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COURSE OF MATHEMATICS.
EUCLID, BOOKS I. II.
1. A point is that which has no parts, or which has no magnitude.
[Thus a point has place (position), but no size (magnitude) : it cannot be parted or divided.]
2. A line is length without breadth.
4. A straight line is that which lies evenly between its extreme points.
[When the direction of a line is known, the line is said to be given in position; also when its length is known, it is said to be given in magnitude.]
5. A superficies (or surface] is that which has only length and breadth.
6. The extremities of a superficies are lines.
7. A plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies.
[All these seven definitions express abstract ideas (you will remember the force of the word abstract in the term abstract noun). Every visible line, for instance, has both length and breadth : we cannot draw a line which has no breadth, but we can reason about lines as having an independent existence, and this is what Euclid requires us to do.]
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.
[This includes the angles formed by two curved lines, as well as that formed by two straight lines, and is not required in the first two books.]
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.
[Thus the inclination of the lines OA, OB is called the angle AOB, or BOA, or the angle at O.
Observe that the magnitude of an angle is entirely independent of the lengths of the two lines which form it, and no change is made in the angle by making these lines longer or shorter. The inclination of the lines to one another, the amount of opening between them, is the measure of its magnitude. It is not very uncommon for a beginner, when asked by his teacher “If from the angle COA you take the angle BOA, what angle remains?' to reply,' If you take away the angle BOA, all that remains is the line CO. Now this is a complete mistake, for angles and lines are altogether different things. If an angle be taken from an angle, it is not a line that remains but an angle (viz. in this case the angle COB), just as when you take money from money, the remainder is money, not ounces or yards. The point at which the lines meet is often called the vertex of the angle.]
10. When a straight line standing on another straight line makes the adjacent angles equal to one another, each of these 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 that which is greater than a right angle.
12. An acute angle is that which is less than a right angle.
13. A term or boundary is the extremity of any. thing.
14. A figure is that which is enclosed by one or more boundaries.
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 to one another.
16. And 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.
[A radius is a straight line drawn from the centre to the circumference, and is therefore half the diameter.]
18. A semicircle is the figure contained by a diameter and the part of the circumference cut off by the diameter.
19. The centre of a semicircle is the same as that of the circle.
20. Rectilineal figures are those which are contained by straight lines.
21. Trilateral figures, or triangles, by three straight lines.
22. Quadrilateral figures, by four straight lines.
23. Multilateral figures, or polygons, by more than four straight lines.
[A polygon is called regular when all its sides are equal and all its angles are also equal.]
24. Of three-sided figures :
An equilateral triangle is that which has three equal sides.
25. An isosceles triangle is that which has two sides equal.
26. A scalene triangle is that which has three unequal sides.
27. A right-angled triangle is that which has a right angle.
[The side opposite the right angle is often called the hypotenuse.)
28. An obtuse-angled triangle is that which has an obtuse angle.