed by a number of points placed close beside one another. Therefore, if two lines meet or cut each other, they will meet or cut in a point; and if a point moves about, it will describe or pass along a line.

4. A right line is the same as a straight line. 5. A superficies is the same as a surface.

6. A surface may be considered as being formed by a number of lines placed close together. Therefore, if two surfaces meet or cut each other, they will meet or cut in a line or lines; and if a line moves or turns about, it will describe a surface.

7. A plane surface is often called simply a plane. A flat surface is a plane.

8 and 9. A plane angle is never used in Euclid. All Euclid's angles are supposed to be contained by straight lines, that is, are rectilineal angles.

B An angle is usually denoted by three letters.

с Thus BOC, COD, and BOD are three different angles at the point 0. If there is only one angle at a point, the angle may be denoted by a single letter.

The vertex is the point where the lines meet and form the angle.

D The arins of the angle are the lines which form the angle. A corner is an angle, and the size of an angle does not depend on the length of the arms.

EXERCISES.

1. What dimensions has a point, a line, a square, a circle, and a marble ?

2. How could you tell with a piece of string whether a certain piece of wood was quite flat ?

3. How many edges and how many plane surfaces has an ordinary brick ?

10. When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle ; and the straight line which stands on the other is called a perpendicular to it.

11. An obtuse angle is an angle which is greater than a right angle.

12. An acute anyle is an angle which is less than a right angle.

13. A term or boundary is the extremity of anything.

14. A figure is that which is contained by one or more boundaries.

C

15. A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference are equal.

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16. This point is called the centre of the circle.

17. A diameter of a circle is a straight line drawn through the centre and terminated both ways by the circumference.

18. A semicircle is a plane figure contained by a diameter of a circle and the part of the circumference that it cuts off.

10. Adjacent angles are such

A as lie next each other. AOC

0 and COB are adjacent angles.

AOC and BOD are opposite vertical angles. с

B A degree (1°) is oth part of a right angle.

The complement of a given angle is an angle which with the given angle makes up a right angle. The complement of an angle of 60° is an angle of 30°, because 60° + 30o = 90°.

The supplement of a given angle is an angle which with the given angle makes up two right angles. The supplement of an angle of 60° is an angle of 120°, because 60° + 120° = 180°.

11. No angle in Euclid is larger than two right angles.

14. The perimeter of a figure is the total length of the lines which bound it.

15. A radius of a circle is a straight line drawn from the centre to the circumference.

EXERCISES.

1. Derive and explain the words adjacent and vertical, as applied to (1) lines and (2) angles.

2. In the figure to the note on Def. 10 name all the adjacent and all the opposite vertical angles.

3. What are the complements of the following angles : 30°, 45°, 90° ? 4. What are the supplements of the following angles : 30°, 45°, 90° ? 5. What kind of angle is the supplement of an acute angle ?

6. In what sense is the diagram in the note on Def. 10 a figure, and in what sense is it not a figure ?

7. If the three sides of a triangle are 2, 3, and 4 feet long, what is the perimeter of the triangle? What is the semi-perimeter ?

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