PART FIRST.

SECTION I.—The properties of straight lines and circular lines.

6. Between two assumed points (as A and B in Fig. 1. fig. 1), several lines may be drawn. Among these there may be one which tends the same way in all its parts; this is the straight line. We therefore define a straight line to be, one which has the same direction throughout its whole extent. It is evident, that the straight line is the shortest which can be drawn between two points. It is also evident, that between two points only one straight line can be drawn.

Fig. 2.

7. A line which continually changes its direction is called a curve line.

Curve lines admit of great variety in the character of their curvature, whereas among straight lines, the only variety is that of length.

Of the various curves, the only one whose properties are made the subject of elementary geometry, is the circular curve. This is the most simple of all curves, its curvature being the same in every part. It is described by one extremity of a straight line, revolving about the other extremity which remains fixed. In fig. 2, if the straight line AC be made to revolve about the point C which does not change its position, the point A will describe the circular curve ABDA. This line, in reference to the circle ABDA, is called its circumference; and it is manifest that every point in this curve is equally distant from the point C, called the centre of the circle. The lines AC, BC, DC, &c, which measure this distance, are called radii of the circle. A straight line passing through the centre and terminating in the circumference, is called a diameter. Any portion of this curve, as BD, is called an arc of the circle.

In the same circle are the radii equal? and why? How does the diameter compare with the radius? 8. A surface to which a straight line can be applied in every direction, so as to touch the surface through the whole extent of the line, is called a plane surface, or simply a plane. A surface which is neither a plane surface nor composed of planes, is called a curve surface. Such is the surface of a ball, a roll of paper, &c.

9. Two straight lines which have the same direction in space, are called parallel lines (fig. 3). As they have Fig. 3. the same direction, they can neither approach towards, nor diverge from each other; parallel lines, therefore, must be throughout at the same distance from each other; and, however far produced, can never meet.

10. Where two straight lines taken in the same plane are not parallel, they must incline the one to the other, and are said to make an angle with each other (fig. 4). Fig. 4. An angle, then, is the inclination of one line to another. The magnitude of the angle depends upon the degree of inclination, and not at all upon the distance or length of the lines.

11. The nearer two lines are to being parallel, the smaller the angle, which becomes nothing when the lines become parallel. It increases as the lines diverge; and when this inclination is the same on both sides (as AB and CD, fig. 5), the angles are called right angles. Fig. 5. In this case, the line AB is said to be perpendicular to the line CD. CD is also perpendicular to AB.

When two lines are neither parallel nor perpendicular to each other, they are said to be oblique to each other. Remark. We have, then, in geometry, four kinds of magnitudes. 1. That of a line. 2. That of a surface. 3. That of a body. 4. That of an (inclination or) angle. Angles may be added, subtracted, multiplied, and divided, like other magnitudes or quantities.

12. Two lines which form an angle in a plane, either meet, or being produced, will meet each other in some point, the point E, for instance, as in fig. 6. This point Fig. 6. of meeting is called the vertex of the angle. We express the angle, usually, by three letters, thus, the angle ÃEC. When there is only one angle at the same vertex, we express it by a single letter; as the angle E (fig. 7), that Fig. 7. is, the angle whose vertex as at E.

13. Suppose the line AB perpendicular to CD (fig. Fig. 8. 8) to move about the point B, into the position A'B, the angle A'BC will be greater than a right angle, and the angle A'BD, will be less than a right angle; but the amount of angular space is not changed by this movement; the angle ABA' is taken from one of the angles and added to the other; the sum of the two angles is the same as before. Hence we say, that, when one straight line meets another straight line, the sum of the two angles is equal to two right angles.

Fig. 8.

Fig. 9.

Fig. 10.

The sum of the angles being equal to two right angles, the exterior lines CB, BD, must be in the same straight line; otherwise something must be added to or subtracted from the angular space to make CBD a straight line; which would make the sum of the two angles greater or less than two right angles.

Remark. An angle which is less than a right angle is called an acute angle; as A'BD. An angle greater than a right angle is called an obtuse angle, as the angle A'BC. Both obtuse and acute angles are called oblique angles.

14. If other lines were drawn to the same point B, on the same side of CD, as in figure 9; the whole amount of angular space, would evidently be neither increased nor diminished by it; but would still be equal to two right angles. And we hence derive this general truth, that,- When a straight line is met by several other straight lines, in the same point, the sum of all the angles on the same side of it, is equal to two right angles.

15. If all these lines be produced through B, it is manifest that the angular space below the line CD, will be equal to the angular space above; therefore,— The sum of all the angles made by several straight lines diverging from the same point, is equal to four right angles.

16. As a straight line has the same direction in all its parts (6), two straight lines must have the same inclination in every part (fig. 10); therefore if they intersect each other, the angles which are opposite, at the vertex, (called vertical angles,) are equal. Each of the two angles AEC and DEB, expresses the inclination of the two straight lines AB and CD, to each other; they are therefore equal angles.

If the two lines are perpendicular to each other, the two angles on the same side of either of them, are equal; but these are respectively equal to their vertical angles (16); it is therefore evident that all right angles are equal.

Of Parallel Lines.

17. The properties of parallel lines are deduced from this fundamental proposition. Two straight lines which

have the same direction in space, must be equally inclined to the same straight line which meets them.

This proposition results directly from the definition of a straight line. If two straight lines have different inclinations to the same straight line, that is, to the same direction, they must be inclined to each other, and therefore cannot be parallel.

18. Suppose the straight line AB (fig. 11) to intersect Fig. 11. the two parallel straight lines CD and FG; it must have the same inclination to both. This inclination is expressed by either of the angles AEC, AHF; they are therefore equal. As these angles have correspondent situations in relation to these two intersections, they are called corresponding angles.* And as nothing in this analysis depends upon any particular inclination of AB to the parallels, we derive this general truth: When a straight line intersects parallel straight lines, the corresponding angles are equal.

19. The sum of the two angles AEC and CEH, is equal to two right angles (13); if instead of the angle AEC, we substitute its equal EHF (18), we shall have the sum of the angles CEH and EHF, equal to two right angles. These angles are called interior on the same side of the cutting line; and we say that-When a straight line cuts two parallels, the sum of the two interior angles, on the same side, is equal to two rightangles. And as the sum of the four angles on the same side of the cutting line, or secant, must be equal to four right angles. The sum of the exterior angles on the same side, is equal to two right angles.

20. The angle EHF is equal to its corresponding angle AEC (18), and the angle DEH is equal to the angle AEC, being vertical with it (16). Therefore the two angles, DEH and EHF, (each equal to AEC,) are equal to one another. These angles are internal with respect to the parallels, but on alternate sides of the cutting line. Whence we derive the general truth,-A straight line cutting parallels, makes the alternate-internal angles equal.

21. The angle GHB is equal to the angle DEH (18), and therefore equal to AEC, vertical with DEH. These

* They are sometimes called internal-external angles.

Fig. 11. two angles AEC and GHB, are exterior with respect to the parallels, and on alternate sides of the secant; they are therefore called alternate-external; and we have this general proposition,-When a straight line intersects two parallel straight lines, the alternate-external angles are equal.

Fig. 12.

22. These five propositions may be summarily enunciated as follows: When a straight line meets two parallel straight lines,

(1.) The corresponding angles are equal.

(2.) The sum of the two interior angles upon the same side, is equal to two right-angles.

(3.) The sum of the two exterior angles on the same side is equal to two right-angles.

(4.) The alternate-internal angles are equal to each other; and

(5.) The alternate-external angles are equal to each

other.

23. If we incline one of the parallels in question, FG for instance, the other two lines remaining the same, it will change the magnitude of each of the angles at H; and neither of the five propositions will be true. We have therefore the converse of these propositions; and we say that, when a straight line meets two straight lines, and the corresponding angles are equal; or the sum of the two interior angles on the same side, or the sum of the two exterior, is equal to two right angles; or the alternateinternal angles equal to each other, or the alternate-external angles equal to each other, the two lines must be parallel.

Cor. If the two interior angles are less than two rightangles, the lines incline towards each other, and will meet on that side, if produced sufficiently far. If the sum of the interior angles on one side is greater than two right angles, the sum of those on the other side, is less than two right-angles (13), and therefore the lines will meet on that side, if produced sufficiently far.

24. Let the two angles ABC, DEF, (fig. 12) have the side AB, parallel to DE, and BC parallel to EF; produce BC to I, and also produce DE till it meet BI in H. Then on account of the parallels EF and BI, the corresponding angles DEF and DHI, are equal (22); and with reference to the parallels DH and AB, the corresponding angles DHI and ABC are equal; that is, ABC